Main:z from

Percentage Accurate: 92.0% → 98.0%

Time: 18.3s

Alternatives: 18

Speedup: 0.2×

Specification

Math

\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 92.0% accurate, 1.0× speedup

Math

\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \mathsf{min}\left(t\_3, t\right)\\ t_5 := \sqrt{t\_4}\\ t_6 := \mathsf{max}\left(t\_3, t\right)\\ t_7 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_8 := \mathsf{max}\left(t\_7, t\_6\right)\\ t_9 := \mathsf{min}\left(t\_7, t\_6\right)\\ t_10 := t\_9 - -1\\ t_11 := \sqrt{t\_8 + 1} - \sqrt{t\_8}\\ t_12 := t\_4 - -1\\ t_13 := \sqrt{t\_2 + 1} - \sqrt{t\_2}\\ \mathbf{if}\;t\_9 \leq 200000000:\\ \;\;\;\;\left(\left(\frac{t\_12 - t\_4}{\sqrt{t\_12} + t\_5} + \frac{\sqrt{t\_10 \cdot t\_10} - \sqrt{t\_9 \cdot t\_9}}{\sqrt{t\_10} + \sqrt{t\_9}}\right) + t\_13\right) + t\_11\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{t\_9 \cdot \sqrt{\frac{1}{t\_9}}}, \frac{1}{t\_5 + \sqrt{1 + t\_4}}\right) + t\_13\right) + t\_11\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (fmin (fmin x y) z))
        (t_4 (fmin t_3 t))
        (t_5 (sqrt t_4))
        (t_6 (fmax t_3 t))
        (t_7 (fmin (fmax x y) t_1))
        (t_8 (fmax t_7 t_6))
        (t_9 (fmin t_7 t_6))
        (t_10 (- t_9 -1.0))
        (t_11 (- (sqrt (+ t_8 1.0)) (sqrt t_8)))
        (t_12 (- t_4 -1.0))
        (t_13 (- (sqrt (+ t_2 1.0)) (sqrt t_2))))
   (if (<= t_9 200000000.0)
     (+
      (+
       (+
        (/ (- t_12 t_4) (+ (sqrt t_12) t_5))
        (/
         (- (sqrt (* t_10 t_10)) (sqrt (* t_9 t_9)))
         (+ (sqrt t_10) (sqrt t_9))))
       t_13)
      t_11)
     (+
      (+
       (fma
        0.5
        (/ 1.0 (* t_9 (sqrt (/ 1.0 t_9))))
        (/ 1.0 (+ t_5 (sqrt (+ 1.0 t_4)))))
       t_13)
      t_11))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = sqrt(t_4);
	double t_6 = fmax(t_3, t);
	double t_7 = fmin(fmax(x, y), t_1);
	double t_8 = fmax(t_7, t_6);
	double t_9 = fmin(t_7, t_6);
	double t_10 = t_9 - -1.0;
	double t_11 = sqrt((t_8 + 1.0)) - sqrt(t_8);
	double t_12 = t_4 - -1.0;
	double t_13 = sqrt((t_2 + 1.0)) - sqrt(t_2);
	double tmp;
	if (t_9 <= 200000000.0) {
		tmp = ((((t_12 - t_4) / (sqrt(t_12) + t_5)) + ((sqrt((t_10 * t_10)) - sqrt((t_9 * t_9))) / (sqrt(t_10) + sqrt(t_9)))) + t_13) + t_11;
	} else {
		tmp = (fma(0.5, (1.0 / (t_9 * sqrt((1.0 / t_9)))), (1.0 / (t_5 + sqrt((1.0 + t_4))))) + t_13) + t_11;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = sqrt(t_4)
	t_6 = fmax(t_3, t)
	t_7 = fmin(fmax(x, y), t_1)
	t_8 = fmax(t_7, t_6)
	t_9 = fmin(t_7, t_6)
	t_10 = Float64(t_9 - -1.0)
	t_11 = Float64(sqrt(Float64(t_8 + 1.0)) - sqrt(t_8))
	t_12 = Float64(t_4 - -1.0)
	t_13 = Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2))
	tmp = 0.0
	if (t_9 <= 200000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(t_12 - t_4) / Float64(sqrt(t_12) + t_5)) + Float64(Float64(sqrt(Float64(t_10 * t_10)) - sqrt(Float64(t_9 * t_9))) / Float64(sqrt(t_10) + sqrt(t_9)))) + t_13) + t_11);
	else
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / Float64(t_9 * sqrt(Float64(1.0 / t_9)))), Float64(1.0 / Float64(t_5 + sqrt(Float64(1.0 + t_4))))) + t_13) + t_11);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[Min[t$95$3, t], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[Max[t$95$3, t], $MachinePrecision]}, Block[{t$95$7 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$7, t$95$6], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$7, t$95$6], $MachinePrecision]}, Block[{t$95$10 = N[(t$95$9 - -1.0), $MachinePrecision]}, Block[{t$95$11 = N[(N[Sqrt[N[(t$95$8 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$8], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$12 = N[(t$95$4 - -1.0), $MachinePrecision]}, Block[{t$95$13 = N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$9, 200000000.0], N[(N[(N[(N[(N[(t$95$12 - t$95$4), $MachinePrecision] / N[(N[Sqrt[t$95$12], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(t$95$10 * t$95$10), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[(t$95$9 * t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$10], $MachinePrecision] + N[Sqrt[t$95$9], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision] + t$95$11), $MachinePrecision], N[(N[(N[(0.5 * N[(1.0 / N[(t$95$9 * N[Sqrt[N[(1.0 / t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$5 + N[Sqrt[N[(1.0 + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision] + t$95$11), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 2e8

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - x}{\sqrt{x - -1} + \sqrt{x}} + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - x}{\sqrt{x - -1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - x}{\sqrt{x - -1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 71.4%

      \[\leadsto \left(\left(\frac{\left(x - -1\right) - x}{\sqrt{x - -1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{\left(y - -1\right) \cdot \left(y - -1\right)} - \sqrt{y \cdot y}}{\sqrt{y - -1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2e8 < y

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{y \cdot \sqrt{\frac{1}{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{1}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{\color{blue}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \color{blue}{\sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 49.2

          \[\leadsto \left(\mathsf{fma}\left(0.5, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 49.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \sqrt{t\_1 + 1} - \sqrt{t\_1}\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \mathsf{max}\left(t\_3, t\right)\\ t_5 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_4\right)\\ t_6 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_4\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := \mathsf{min}\left(t\_3, t\right)\\ t_9 := \sqrt{t\_8}\\ t_10 := \sqrt{t\_5 + 1} - \sqrt{t\_5}\\ t_11 := t\_6 - -1\\ \mathbf{if}\;t\_6 \leq 54000000:\\ \;\;\;\;\left(\left(\left(\sqrt{t\_8 + 1} - t\_9\right) + \frac{t\_11 - t\_7 \cdot t\_7}{\sqrt{t\_11} + t\_7}\right) + t\_2\right) + t\_10\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{t\_6 \cdot \sqrt{\frac{1}{t\_6}}}, \frac{1}{t\_9 + \sqrt{1 + t\_8}}\right) + t\_2\right) + t\_10\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (- (sqrt (+ t_1 1.0)) (sqrt t_1)))
        (t_3 (fmin (fmin x y) z))
        (t_4 (fmax t_3 t))
        (t_5 (fmax (fmax x y) t_4))
        (t_6 (fmin (fmax x y) t_4))
        (t_7 (sqrt t_6))
        (t_8 (fmin t_3 t))
        (t_9 (sqrt t_8))
        (t_10 (- (sqrt (+ t_5 1.0)) (sqrt t_5)))
        (t_11 (- t_6 -1.0)))
   (if (<= t_6 54000000.0)
     (+
      (+
       (+
        (- (sqrt (+ t_8 1.0)) t_9)
        (/ (- t_11 (* t_7 t_7)) (+ (sqrt t_11) t_7)))
       t_2)
      t_10)
     (+
      (+
       (fma
        0.5
        (/ 1.0 (* t_6 (sqrt (/ 1.0 t_6))))
        (/ 1.0 (+ t_9 (sqrt (+ 1.0 t_8)))))
       t_2)
      t_10))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = sqrt((t_1 + 1.0)) - sqrt(t_1);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = fmax(t_3, t);
	double t_5 = fmax(fmax(x, y), t_4);
	double t_6 = fmin(fmax(x, y), t_4);
	double t_7 = sqrt(t_6);
	double t_8 = fmin(t_3, t);
	double t_9 = sqrt(t_8);
	double t_10 = sqrt((t_5 + 1.0)) - sqrt(t_5);
	double t_11 = t_6 - -1.0;
	double tmp;
	if (t_6 <= 54000000.0) {
		tmp = (((sqrt((t_8 + 1.0)) - t_9) + ((t_11 - (t_7 * t_7)) / (sqrt(t_11) + t_7))) + t_2) + t_10;
	} else {
		tmp = (fma(0.5, (1.0 / (t_6 * sqrt((1.0 / t_6)))), (1.0 / (t_9 + sqrt((1.0 + t_8))))) + t_2) + t_10;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = Float64(sqrt(Float64(t_1 + 1.0)) - sqrt(t_1))
	t_3 = fmin(fmin(x, y), z)
	t_4 = fmax(t_3, t)
	t_5 = fmax(fmax(x, y), t_4)
	t_6 = fmin(fmax(x, y), t_4)
	t_7 = sqrt(t_6)
	t_8 = fmin(t_3, t)
	t_9 = sqrt(t_8)
	t_10 = Float64(sqrt(Float64(t_5 + 1.0)) - sqrt(t_5))
	t_11 = Float64(t_6 - -1.0)
	tmp = 0.0
	if (t_6 <= 54000000.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(t_8 + 1.0)) - t_9) + Float64(Float64(t_11 - Float64(t_7 * t_7)) / Float64(sqrt(t_11) + t_7))) + t_2) + t_10);
	else
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / Float64(t_6 * sqrt(Float64(1.0 / t_6)))), Float64(1.0 / Float64(t_9 + sqrt(Float64(1.0 + t_8))))) + t_2) + t_10);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[Max[t$95$3, t], $MachinePrecision]}, Block[{t$95$5 = N[Max[N[Max[x, y], $MachinePrecision], t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[Min[N[Max[x, y], $MachinePrecision], t$95$4], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Min[t$95$3, t], $MachinePrecision]}, Block[{t$95$9 = N[Sqrt[t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[(t$95$6 - -1.0), $MachinePrecision]}, If[LessEqual[t$95$6, 54000000.0], N[(N[(N[(N[(N[Sqrt[N[(t$95$8 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$9), $MachinePrecision] + N[(N[(t$95$11 - N[(t$95$7 * t$95$7), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$11], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$10), $MachinePrecision], N[(N[(N[(0.5 * N[(1.0 / N[(t$95$6 * N[Sqrt[N[(1.0 / t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$9 + N[Sqrt[N[(1.0 + t$95$8), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$10), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4e7

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1} \cdot \sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1} \cdot \sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y - \color{blue}{-1}\right) - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y - -1\right) - \color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y - -1\right) - \sqrt{y} \cdot \sqrt{y}}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y - -1\right) - \sqrt{y} \cdot \sqrt{y}}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y - -1\right) - \sqrt{y} \cdot \sqrt{y}}{\sqrt{\color{blue}{y - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y - -1\right) - \sqrt{y} \cdot \sqrt{y}}{\sqrt{\color{blue}{y - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y - -1\right) - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y - \color{blue}{-1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y - -1\right) - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y - -1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.4e7 < y

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{y \cdot \sqrt{\frac{1}{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{1}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{\color{blue}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \color{blue}{\sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 49.2

          \[\leadsto \left(\mathsf{fma}\left(0.5, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 49.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_5 := \sqrt{t\_2 + 1} - t\_3\\ t_6 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_7 := \mathsf{min}\left(t\_6, t\right)\\ t_8 := \sqrt{t\_7}\\ t_9 := \mathsf{max}\left(t\_6, t\right)\\ t_10 := \mathsf{max}\left(t\_4, t\_9\right)\\ t_11 := \sqrt{t\_10}\\ t_12 := \sqrt{t\_10 + 1} - t\_11\\ t_13 := \mathsf{min}\left(t\_4, t\_9\right)\\ t_14 := \sqrt{t\_13}\\ \mathbf{if}\;\left(\left(\left(\sqrt{t\_7 + 1} - t\_8\right) + \left(\sqrt{t\_13 + 1} - t\_14\right)\right) + t\_5\right) + t\_12 \leq 1.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{t\_13 \cdot \sqrt{\frac{1}{t\_13}}}, \frac{1}{t\_8 + \sqrt{1 + t\_7}}\right) + t\_5\right) + t\_12\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{t\_2 - -1} - t\_3\right) + \left(\sqrt{t\_13 - -1} - \left(t\_8 - \sqrt{t\_7 - -1}\right)\right)\right) - \left(t\_14 + \left(t\_11 - \sqrt{t\_10 - -1}\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (sqrt t_2))
        (t_4 (fmin (fmax x y) t_1))
        (t_5 (- (sqrt (+ t_2 1.0)) t_3))
        (t_6 (fmin (fmin x y) z))
        (t_7 (fmin t_6 t))
        (t_8 (sqrt t_7))
        (t_9 (fmax t_6 t))
        (t_10 (fmax t_4 t_9))
        (t_11 (sqrt t_10))
        (t_12 (- (sqrt (+ t_10 1.0)) t_11))
        (t_13 (fmin t_4 t_9))
        (t_14 (sqrt t_13)))
   (if (<=
        (+
         (+ (+ (- (sqrt (+ t_7 1.0)) t_8) (- (sqrt (+ t_13 1.0)) t_14)) t_5)
         t_12)
        1.0002)
     (+
      (+
       (fma
        0.5
        (/ 1.0 (* t_13 (sqrt (/ 1.0 t_13))))
        (/ 1.0 (+ t_8 (sqrt (+ 1.0 t_7)))))
       t_5)
      t_12)
     (-
      (+
       (- (sqrt (- t_2 -1.0)) t_3)
       (- (sqrt (- t_13 -1.0)) (- t_8 (sqrt (- t_7 -1.0)))))
      (+ t_14 (- t_11 (sqrt (- t_10 -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = sqrt(t_2);
	double t_4 = fmin(fmax(x, y), t_1);
	double t_5 = sqrt((t_2 + 1.0)) - t_3;
	double t_6 = fmin(fmin(x, y), z);
	double t_7 = fmin(t_6, t);
	double t_8 = sqrt(t_7);
	double t_9 = fmax(t_6, t);
	double t_10 = fmax(t_4, t_9);
	double t_11 = sqrt(t_10);
	double t_12 = sqrt((t_10 + 1.0)) - t_11;
	double t_13 = fmin(t_4, t_9);
	double t_14 = sqrt(t_13);
	double tmp;
	if (((((sqrt((t_7 + 1.0)) - t_8) + (sqrt((t_13 + 1.0)) - t_14)) + t_5) + t_12) <= 1.0002) {
		tmp = (fma(0.5, (1.0 / (t_13 * sqrt((1.0 / t_13)))), (1.0 / (t_8 + sqrt((1.0 + t_7))))) + t_5) + t_12;
	} else {
		tmp = ((sqrt((t_2 - -1.0)) - t_3) + (sqrt((t_13 - -1.0)) - (t_8 - sqrt((t_7 - -1.0))))) - (t_14 + (t_11 - sqrt((t_10 - -1.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = sqrt(t_2)
	t_4 = fmin(fmax(x, y), t_1)
	t_5 = Float64(sqrt(Float64(t_2 + 1.0)) - t_3)
	t_6 = fmin(fmin(x, y), z)
	t_7 = fmin(t_6, t)
	t_8 = sqrt(t_7)
	t_9 = fmax(t_6, t)
	t_10 = fmax(t_4, t_9)
	t_11 = sqrt(t_10)
	t_12 = Float64(sqrt(Float64(t_10 + 1.0)) - t_11)
	t_13 = fmin(t_4, t_9)
	t_14 = sqrt(t_13)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(t_7 + 1.0)) - t_8) + Float64(sqrt(Float64(t_13 + 1.0)) - t_14)) + t_5) + t_12) <= 1.0002)
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / Float64(t_13 * sqrt(Float64(1.0 / t_13)))), Float64(1.0 / Float64(t_8 + sqrt(Float64(1.0 + t_7))))) + t_5) + t_12);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(t_2 - -1.0)) - t_3) + Float64(sqrt(Float64(t_13 - -1.0)) - Float64(t_8 - sqrt(Float64(t_7 - -1.0))))) - Float64(t_14 + Float64(t_11 - sqrt(Float64(t_10 - -1.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$7 = N[Min[t$95$6, t], $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$6, t], $MachinePrecision]}, Block[{t$95$10 = N[Max[t$95$4, t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Sqrt[t$95$10], $MachinePrecision]}, Block[{t$95$12 = N[(N[Sqrt[N[(t$95$10 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$11), $MachinePrecision]}, Block[{t$95$13 = N[Min[t$95$4, t$95$9], $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[t$95$13], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$8), $MachinePrecision] + N[(N[Sqrt[N[(t$95$13 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$14), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$12), $MachinePrecision], 1.0002], N[(N[(N[(0.5 * N[(1.0 / N[(t$95$13 * N[Sqrt[N[(1.0 / t$95$13), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$8 + N[Sqrt[N[(1.0 + t$95$7), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$12), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(t$95$2 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision] + N[(N[Sqrt[N[(t$95$13 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$8 - N[Sqrt[N[(t$95$7 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$14 + N[(t$95$11 - N[Sqrt[N[(t$95$10 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0002

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{y \cdot \sqrt{\frac{1}{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{1}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{\color{blue}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \color{blue}{\sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 49.2

          \[\leadsto \left(\mathsf{fma}\left(0.5, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 49.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.0002 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      7. associate-+r- N/A

         \[\leadsto \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)}\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      8. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \sqrt{y}\right)} - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   3. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z - -1} - \sqrt{z}\right) + \left(\sqrt{y - -1} - \left(\sqrt{x} - \sqrt{x - -1}\right)\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t - -1}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_5 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_6 := \mathsf{min}\left(t\_5, t\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := \mathsf{max}\left(t\_5, t\right)\\ t_9 := \mathsf{max}\left(t\_4, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \sqrt{t\_9 + 1} - t\_10\\ t_12 := \mathsf{min}\left(t\_4, t\_8\right)\\ t_13 := \sqrt{t\_12}\\ \mathbf{if}\;\left(\left(\left(\sqrt{t\_6 + 1} - t\_7\right) + \left(\sqrt{t\_12 + 1} - t\_13\right)\right) + \left(\sqrt{t\_2 + 1} - t\_3\right)\right) + t\_11 \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{t\_12 \cdot \sqrt{\frac{1}{t\_12}}}, \frac{1}{t\_7 + \sqrt{1 + t\_6}}\right) + t\_11\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{t\_2 - -1} - t\_3\right) + \left(\sqrt{t\_12 - -1} - \left(t\_7 - \sqrt{t\_6 - -1}\right)\right)\right) - \left(t\_13 + \left(t\_10 - \sqrt{t\_9 - -1}\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (sqrt t_2))
        (t_4 (fmin (fmax x y) t_1))
        (t_5 (fmin (fmin x y) z))
        (t_6 (fmin t_5 t))
        (t_7 (sqrt t_6))
        (t_8 (fmax t_5 t))
        (t_9 (fmax t_4 t_8))
        (t_10 (sqrt t_9))
        (t_11 (- (sqrt (+ t_9 1.0)) t_10))
        (t_12 (fmin t_4 t_8))
        (t_13 (sqrt t_12)))
   (if (<=
        (+
         (+
          (+ (- (sqrt (+ t_6 1.0)) t_7) (- (sqrt (+ t_12 1.0)) t_13))
          (- (sqrt (+ t_2 1.0)) t_3))
         t_11)
        1.0002)
     (+
      (fma
       0.5
       (/ 1.0 (* t_12 (sqrt (/ 1.0 t_12))))
       (/ 1.0 (+ t_7 (sqrt (+ 1.0 t_6)))))
      t_11)
     (-
      (+
       (- (sqrt (- t_2 -1.0)) t_3)
       (- (sqrt (- t_12 -1.0)) (- t_7 (sqrt (- t_6 -1.0)))))
      (+ t_13 (- t_10 (sqrt (- t_9 -1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = sqrt(t_2);
	double t_4 = fmin(fmax(x, y), t_1);
	double t_5 = fmin(fmin(x, y), z);
	double t_6 = fmin(t_5, t);
	double t_7 = sqrt(t_6);
	double t_8 = fmax(t_5, t);
	double t_9 = fmax(t_4, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = sqrt((t_9 + 1.0)) - t_10;
	double t_12 = fmin(t_4, t_8);
	double t_13 = sqrt(t_12);
	double tmp;
	if (((((sqrt((t_6 + 1.0)) - t_7) + (sqrt((t_12 + 1.0)) - t_13)) + (sqrt((t_2 + 1.0)) - t_3)) + t_11) <= 1.0002) {
		tmp = fma(0.5, (1.0 / (t_12 * sqrt((1.0 / t_12)))), (1.0 / (t_7 + sqrt((1.0 + t_6))))) + t_11;
	} else {
		tmp = ((sqrt((t_2 - -1.0)) - t_3) + (sqrt((t_12 - -1.0)) - (t_7 - sqrt((t_6 - -1.0))))) - (t_13 + (t_10 - sqrt((t_9 - -1.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = sqrt(t_2)
	t_4 = fmin(fmax(x, y), t_1)
	t_5 = fmin(fmin(x, y), z)
	t_6 = fmin(t_5, t)
	t_7 = sqrt(t_6)
	t_8 = fmax(t_5, t)
	t_9 = fmax(t_4, t_8)
	t_10 = sqrt(t_9)
	t_11 = Float64(sqrt(Float64(t_9 + 1.0)) - t_10)
	t_12 = fmin(t_4, t_8)
	t_13 = sqrt(t_12)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(t_6 + 1.0)) - t_7) + Float64(sqrt(Float64(t_12 + 1.0)) - t_13)) + Float64(sqrt(Float64(t_2 + 1.0)) - t_3)) + t_11) <= 1.0002)
		tmp = Float64(fma(0.5, Float64(1.0 / Float64(t_12 * sqrt(Float64(1.0 / t_12)))), Float64(1.0 / Float64(t_7 + sqrt(Float64(1.0 + t_6))))) + t_11);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(t_2 - -1.0)) - t_3) + Float64(sqrt(Float64(t_12 - -1.0)) - Float64(t_7 - sqrt(Float64(t_6 - -1.0))))) - Float64(t_13 + Float64(t_10 - sqrt(Float64(t_9 - -1.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$5 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$5, t], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$5, t], $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$4, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]}, Block[{t$95$12 = N[Min[t$95$4, t$95$8], $MachinePrecision]}, Block[{t$95$13 = N[Sqrt[t$95$12], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(t$95$6 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision] + N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$13), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$11), $MachinePrecision], 1.0002], N[(N[(0.5 * N[(1.0 / N[(t$95$12 * N[Sqrt[N[(1.0 / t$95$12), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$7 + N[Sqrt[N[(1.0 + t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$11), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(t$95$2 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision] + N[(N[Sqrt[N[(t$95$12 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(t$95$7 - N[Sqrt[N[(t$95$6 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$13 + N[(t$95$10 - N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0002

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\frac{1}{2} \cdot \frac{1}{y \cdot \sqrt{\frac{1}{y}}} + \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{\color{blue}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \color{blue}{\sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. lower-+.f64 28.7

           \[\leadsto \mathsf{fma}\left(0.5, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 28.7%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{1}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.0002 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right)\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      7. associate-+r- N/A

         \[\leadsto \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right) - \sqrt{y}\right)}\right) - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      8. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{y + 1}\right)\right) - \sqrt{y}\right)} - \left(\mathsf{neg}\left(\left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   3. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z - -1} - \sqrt{z}\right) + \left(\sqrt{y - -1} - \left(\sqrt{x} - \sqrt{x - -1}\right)\right)\right) - \left(\sqrt{y} + \left(\sqrt{t} - \sqrt{t - -1}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \sqrt{t\_2 + 1} - \sqrt{t\_2}\\ t_6 := \mathsf{min}\left(t\_4, t\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := \mathsf{max}\left(t\_4, t\right)\\ t_9 := \mathsf{max}\left(t\_3, t\_8\right)\\ t_10 := \sqrt{t\_9 + 1} - \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_8\right)\\ t_12 := \sqrt{t\_11 + 1} - \sqrt{t\_11}\\ \mathbf{if}\;\left(\left(\left(\sqrt{t\_6 + 1} - t\_7\right) + t\_12\right) + t\_5\right) + t\_10 \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{t\_11 \cdot \sqrt{\frac{1}{t\_11}}}, \frac{1}{t\_7 + \sqrt{1 + t\_6}}\right) + t\_10\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{1 + t\_7} + t\_12\right) + t\_5\right) + t\_10\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (fmin (fmax x y) t_1))
        (t_4 (fmin (fmin x y) z))
        (t_5 (- (sqrt (+ t_2 1.0)) (sqrt t_2)))
        (t_6 (fmin t_4 t))
        (t_7 (sqrt t_6))
        (t_8 (fmax t_4 t))
        (t_9 (fmax t_3 t_8))
        (t_10 (- (sqrt (+ t_9 1.0)) (sqrt t_9)))
        (t_11 (fmin t_3 t_8))
        (t_12 (- (sqrt (+ t_11 1.0)) (sqrt t_11))))
   (if (<= (+ (+ (+ (- (sqrt (+ t_6 1.0)) t_7) t_12) t_5) t_10) 1.0002)
     (+
      (fma
       0.5
       (/ 1.0 (* t_11 (sqrt (/ 1.0 t_11))))
       (/ 1.0 (+ t_7 (sqrt (+ 1.0 t_6)))))
      t_10)
     (+ (+ (+ (/ 1.0 (+ 1.0 t_7)) t_12) t_5) t_10))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = sqrt((t_2 + 1.0)) - sqrt(t_2);
	double t_6 = fmin(t_4, t);
	double t_7 = sqrt(t_6);
	double t_8 = fmax(t_4, t);
	double t_9 = fmax(t_3, t_8);
	double t_10 = sqrt((t_9 + 1.0)) - sqrt(t_9);
	double t_11 = fmin(t_3, t_8);
	double t_12 = sqrt((t_11 + 1.0)) - sqrt(t_11);
	double tmp;
	if (((((sqrt((t_6 + 1.0)) - t_7) + t_12) + t_5) + t_10) <= 1.0002) {
		tmp = fma(0.5, (1.0 / (t_11 * sqrt((1.0 / t_11)))), (1.0 / (t_7 + sqrt((1.0 + t_6))))) + t_10;
	} else {
		tmp = (((1.0 / (1.0 + t_7)) + t_12) + t_5) + t_10;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2))
	t_6 = fmin(t_4, t)
	t_7 = sqrt(t_6)
	t_8 = fmax(t_4, t)
	t_9 = fmax(t_3, t_8)
	t_10 = Float64(sqrt(Float64(t_9 + 1.0)) - sqrt(t_9))
	t_11 = fmin(t_3, t_8)
	t_12 = Float64(sqrt(Float64(t_11 + 1.0)) - sqrt(t_11))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(t_6 + 1.0)) - t_7) + t_12) + t_5) + t_10) <= 1.0002)
		tmp = Float64(fma(0.5, Float64(1.0 / Float64(t_11 * sqrt(Float64(1.0 / t_11)))), Float64(1.0 / Float64(t_7 + sqrt(Float64(1.0 + t_6))))) + t_10);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(1.0 + t_7)) + t_12) + t_5) + t_10);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$9 = N[Max[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$9], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$8], $MachinePrecision]}, Block[{t$95$12 = N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$11], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(t$95$6 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision] + t$95$12), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$10), $MachinePrecision], 1.0002], N[(N[(0.5 * N[(1.0 / N[(t$95$11 * N[Sqrt[N[(1.0 / t$95$11), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$7 + N[Sqrt[N[(1.0 + t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$10), $MachinePrecision], N[(N[(N[(N[(1.0 / N[(1.0 + t$95$7), $MachinePrecision]), $MachinePrecision] + t$95$12), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$10), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0002

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\frac{1}{2} \cdot \frac{1}{y \cdot \sqrt{\frac{1}{y}}} + \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{\color{blue}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \color{blue}{\sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. lower-+.f64 28.7

           \[\leadsto \mathsf{fma}\left(0.5, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 28.7%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{1}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.0002 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\frac{1}{1 + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{1 + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{1 + \color{blue}{\sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 90.5

         \[\leadsto \left(\left(\frac{1}{1 + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 90.5%

      \[\leadsto \left(\left(\color{blue}{\frac{1}{1 + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_5 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_6 := \mathsf{min}\left(t\_5, t\_2\right)\\ t_7 := \mathsf{max}\left(t\_5, t\_2\right)\\ t_8 := \sqrt{t\_7 + 1} - \sqrt{t\_7}\\ t_9 := \mathsf{min}\left(t\_1, t\right)\\ t_10 := \sqrt{t\_9}\\ \mathbf{if}\;t\_6 \leq 1.2:\\ \;\;\;\;\left(\left(\left(2 + 0.5 \cdot t\_6\right) - \left(t\_10 + \sqrt{t\_6}\right)\right) + \left(\sqrt{t\_4 + 1} - \sqrt{t\_4}\right)\right) + t\_8\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{t\_6 \cdot \sqrt{\frac{1}{t\_6}}}, \frac{1}{t\_10 + \sqrt{1 + t\_9}}\right) + t\_8\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmin (fmin x y) z))
        (t_2 (fmax t_1 t))
        (t_3 (fmax (fmin x y) z))
        (t_4 (fmax (fmax x y) t_3))
        (t_5 (fmin (fmax x y) t_3))
        (t_6 (fmin t_5 t_2))
        (t_7 (fmax t_5 t_2))
        (t_8 (- (sqrt (+ t_7 1.0)) (sqrt t_7)))
        (t_9 (fmin t_1 t))
        (t_10 (sqrt t_9)))
   (if (<= t_6 1.2)
     (+
      (+
       (- (+ 2.0 (* 0.5 t_6)) (+ t_10 (sqrt t_6)))
       (- (sqrt (+ t_4 1.0)) (sqrt t_4)))
      t_8)
     (+
      (fma
       0.5
       (/ 1.0 (* t_6 (sqrt (/ 1.0 t_6))))
       (/ 1.0 (+ t_10 (sqrt (+ 1.0 t_9)))))
      t_8))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmax(fmin(x, y), z);
	double t_4 = fmax(fmax(x, y), t_3);
	double t_5 = fmin(fmax(x, y), t_3);
	double t_6 = fmin(t_5, t_2);
	double t_7 = fmax(t_5, t_2);
	double t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	double t_9 = fmin(t_1, t);
	double t_10 = sqrt(t_9);
	double tmp;
	if (t_6 <= 1.2) {
		tmp = (((2.0 + (0.5 * t_6)) - (t_10 + sqrt(t_6))) + (sqrt((t_4 + 1.0)) - sqrt(t_4))) + t_8;
	} else {
		tmp = fma(0.5, (1.0 / (t_6 * sqrt((1.0 / t_6)))), (1.0 / (t_10 + sqrt((1.0 + t_9))))) + t_8;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmax(fmin(x, y), z)
	t_4 = fmax(fmax(x, y), t_3)
	t_5 = fmin(fmax(x, y), t_3)
	t_6 = fmin(t_5, t_2)
	t_7 = fmax(t_5, t_2)
	t_8 = Float64(sqrt(Float64(t_7 + 1.0)) - sqrt(t_7))
	t_9 = fmin(t_1, t)
	t_10 = sqrt(t_9)
	tmp = 0.0
	if (t_6 <= 1.2)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(0.5 * t_6)) - Float64(t_10 + sqrt(t_6))) + Float64(sqrt(Float64(t_4 + 1.0)) - sqrt(t_4))) + t_8);
	else
		tmp = Float64(fma(0.5, Float64(1.0 / Float64(t_6 * sqrt(Float64(1.0 / t_6)))), Float64(1.0 / Float64(t_10 + sqrt(Float64(1.0 + t_9))))) + t_8);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Max[x, y], $MachinePrecision], t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Min[N[Max[x, y], $MachinePrecision], t$95$3], $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$5, t$95$2], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$5, t$95$2], $MachinePrecision]}, Block[{t$95$8 = N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, If[LessEqual[t$95$6, 1.2], N[(N[(N[(N[(2.0 + N[(0.5 * t$95$6), $MachinePrecision]), $MachinePrecision] - N[(t$95$10 + N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision], N[(N[(0.5 * N[(1.0 / N[(t$95$6 * N[Sqrt[N[(1.0 / t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$10 + N[Sqrt[N[(1.0 + t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.19999999999999996

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{\color{blue}{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 36.0

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + 0.5 \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 36.0%

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + 0.5 \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\left(\left(2 + \frac{1}{2} \cdot y\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(2 + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 27.1

         \[\leadsto \left(\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 27.1%

      \[\leadsto \left(\left(\left(2 + 0.5 \cdot y\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.19999999999999996 < y

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\frac{1}{2} \cdot \frac{1}{y \cdot \sqrt{\frac{1}{y}}} + \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{\color{blue}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \color{blue}{\sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       7. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       10. lower-+.f64 28.7

           \[\leadsto \mathsf{fma}\left(0.5, \frac{1}{y \cdot \sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 28.7%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{1}{y \cdot \sqrt{\frac{1}{y}}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_5 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_3\right)\\ t_6 := \mathsf{min}\left(t\_5, t\_2\right)\\ t_7 := \sqrt{t\_6}\\ t_8 := \mathsf{max}\left(t\_5, t\_2\right)\\ t_9 := \sqrt{t\_8 + 1} - \sqrt{t\_8}\\ t_10 := \mathsf{min}\left(t\_1, t\right)\\ t_11 := \sqrt{t\_10}\\ \mathbf{if}\;t\_6 \leq 0.18:\\ \;\;\;\;\left(\left(\left(2 + 0.5 \cdot t\_6\right) - \left(t\_11 + t\_7\right)\right) + \left(\sqrt{t\_4 + 1} - \sqrt{t\_4}\right)\right) + t\_9\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_11 + \sqrt{t\_10 - -1}} + \left(\sqrt{t\_6 - -1} - t\_7\right)\right) + t\_9\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmin (fmin x y) z))
        (t_2 (fmax t_1 t))
        (t_3 (fmax (fmin x y) z))
        (t_4 (fmax (fmax x y) t_3))
        (t_5 (fmin (fmax x y) t_3))
        (t_6 (fmin t_5 t_2))
        (t_7 (sqrt t_6))
        (t_8 (fmax t_5 t_2))
        (t_9 (- (sqrt (+ t_8 1.0)) (sqrt t_8)))
        (t_10 (fmin t_1 t))
        (t_11 (sqrt t_10)))
   (if (<= t_6 0.18)
     (+
      (+
       (- (+ 2.0 (* 0.5 t_6)) (+ t_11 t_7))
       (- (sqrt (+ t_4 1.0)) (sqrt t_4)))
      t_9)
     (+
      (+ (/ 1.0 (+ t_11 (sqrt (- t_10 -1.0)))) (- (sqrt (- t_6 -1.0)) t_7))
      t_9))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmax(fmin(x, y), z);
	double t_4 = fmax(fmax(x, y), t_3);
	double t_5 = fmin(fmax(x, y), t_3);
	double t_6 = fmin(t_5, t_2);
	double t_7 = sqrt(t_6);
	double t_8 = fmax(t_5, t_2);
	double t_9 = sqrt((t_8 + 1.0)) - sqrt(t_8);
	double t_10 = fmin(t_1, t);
	double t_11 = sqrt(t_10);
	double tmp;
	if (t_6 <= 0.18) {
		tmp = (((2.0 + (0.5 * t_6)) - (t_11 + t_7)) + (sqrt((t_4 + 1.0)) - sqrt(t_4))) + t_9;
	} else {
		tmp = ((1.0 / (t_11 + sqrt((t_10 - -1.0)))) + (sqrt((t_6 - -1.0)) - t_7)) + t_9;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmax(t_1, t)
    t_3 = fmax(fmin(x, y), z)
    t_4 = fmax(fmax(x, y), t_3)
    t_5 = fmin(fmax(x, y), t_3)
    t_6 = fmin(t_5, t_2)
    t_7 = sqrt(t_6)
    t_8 = fmax(t_5, t_2)
    t_9 = sqrt((t_8 + 1.0d0)) - sqrt(t_8)
    t_10 = fmin(t_1, t)
    t_11 = sqrt(t_10)
    if (t_6 <= 0.18d0) then
        tmp = (((2.0d0 + (0.5d0 * t_6)) - (t_11 + t_7)) + (sqrt((t_4 + 1.0d0)) - sqrt(t_4))) + t_9
    else
        tmp = ((1.0d0 / (t_11 + sqrt((t_10 - (-1.0d0))))) + (sqrt((t_6 - (-1.0d0))) - t_7)) + t_9
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmax(fmin(x, y), z);
	double t_4 = fmax(fmax(x, y), t_3);
	double t_5 = fmin(fmax(x, y), t_3);
	double t_6 = fmin(t_5, t_2);
	double t_7 = Math.sqrt(t_6);
	double t_8 = fmax(t_5, t_2);
	double t_9 = Math.sqrt((t_8 + 1.0)) - Math.sqrt(t_8);
	double t_10 = fmin(t_1, t);
	double t_11 = Math.sqrt(t_10);
	double tmp;
	if (t_6 <= 0.18) {
		tmp = (((2.0 + (0.5 * t_6)) - (t_11 + t_7)) + (Math.sqrt((t_4 + 1.0)) - Math.sqrt(t_4))) + t_9;
	} else {
		tmp = ((1.0 / (t_11 + Math.sqrt((t_10 - -1.0)))) + (Math.sqrt((t_6 - -1.0)) - t_7)) + t_9;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmax(fmin(x, y), z)
	t_4 = fmax(fmax(x, y), t_3)
	t_5 = fmin(fmax(x, y), t_3)
	t_6 = fmin(t_5, t_2)
	t_7 = math.sqrt(t_6)
	t_8 = fmax(t_5, t_2)
	t_9 = math.sqrt((t_8 + 1.0)) - math.sqrt(t_8)
	t_10 = fmin(t_1, t)
	t_11 = math.sqrt(t_10)
	tmp = 0
	if t_6 <= 0.18:
		tmp = (((2.0 + (0.5 * t_6)) - (t_11 + t_7)) + (math.sqrt((t_4 + 1.0)) - math.sqrt(t_4))) + t_9
	else:
		tmp = ((1.0 / (t_11 + math.sqrt((t_10 - -1.0)))) + (math.sqrt((t_6 - -1.0)) - t_7)) + t_9
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmax(fmin(x, y), z)
	t_4 = fmax(fmax(x, y), t_3)
	t_5 = fmin(fmax(x, y), t_3)
	t_6 = fmin(t_5, t_2)
	t_7 = sqrt(t_6)
	t_8 = fmax(t_5, t_2)
	t_9 = Float64(sqrt(Float64(t_8 + 1.0)) - sqrt(t_8))
	t_10 = fmin(t_1, t)
	t_11 = sqrt(t_10)
	tmp = 0.0
	if (t_6 <= 0.18)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(0.5 * t_6)) - Float64(t_11 + t_7)) + Float64(sqrt(Float64(t_4 + 1.0)) - sqrt(t_4))) + t_9);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_11 + sqrt(Float64(t_10 - -1.0)))) + Float64(sqrt(Float64(t_6 - -1.0)) - t_7)) + t_9);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = max(t_1, t);
	t_3 = max(min(x, y), z);
	t_4 = max(max(x, y), t_3);
	t_5 = min(max(x, y), t_3);
	t_6 = min(t_5, t_2);
	t_7 = sqrt(t_6);
	t_8 = max(t_5, t_2);
	t_9 = sqrt((t_8 + 1.0)) - sqrt(t_8);
	t_10 = min(t_1, t);
	t_11 = sqrt(t_10);
	tmp = 0.0;
	if (t_6 <= 0.18)
		tmp = (((2.0 + (0.5 * t_6)) - (t_11 + t_7)) + (sqrt((t_4 + 1.0)) - sqrt(t_4))) + t_9;
	else
		tmp = ((1.0 / (t_11 + sqrt((t_10 - -1.0)))) + (sqrt((t_6 - -1.0)) - t_7)) + t_9;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[Max[N[Max[x, y], $MachinePrecision], t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Min[N[Max[x, y], $MachinePrecision], t$95$3], $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$5, t$95$2], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$5, t$95$2], $MachinePrecision]}, Block[{t$95$9 = N[(N[Sqrt[N[(t$95$8 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$8], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$11 = N[Sqrt[t$95$10], $MachinePrecision]}, If[LessEqual[t$95$6, 0.18], N[(N[(N[(N[(2.0 + N[(0.5 * t$95$6), $MachinePrecision]), $MachinePrecision] - N[(t$95$11 + t$95$7), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$9), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$11 + N[Sqrt[N[(t$95$10 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$6 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision] + t$95$9), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.17999999999999999

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{2} \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{\color{blue}{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 36.0

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + x} + 0.5 \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 36.0%

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + 0.5 \cdot y\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\left(\left(2 + \frac{1}{2} \cdot y\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(2 + \frac{1}{2} \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 27.1

         \[\leadsto \left(\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 27.1%

      \[\leadsto \left(\left(\left(2 + 0.5 \cdot y\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.17999999999999999 < y

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \sqrt{1 + y}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + y} - \sqrt{\color{blue}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. add-flip N/A

          \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{x - \left(\mathsf{neg}\left(1\right)\right)}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{x - -1}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{x - -1}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 52.9%

       \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{x - -1}} + \color{blue}{\left(\sqrt{y - -1} - \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \mathsf{min}\left(t\_3, t\right)\\ t_5 := \mathsf{max}\left(t\_3, t\right)\\ t_6 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_7 := \mathsf{max}\left(t\_6, t\_5\right)\\ t_8 := \sqrt{t\_7 + 1} - \sqrt{t\_7}\\ t_9 := \mathsf{min}\left(t\_6, t\_5\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \sqrt{t\_4}\\ \mathbf{if}\;t\_9 \leq 1.15 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(2 - \left(t\_11 + t\_10\right)\right) + \left(\sqrt{t\_2 + 1} - \sqrt{t\_2}\right)\right) + t\_8\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_11 + \sqrt{t\_4 - -1}} + \left(\sqrt{t\_9 - -1} - t\_10\right)\right) + t\_8\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (fmin (fmin x y) z))
        (t_4 (fmin t_3 t))
        (t_5 (fmax t_3 t))
        (t_6 (fmin (fmax x y) t_1))
        (t_7 (fmax t_6 t_5))
        (t_8 (- (sqrt (+ t_7 1.0)) (sqrt t_7)))
        (t_9 (fmin t_6 t_5))
        (t_10 (sqrt t_9))
        (t_11 (sqrt t_4)))
   (if (<= t_9 1.15e-13)
     (+ (+ (- 2.0 (+ t_11 t_10)) (- (sqrt (+ t_2 1.0)) (sqrt t_2))) t_8)
     (+
      (+ (/ 1.0 (+ t_11 (sqrt (- t_4 -1.0)))) (- (sqrt (- t_9 -1.0)) t_10))
      t_8))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = fmax(t_3, t);
	double t_6 = fmin(fmax(x, y), t_1);
	double t_7 = fmax(t_6, t_5);
	double t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	double t_9 = fmin(t_6, t_5);
	double t_10 = sqrt(t_9);
	double t_11 = sqrt(t_4);
	double tmp;
	if (t_9 <= 1.15e-13) {
		tmp = ((2.0 - (t_11 + t_10)) + (sqrt((t_2 + 1.0)) - sqrt(t_2))) + t_8;
	} else {
		tmp = ((1.0 / (t_11 + sqrt((t_4 - -1.0)))) + (sqrt((t_9 - -1.0)) - t_10)) + t_8;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmin(x, y), z)
    t_4 = fmin(t_3, t)
    t_5 = fmax(t_3, t)
    t_6 = fmin(fmax(x, y), t_1)
    t_7 = fmax(t_6, t_5)
    t_8 = sqrt((t_7 + 1.0d0)) - sqrt(t_7)
    t_9 = fmin(t_6, t_5)
    t_10 = sqrt(t_9)
    t_11 = sqrt(t_4)
    if (t_9 <= 1.15d-13) then
        tmp = ((2.0d0 - (t_11 + t_10)) + (sqrt((t_2 + 1.0d0)) - sqrt(t_2))) + t_8
    else
        tmp = ((1.0d0 / (t_11 + sqrt((t_4 - (-1.0d0))))) + (sqrt((t_9 - (-1.0d0))) - t_10)) + t_8
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = fmax(t_3, t);
	double t_6 = fmin(fmax(x, y), t_1);
	double t_7 = fmax(t_6, t_5);
	double t_8 = Math.sqrt((t_7 + 1.0)) - Math.sqrt(t_7);
	double t_9 = fmin(t_6, t_5);
	double t_10 = Math.sqrt(t_9);
	double t_11 = Math.sqrt(t_4);
	double tmp;
	if (t_9 <= 1.15e-13) {
		tmp = ((2.0 - (t_11 + t_10)) + (Math.sqrt((t_2 + 1.0)) - Math.sqrt(t_2))) + t_8;
	} else {
		tmp = ((1.0 / (t_11 + Math.sqrt((t_4 - -1.0)))) + (Math.sqrt((t_9 - -1.0)) - t_10)) + t_8;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = fmax(t_3, t)
	t_6 = fmin(fmax(x, y), t_1)
	t_7 = fmax(t_6, t_5)
	t_8 = math.sqrt((t_7 + 1.0)) - math.sqrt(t_7)
	t_9 = fmin(t_6, t_5)
	t_10 = math.sqrt(t_9)
	t_11 = math.sqrt(t_4)
	tmp = 0
	if t_9 <= 1.15e-13:
		tmp = ((2.0 - (t_11 + t_10)) + (math.sqrt((t_2 + 1.0)) - math.sqrt(t_2))) + t_8
	else:
		tmp = ((1.0 / (t_11 + math.sqrt((t_4 - -1.0)))) + (math.sqrt((t_9 - -1.0)) - t_10)) + t_8
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = fmax(t_3, t)
	t_6 = fmin(fmax(x, y), t_1)
	t_7 = fmax(t_6, t_5)
	t_8 = Float64(sqrt(Float64(t_7 + 1.0)) - sqrt(t_7))
	t_9 = fmin(t_6, t_5)
	t_10 = sqrt(t_9)
	t_11 = sqrt(t_4)
	tmp = 0.0
	if (t_9 <= 1.15e-13)
		tmp = Float64(Float64(Float64(2.0 - Float64(t_11 + t_10)) + Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2))) + t_8);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_11 + sqrt(Float64(t_4 - -1.0)))) + Float64(sqrt(Float64(t_9 - -1.0)) - t_10)) + t_8);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(min(x, y), z);
	t_4 = min(t_3, t);
	t_5 = max(t_3, t);
	t_6 = min(max(x, y), t_1);
	t_7 = max(t_6, t_5);
	t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	t_9 = min(t_6, t_5);
	t_10 = sqrt(t_9);
	t_11 = sqrt(t_4);
	tmp = 0.0;
	if (t_9 <= 1.15e-13)
		tmp = ((2.0 - (t_11 + t_10)) + (sqrt((t_2 + 1.0)) - sqrt(t_2))) + t_8;
	else
		tmp = ((1.0 / (t_11 + sqrt((t_4 - -1.0)))) + (sqrt((t_9 - -1.0)) - t_10)) + t_8;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[Min[t$95$3, t], $MachinePrecision]}, Block[{t$95$5 = N[Max[t$95$3, t], $MachinePrecision]}, Block[{t$95$6 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$6, t$95$5], $MachinePrecision]}, Block[{t$95$8 = N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$6, t$95$5], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Sqrt[t$95$4], $MachinePrecision]}, If[LessEqual[t$95$9, 1.15e-13], N[(N[(N[(2.0 - N[(t$95$11 + t$95$10), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$11 + N[Sqrt[N[(t$95$4 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1499999999999999e-13

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{\color{blue}{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 37.8

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 37.8%

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\left(2 - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 25.2%

      \[\leadsto \left(\left(2 - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.1499999999999999e-13 < y

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \sqrt{1 + y}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. associate--l+ N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + y} - \sqrt{\color{blue}{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. add-flip N/A

          \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{x - \left(\mathsf{neg}\left(1\right)\right)}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{x - -1}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lift--.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{x - -1}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 52.9%

       \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{x - -1}} + \color{blue}{\left(\sqrt{y - -1} - \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 95.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_4 := \mathsf{min}\left(t\_3, t\right)\\ t_5 := \mathsf{max}\left(t\_3, t\right)\\ t_6 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_7 := \mathsf{max}\left(t\_6, t\_5\right)\\ t_8 := \sqrt{t\_7 + 1} - \sqrt{t\_7}\\ t_9 := \mathsf{min}\left(t\_6, t\_5\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \sqrt{t\_4}\\ \mathbf{if}\;t\_9 \leq 1.15 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(2 - \left(t\_11 + t\_10\right)\right) + \left(\sqrt{t\_2 + 1} - \sqrt{t\_2}\right)\right) + t\_8\\ \mathbf{elif}\;t\_9 \leq 1600000000000:\\ \;\;\;\;\left(\left(\sqrt{1 + t\_9} + \frac{1}{1 + t\_11}\right) - t\_10\right) + t\_8\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_11 + \sqrt{1 + t\_4}} + t\_8\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (fmin (fmin x y) z))
        (t_4 (fmin t_3 t))
        (t_5 (fmax t_3 t))
        (t_6 (fmin (fmax x y) t_1))
        (t_7 (fmax t_6 t_5))
        (t_8 (- (sqrt (+ t_7 1.0)) (sqrt t_7)))
        (t_9 (fmin t_6 t_5))
        (t_10 (sqrt t_9))
        (t_11 (sqrt t_4)))
   (if (<= t_9 1.15e-13)
     (+ (+ (- 2.0 (+ t_11 t_10)) (- (sqrt (+ t_2 1.0)) (sqrt t_2))) t_8)
     (if (<= t_9 1600000000000.0)
       (+ (- (+ (sqrt (+ 1.0 t_9)) (/ 1.0 (+ 1.0 t_11))) t_10) t_8)
       (+ (/ 1.0 (+ t_11 (sqrt (+ 1.0 t_4)))) t_8)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = fmax(t_3, t);
	double t_6 = fmin(fmax(x, y), t_1);
	double t_7 = fmax(t_6, t_5);
	double t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	double t_9 = fmin(t_6, t_5);
	double t_10 = sqrt(t_9);
	double t_11 = sqrt(t_4);
	double tmp;
	if (t_9 <= 1.15e-13) {
		tmp = ((2.0 - (t_11 + t_10)) + (sqrt((t_2 + 1.0)) - sqrt(t_2))) + t_8;
	} else if (t_9 <= 1600000000000.0) {
		tmp = ((sqrt((1.0 + t_9)) + (1.0 / (1.0 + t_11))) - t_10) + t_8;
	} else {
		tmp = (1.0 / (t_11 + sqrt((1.0 + t_4)))) + t_8;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmin(x, y), z)
    t_4 = fmin(t_3, t)
    t_5 = fmax(t_3, t)
    t_6 = fmin(fmax(x, y), t_1)
    t_7 = fmax(t_6, t_5)
    t_8 = sqrt((t_7 + 1.0d0)) - sqrt(t_7)
    t_9 = fmin(t_6, t_5)
    t_10 = sqrt(t_9)
    t_11 = sqrt(t_4)
    if (t_9 <= 1.15d-13) then
        tmp = ((2.0d0 - (t_11 + t_10)) + (sqrt((t_2 + 1.0d0)) - sqrt(t_2))) + t_8
    else if (t_9 <= 1600000000000.0d0) then
        tmp = ((sqrt((1.0d0 + t_9)) + (1.0d0 / (1.0d0 + t_11))) - t_10) + t_8
    else
        tmp = (1.0d0 / (t_11 + sqrt((1.0d0 + t_4)))) + t_8
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmin(x, y), z);
	double t_4 = fmin(t_3, t);
	double t_5 = fmax(t_3, t);
	double t_6 = fmin(fmax(x, y), t_1);
	double t_7 = fmax(t_6, t_5);
	double t_8 = Math.sqrt((t_7 + 1.0)) - Math.sqrt(t_7);
	double t_9 = fmin(t_6, t_5);
	double t_10 = Math.sqrt(t_9);
	double t_11 = Math.sqrt(t_4);
	double tmp;
	if (t_9 <= 1.15e-13) {
		tmp = ((2.0 - (t_11 + t_10)) + (Math.sqrt((t_2 + 1.0)) - Math.sqrt(t_2))) + t_8;
	} else if (t_9 <= 1600000000000.0) {
		tmp = ((Math.sqrt((1.0 + t_9)) + (1.0 / (1.0 + t_11))) - t_10) + t_8;
	} else {
		tmp = (1.0 / (t_11 + Math.sqrt((1.0 + t_4)))) + t_8;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = fmax(t_3, t)
	t_6 = fmin(fmax(x, y), t_1)
	t_7 = fmax(t_6, t_5)
	t_8 = math.sqrt((t_7 + 1.0)) - math.sqrt(t_7)
	t_9 = fmin(t_6, t_5)
	t_10 = math.sqrt(t_9)
	t_11 = math.sqrt(t_4)
	tmp = 0
	if t_9 <= 1.15e-13:
		tmp = ((2.0 - (t_11 + t_10)) + (math.sqrt((t_2 + 1.0)) - math.sqrt(t_2))) + t_8
	elif t_9 <= 1600000000000.0:
		tmp = ((math.sqrt((1.0 + t_9)) + (1.0 / (1.0 + t_11))) - t_10) + t_8
	else:
		tmp = (1.0 / (t_11 + math.sqrt((1.0 + t_4)))) + t_8
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmin(x, y), z)
	t_4 = fmin(t_3, t)
	t_5 = fmax(t_3, t)
	t_6 = fmin(fmax(x, y), t_1)
	t_7 = fmax(t_6, t_5)
	t_8 = Float64(sqrt(Float64(t_7 + 1.0)) - sqrt(t_7))
	t_9 = fmin(t_6, t_5)
	t_10 = sqrt(t_9)
	t_11 = sqrt(t_4)
	tmp = 0.0
	if (t_9 <= 1.15e-13)
		tmp = Float64(Float64(Float64(2.0 - Float64(t_11 + t_10)) + Float64(sqrt(Float64(t_2 + 1.0)) - sqrt(t_2))) + t_8);
	elseif (t_9 <= 1600000000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t_9)) + Float64(1.0 / Float64(1.0 + t_11))) - t_10) + t_8);
	else
		tmp = Float64(Float64(1.0 / Float64(t_11 + sqrt(Float64(1.0 + t_4)))) + t_8);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(min(x, y), z);
	t_4 = min(t_3, t);
	t_5 = max(t_3, t);
	t_6 = min(max(x, y), t_1);
	t_7 = max(t_6, t_5);
	t_8 = sqrt((t_7 + 1.0)) - sqrt(t_7);
	t_9 = min(t_6, t_5);
	t_10 = sqrt(t_9);
	t_11 = sqrt(t_4);
	tmp = 0.0;
	if (t_9 <= 1.15e-13)
		tmp = ((2.0 - (t_11 + t_10)) + (sqrt((t_2 + 1.0)) - sqrt(t_2))) + t_8;
	elseif (t_9 <= 1600000000000.0)
		tmp = ((sqrt((1.0 + t_9)) + (1.0 / (1.0 + t_11))) - t_10) + t_8;
	else
		tmp = (1.0 / (t_11 + sqrt((1.0 + t_4)))) + t_8;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$4 = N[Min[t$95$3, t], $MachinePrecision]}, Block[{t$95$5 = N[Max[t$95$3, t], $MachinePrecision]}, Block[{t$95$6 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$6, t$95$5], $MachinePrecision]}, Block[{t$95$8 = N[(N[Sqrt[N[(t$95$7 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$6, t$95$5], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Sqrt[t$95$4], $MachinePrecision]}, If[LessEqual[t$95$9, 1.15e-13], N[(N[(N[(2.0 - N[(t$95$11 + t$95$10), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision], If[LessEqual[t$95$9, 1600000000000.0], N[(N[(N[(N[Sqrt[N[(1.0 + t$95$9), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$10), $MachinePrecision] + t$95$8), $MachinePrecision], N[(N[(1.0 / N[(t$95$11 + N[Sqrt[N[(1.0 + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.1499999999999999e-13

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. Taylor expanded in y around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{\color{blue}{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 37.8

         \[\leadsto \left(\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 37.8%

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\left(2 - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 25.2%

      \[\leadsto \left(\left(2 - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.1499999999999999e-13 < y < 1.6e12

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. lower-sqrt.f64 39.9

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 39.9%

       \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.6e12 < y

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 31.3

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 31.3%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_13 := \sqrt{t\_12 + 1} - \sqrt{t\_12}\\ t_14 := \sqrt{t\_11}\\ t_15 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{t\_11 + 1} - t\_14\right)\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + t\_13\\ \mathbf{if}\;t\_15 \leq 1:\\ \;\;\;\;\frac{1}{t\_6 + \sqrt{1 + t\_5}} + t\_13\\ \mathbf{elif}\;t\_15 \leq 1.9999999999999996:\\ \;\;\;\;\left(\left(\sqrt{1 + t\_11} + \frac{1}{1 + t\_6}\right) - t\_14\right) + t\_13\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t\_5 - -1} + \sqrt{t\_11 - -1}\right) + \left(\sqrt{t\_9 - -1} - \left(\left(t\_10 + t\_14\right) + t\_6\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (fmin (fmax x y) t_1))
        (t_4 (fmin (fmin x y) z))
        (t_5 (fmin t_4 t))
        (t_6 (sqrt t_5))
        (t_7 (fmax t_4 t))
        (t_8 (fmax t_3 t_7))
        (t_9 (fmin t_2 t_8))
        (t_10 (sqrt t_9))
        (t_11 (fmin t_3 t_7))
        (t_12 (fmax t_2 t_8))
        (t_13 (- (sqrt (+ t_12 1.0)) (sqrt t_12)))
        (t_14 (sqrt t_11))
        (t_15
         (+
          (+
           (+ (- (sqrt (+ t_5 1.0)) t_6) (- (sqrt (+ t_11 1.0)) t_14))
           (- (sqrt (+ t_9 1.0)) t_10))
          t_13)))
   (if (<= t_15 1.0)
     (+ (/ 1.0 (+ t_6 (sqrt (+ 1.0 t_5)))) t_13)
     (if (<= t_15 1.9999999999999996)
       (+ (- (+ (sqrt (+ 1.0 t_11)) (/ 1.0 (+ 1.0 t_6))) t_14) t_13)
       (+
        (+ (sqrt (- t_5 -1.0)) (sqrt (- t_11 -1.0)))
        (- (sqrt (- t_9 -1.0)) (+ (+ t_10 t_14) t_6)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	double t_14 = sqrt(t_11);
	double t_15 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_14)) + (sqrt((t_9 + 1.0)) - t_10)) + t_13;
	double tmp;
	if (t_15 <= 1.0) {
		tmp = (1.0 / (t_6 + sqrt((1.0 + t_5)))) + t_13;
	} else if (t_15 <= 1.9999999999999996) {
		tmp = ((sqrt((1.0 + t_11)) + (1.0 / (1.0 + t_6))) - t_14) + t_13;
	} else {
		tmp = (sqrt((t_5 - -1.0)) + sqrt((t_11 - -1.0))) + (sqrt((t_9 - -1.0)) - ((t_10 + t_14) + t_6));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = fmin(t_2, t_8)
    t_10 = sqrt(t_9)
    t_11 = fmin(t_3, t_7)
    t_12 = fmax(t_2, t_8)
    t_13 = sqrt((t_12 + 1.0d0)) - sqrt(t_12)
    t_14 = sqrt(t_11)
    t_15 = (((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_11 + 1.0d0)) - t_14)) + (sqrt((t_9 + 1.0d0)) - t_10)) + t_13
    if (t_15 <= 1.0d0) then
        tmp = (1.0d0 / (t_6 + sqrt((1.0d0 + t_5)))) + t_13
    else if (t_15 <= 1.9999999999999996d0) then
        tmp = ((sqrt((1.0d0 + t_11)) + (1.0d0 / (1.0d0 + t_6))) - t_14) + t_13
    else
        tmp = (sqrt((t_5 - (-1.0d0))) + sqrt((t_11 - (-1.0d0)))) + (sqrt((t_9 - (-1.0d0))) - ((t_10 + t_14) + t_6))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = Math.sqrt((t_12 + 1.0)) - Math.sqrt(t_12);
	double t_14 = Math.sqrt(t_11);
	double t_15 = (((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((t_11 + 1.0)) - t_14)) + (Math.sqrt((t_9 + 1.0)) - t_10)) + t_13;
	double tmp;
	if (t_15 <= 1.0) {
		tmp = (1.0 / (t_6 + Math.sqrt((1.0 + t_5)))) + t_13;
	} else if (t_15 <= 1.9999999999999996) {
		tmp = ((Math.sqrt((1.0 + t_11)) + (1.0 / (1.0 + t_6))) - t_14) + t_13;
	} else {
		tmp = (Math.sqrt((t_5 - -1.0)) + Math.sqrt((t_11 - -1.0))) + (Math.sqrt((t_9 - -1.0)) - ((t_10 + t_14) + t_6));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = math.sqrt((t_12 + 1.0)) - math.sqrt(t_12)
	t_14 = math.sqrt(t_11)
	t_15 = (((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((t_11 + 1.0)) - t_14)) + (math.sqrt((t_9 + 1.0)) - t_10)) + t_13
	tmp = 0
	if t_15 <= 1.0:
		tmp = (1.0 / (t_6 + math.sqrt((1.0 + t_5)))) + t_13
	elif t_15 <= 1.9999999999999996:
		tmp = ((math.sqrt((1.0 + t_11)) + (1.0 / (1.0 + t_6))) - t_14) + t_13
	else:
		tmp = (math.sqrt((t_5 - -1.0)) + math.sqrt((t_11 - -1.0))) + (math.sqrt((t_9 - -1.0)) - ((t_10 + t_14) + t_6))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = Float64(sqrt(Float64(t_12 + 1.0)) - sqrt(t_12))
	t_14 = sqrt(t_11)
	t_15 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(t_11 + 1.0)) - t_14)) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + t_13)
	tmp = 0.0
	if (t_15 <= 1.0)
		tmp = Float64(Float64(1.0 / Float64(t_6 + sqrt(Float64(1.0 + t_5)))) + t_13);
	elseif (t_15 <= 1.9999999999999996)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t_11)) + Float64(1.0 / Float64(1.0 + t_6))) - t_14) + t_13);
	else
		tmp = Float64(Float64(sqrt(Float64(t_5 - -1.0)) + sqrt(Float64(t_11 - -1.0))) + Float64(sqrt(Float64(t_9 - -1.0)) - Float64(Float64(t_10 + t_14) + t_6)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_3, t_7);
	t_9 = min(t_2, t_8);
	t_10 = sqrt(t_9);
	t_11 = min(t_3, t_7);
	t_12 = max(t_2, t_8);
	t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	t_14 = sqrt(t_11);
	t_15 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_14)) + (sqrt((t_9 + 1.0)) - t_10)) + t_13;
	tmp = 0.0;
	if (t_15 <= 1.0)
		tmp = (1.0 / (t_6 + sqrt((1.0 + t_5)))) + t_13;
	elseif (t_15 <= 1.9999999999999996)
		tmp = ((sqrt((1.0 + t_11)) + (1.0 / (1.0 + t_6))) - t_14) + t_13;
	else
		tmp = (sqrt((t_5 - -1.0)) + sqrt((t_11 - -1.0))) + (sqrt((t_9 - -1.0)) - ((t_10 + t_14) + t_6));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Max[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$13 = N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$12], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$15 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$14), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision]}, If[LessEqual[t$95$15, 1.0], N[(N[(1.0 / N[(t$95$6 + N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision], If[LessEqual[t$95$15, 1.9999999999999996], N[(N[(N[(N[Sqrt[N[(1.0 + t$95$11), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$14), $MachinePrecision] + t$95$13), $MachinePrecision], N[(N[(N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(t$95$11 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$10 + t$95$14), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 31.3

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 31.3%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.9999999999999996

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. lower-sqrt.f64 39.9

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 39.9%

       \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.9999999999999996 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{\color{blue}{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   4. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{z + 1}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. add-flip N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{z - \left(\mathsf{neg}\left(1\right)\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{z - -1}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{z - -1}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\left(\sqrt{z - -1} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\left(\sqrt{z - -1} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Applied rewrites 18.8%

      \[\leadsto \left(\sqrt{x - -1} + \sqrt{y - -1}\right) + \color{blue}{\left(\sqrt{z - -1} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_13 := \sqrt{t\_12 + 1} - \sqrt{t\_12}\\ t_14 := \sqrt{t\_11}\\ t_15 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{t\_11 + 1} - t\_14\right)\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + t\_13\\ \mathbf{if}\;t\_15 \leq 1:\\ \;\;\;\;\frac{1}{t\_6 + \sqrt{1 + t\_5}} + t\_13\\ \mathbf{elif}\;t\_15 \leq 2:\\ \;\;\;\;\left(\left(\sqrt{1 + t\_11} + \frac{1}{1 + t\_6}\right) - t\_14\right) + t\_13\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\sqrt{t\_5 - -1} + \sqrt{t\_9 - -1}\right) + \sqrt{t\_11 - -1}\right) - t\_6\right) - t\_14\right) - t\_10\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (fmin (fmax x y) t_1))
        (t_4 (fmin (fmin x y) z))
        (t_5 (fmin t_4 t))
        (t_6 (sqrt t_5))
        (t_7 (fmax t_4 t))
        (t_8 (fmax t_3 t_7))
        (t_9 (fmin t_2 t_8))
        (t_10 (sqrt t_9))
        (t_11 (fmin t_3 t_7))
        (t_12 (fmax t_2 t_8))
        (t_13 (- (sqrt (+ t_12 1.0)) (sqrt t_12)))
        (t_14 (sqrt t_11))
        (t_15
         (+
          (+
           (+ (- (sqrt (+ t_5 1.0)) t_6) (- (sqrt (+ t_11 1.0)) t_14))
           (- (sqrt (+ t_9 1.0)) t_10))
          t_13)))
   (if (<= t_15 1.0)
     (+ (/ 1.0 (+ t_6 (sqrt (+ 1.0 t_5)))) t_13)
     (if (<= t_15 2.0)
       (+ (- (+ (sqrt (+ 1.0 t_11)) (/ 1.0 (+ 1.0 t_6))) t_14) t_13)
       (-
        (-
         (-
          (+ (+ (sqrt (- t_5 -1.0)) (sqrt (- t_9 -1.0))) (sqrt (- t_11 -1.0)))
          t_6)
         t_14)
        t_10)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	double t_14 = sqrt(t_11);
	double t_15 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_14)) + (sqrt((t_9 + 1.0)) - t_10)) + t_13;
	double tmp;
	if (t_15 <= 1.0) {
		tmp = (1.0 / (t_6 + sqrt((1.0 + t_5)))) + t_13;
	} else if (t_15 <= 2.0) {
		tmp = ((sqrt((1.0 + t_11)) + (1.0 / (1.0 + t_6))) - t_14) + t_13;
	} else {
		tmp = ((((sqrt((t_5 - -1.0)) + sqrt((t_9 - -1.0))) + sqrt((t_11 - -1.0))) - t_6) - t_14) - t_10;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = fmin(t_2, t_8)
    t_10 = sqrt(t_9)
    t_11 = fmin(t_3, t_7)
    t_12 = fmax(t_2, t_8)
    t_13 = sqrt((t_12 + 1.0d0)) - sqrt(t_12)
    t_14 = sqrt(t_11)
    t_15 = (((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_11 + 1.0d0)) - t_14)) + (sqrt((t_9 + 1.0d0)) - t_10)) + t_13
    if (t_15 <= 1.0d0) then
        tmp = (1.0d0 / (t_6 + sqrt((1.0d0 + t_5)))) + t_13
    else if (t_15 <= 2.0d0) then
        tmp = ((sqrt((1.0d0 + t_11)) + (1.0d0 / (1.0d0 + t_6))) - t_14) + t_13
    else
        tmp = ((((sqrt((t_5 - (-1.0d0))) + sqrt((t_9 - (-1.0d0)))) + sqrt((t_11 - (-1.0d0)))) - t_6) - t_14) - t_10
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = fmax(t_2, t_8);
	double t_13 = Math.sqrt((t_12 + 1.0)) - Math.sqrt(t_12);
	double t_14 = Math.sqrt(t_11);
	double t_15 = (((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((t_11 + 1.0)) - t_14)) + (Math.sqrt((t_9 + 1.0)) - t_10)) + t_13;
	double tmp;
	if (t_15 <= 1.0) {
		tmp = (1.0 / (t_6 + Math.sqrt((1.0 + t_5)))) + t_13;
	} else if (t_15 <= 2.0) {
		tmp = ((Math.sqrt((1.0 + t_11)) + (1.0 / (1.0 + t_6))) - t_14) + t_13;
	} else {
		tmp = ((((Math.sqrt((t_5 - -1.0)) + Math.sqrt((t_9 - -1.0))) + Math.sqrt((t_11 - -1.0))) - t_6) - t_14) - t_10;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = math.sqrt((t_12 + 1.0)) - math.sqrt(t_12)
	t_14 = math.sqrt(t_11)
	t_15 = (((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((t_11 + 1.0)) - t_14)) + (math.sqrt((t_9 + 1.0)) - t_10)) + t_13
	tmp = 0
	if t_15 <= 1.0:
		tmp = (1.0 / (t_6 + math.sqrt((1.0 + t_5)))) + t_13
	elif t_15 <= 2.0:
		tmp = ((math.sqrt((1.0 + t_11)) + (1.0 / (1.0 + t_6))) - t_14) + t_13
	else:
		tmp = ((((math.sqrt((t_5 - -1.0)) + math.sqrt((t_9 - -1.0))) + math.sqrt((t_11 - -1.0))) - t_6) - t_14) - t_10
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = fmax(t_2, t_8)
	t_13 = Float64(sqrt(Float64(t_12 + 1.0)) - sqrt(t_12))
	t_14 = sqrt(t_11)
	t_15 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(t_11 + 1.0)) - t_14)) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + t_13)
	tmp = 0.0
	if (t_15 <= 1.0)
		tmp = Float64(Float64(1.0 / Float64(t_6 + sqrt(Float64(1.0 + t_5)))) + t_13);
	elseif (t_15 <= 2.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t_11)) + Float64(1.0 / Float64(1.0 + t_6))) - t_14) + t_13);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_5 - -1.0)) + sqrt(Float64(t_9 - -1.0))) + sqrt(Float64(t_11 - -1.0))) - t_6) - t_14) - t_10);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_3, t_7);
	t_9 = min(t_2, t_8);
	t_10 = sqrt(t_9);
	t_11 = min(t_3, t_7);
	t_12 = max(t_2, t_8);
	t_13 = sqrt((t_12 + 1.0)) - sqrt(t_12);
	t_14 = sqrt(t_11);
	t_15 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_14)) + (sqrt((t_9 + 1.0)) - t_10)) + t_13;
	tmp = 0.0;
	if (t_15 <= 1.0)
		tmp = (1.0 / (t_6 + sqrt((1.0 + t_5)))) + t_13;
	elseif (t_15 <= 2.0)
		tmp = ((sqrt((1.0 + t_11)) + (1.0 / (1.0 + t_6))) - t_14) + t_13;
	else
		tmp = ((((sqrt((t_5 - -1.0)) + sqrt((t_9 - -1.0))) + sqrt((t_11 - -1.0))) - t_6) - t_14) - t_10;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Max[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$13 = N[(N[Sqrt[N[(t$95$12 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$12], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$14 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$15 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$14), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision]}, If[LessEqual[t$95$15, 1.0], N[(N[(1.0 / N[(t$95$6 + N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$13), $MachinePrecision], If[LessEqual[t$95$15, 2.0], N[(N[(N[(N[Sqrt[N[(1.0 + t$95$11), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$14), $MachinePrecision] + t$95$13), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$5 - -1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(t$95$9 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(t$95$11 - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] - t$95$14), $MachinePrecision] - t$95$10), $MachinePrecision]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 31.3

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 31.3%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. lower-sqrt.f64 39.9

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 39.9%

       \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{\color{blue}{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   4. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      3. associate--r+ N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{x}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right) \]

      5. associate--r+ N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{x}\right) - \sqrt{y}\right) - \color{blue}{\sqrt{z}} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{x}\right) - \sqrt{y}\right) - \color{blue}{\sqrt{z}} \]

   6. Applied rewrites 11.5%

      \[\leadsto \left(\left(\left(\left(\sqrt{x - -1} + \sqrt{z - -1}\right) + \sqrt{y - -1}\right) - \sqrt{x}\right) - \sqrt{y}\right) - \color{blue}{\sqrt{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \sqrt{1 + t\_11}\\ t_13 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_14 := \sqrt{t\_13 + 1} - \sqrt{t\_13}\\ t_15 := \sqrt{t\_11}\\ t_16 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{t\_11 + 1} - t\_15\right)\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + t\_14\\ \mathbf{if}\;t\_16 \leq 1:\\ \;\;\;\;\frac{1}{t\_6 + \sqrt{1 + t\_5}} + t\_14\\ \mathbf{elif}\;t\_16 \leq 2:\\ \;\;\;\;\left(\left(t\_12 + \frac{1}{1 + t\_6}\right) - t\_15\right) + t\_14\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(t\_12 + \sqrt{1 + t\_9}\right)\right) - \left(t\_6 + \left(t\_15 + t\_10\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (fmin (fmax x y) t_1))
        (t_4 (fmin (fmin x y) z))
        (t_5 (fmin t_4 t))
        (t_6 (sqrt t_5))
        (t_7 (fmax t_4 t))
        (t_8 (fmax t_3 t_7))
        (t_9 (fmin t_2 t_8))
        (t_10 (sqrt t_9))
        (t_11 (fmin t_3 t_7))
        (t_12 (sqrt (+ 1.0 t_11)))
        (t_13 (fmax t_2 t_8))
        (t_14 (- (sqrt (+ t_13 1.0)) (sqrt t_13)))
        (t_15 (sqrt t_11))
        (t_16
         (+
          (+
           (+ (- (sqrt (+ t_5 1.0)) t_6) (- (sqrt (+ t_11 1.0)) t_15))
           (- (sqrt (+ t_9 1.0)) t_10))
          t_14)))
   (if (<= t_16 1.0)
     (+ (/ 1.0 (+ t_6 (sqrt (+ 1.0 t_5)))) t_14)
     (if (<= t_16 2.0)
       (+ (- (+ t_12 (/ 1.0 (+ 1.0 t_6))) t_15) t_14)
       (- (+ 1.0 (+ t_12 (sqrt (+ 1.0 t_9)))) (+ t_6 (+ t_15 t_10)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = sqrt((1.0 + t_11));
	double t_13 = fmax(t_2, t_8);
	double t_14 = sqrt((t_13 + 1.0)) - sqrt(t_13);
	double t_15 = sqrt(t_11);
	double t_16 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_15)) + (sqrt((t_9 + 1.0)) - t_10)) + t_14;
	double tmp;
	if (t_16 <= 1.0) {
		tmp = (1.0 / (t_6 + sqrt((1.0 + t_5)))) + t_14;
	} else if (t_16 <= 2.0) {
		tmp = ((t_12 + (1.0 / (1.0 + t_6))) - t_15) + t_14;
	} else {
		tmp = (1.0 + (t_12 + sqrt((1.0 + t_9)))) - (t_6 + (t_15 + t_10));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_16
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = fmin(t_2, t_8)
    t_10 = sqrt(t_9)
    t_11 = fmin(t_3, t_7)
    t_12 = sqrt((1.0d0 + t_11))
    t_13 = fmax(t_2, t_8)
    t_14 = sqrt((t_13 + 1.0d0)) - sqrt(t_13)
    t_15 = sqrt(t_11)
    t_16 = (((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_11 + 1.0d0)) - t_15)) + (sqrt((t_9 + 1.0d0)) - t_10)) + t_14
    if (t_16 <= 1.0d0) then
        tmp = (1.0d0 / (t_6 + sqrt((1.0d0 + t_5)))) + t_14
    else if (t_16 <= 2.0d0) then
        tmp = ((t_12 + (1.0d0 / (1.0d0 + t_6))) - t_15) + t_14
    else
        tmp = (1.0d0 + (t_12 + sqrt((1.0d0 + t_9)))) - (t_6 + (t_15 + t_10))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = Math.sqrt((1.0 + t_11));
	double t_13 = fmax(t_2, t_8);
	double t_14 = Math.sqrt((t_13 + 1.0)) - Math.sqrt(t_13);
	double t_15 = Math.sqrt(t_11);
	double t_16 = (((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((t_11 + 1.0)) - t_15)) + (Math.sqrt((t_9 + 1.0)) - t_10)) + t_14;
	double tmp;
	if (t_16 <= 1.0) {
		tmp = (1.0 / (t_6 + Math.sqrt((1.0 + t_5)))) + t_14;
	} else if (t_16 <= 2.0) {
		tmp = ((t_12 + (1.0 / (1.0 + t_6))) - t_15) + t_14;
	} else {
		tmp = (1.0 + (t_12 + Math.sqrt((1.0 + t_9)))) - (t_6 + (t_15 + t_10));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = math.sqrt((1.0 + t_11))
	t_13 = fmax(t_2, t_8)
	t_14 = math.sqrt((t_13 + 1.0)) - math.sqrt(t_13)
	t_15 = math.sqrt(t_11)
	t_16 = (((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((t_11 + 1.0)) - t_15)) + (math.sqrt((t_9 + 1.0)) - t_10)) + t_14
	tmp = 0
	if t_16 <= 1.0:
		tmp = (1.0 / (t_6 + math.sqrt((1.0 + t_5)))) + t_14
	elif t_16 <= 2.0:
		tmp = ((t_12 + (1.0 / (1.0 + t_6))) - t_15) + t_14
	else:
		tmp = (1.0 + (t_12 + math.sqrt((1.0 + t_9)))) - (t_6 + (t_15 + t_10))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = sqrt(Float64(1.0 + t_11))
	t_13 = fmax(t_2, t_8)
	t_14 = Float64(sqrt(Float64(t_13 + 1.0)) - sqrt(t_13))
	t_15 = sqrt(t_11)
	t_16 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(t_11 + 1.0)) - t_15)) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + t_14)
	tmp = 0.0
	if (t_16 <= 1.0)
		tmp = Float64(Float64(1.0 / Float64(t_6 + sqrt(Float64(1.0 + t_5)))) + t_14);
	elseif (t_16 <= 2.0)
		tmp = Float64(Float64(Float64(t_12 + Float64(1.0 / Float64(1.0 + t_6))) - t_15) + t_14);
	else
		tmp = Float64(Float64(1.0 + Float64(t_12 + sqrt(Float64(1.0 + t_9)))) - Float64(t_6 + Float64(t_15 + t_10)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_3, t_7);
	t_9 = min(t_2, t_8);
	t_10 = sqrt(t_9);
	t_11 = min(t_3, t_7);
	t_12 = sqrt((1.0 + t_11));
	t_13 = max(t_2, t_8);
	t_14 = sqrt((t_13 + 1.0)) - sqrt(t_13);
	t_15 = sqrt(t_11);
	t_16 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_15)) + (sqrt((t_9 + 1.0)) - t_10)) + t_14;
	tmp = 0.0;
	if (t_16 <= 1.0)
		tmp = (1.0 / (t_6 + sqrt((1.0 + t_5)))) + t_14;
	elseif (t_16 <= 2.0)
		tmp = ((t_12 + (1.0 / (1.0 + t_6))) - t_15) + t_14;
	else
		tmp = (1.0 + (t_12 + sqrt((1.0 + t_9)))) - (t_6 + (t_15 + t_10));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[N[(1.0 + t$95$11), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$13 = N[Max[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$14 = N[(N[Sqrt[N[(t$95$13 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$13], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$15 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$16 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$15), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$14), $MachinePrecision]}, If[LessEqual[t$95$16, 1.0], N[(N[(1.0 / N[(t$95$6 + N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$14), $MachinePrecision], If[LessEqual[t$95$16, 2.0], N[(N[(N[(t$95$12 + N[(1.0 / N[(1.0 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$15), $MachinePrecision] + t$95$14), $MachinePrecision], N[(N[(1.0 + N[(t$95$12 + N[Sqrt[N[(1.0 + t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$6 + N[(t$95$15 + t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 31.3

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 31.3%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       6. lower-sqrt.f64 39.9

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 39.9%

       \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{1 + \sqrt{x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{\color{blue}{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   4. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 10.8

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. Applied rewrites 10.8%

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_4 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_5 := \mathsf{min}\left(t\_4, t\right)\\ t_6 := \sqrt{t\_5}\\ t_7 := \mathsf{max}\left(t\_4, t\right)\\ t_8 := \mathsf{max}\left(t\_3, t\_7\right)\\ t_9 := \mathsf{min}\left(t\_2, t\_8\right)\\ t_10 := \sqrt{t\_9}\\ t_11 := \mathsf{min}\left(t\_3, t\_7\right)\\ t_12 := \sqrt{1 + t\_11}\\ t_13 := \mathsf{max}\left(t\_2, t\_8\right)\\ t_14 := \sqrt{t\_13 + 1} - \sqrt{t\_13}\\ t_15 := \sqrt{t\_11}\\ t_16 := \left(\left(\left(\sqrt{t\_5 + 1} - t\_6\right) + \left(\sqrt{t\_11 + 1} - t\_15\right)\right) + \left(\sqrt{t\_9 + 1} - t\_10\right)\right) + t\_14\\ t_17 := \sqrt{1 + t\_5}\\ \mathbf{if}\;t\_16 \leq 1:\\ \;\;\;\;\frac{1}{t\_6 + t\_17} + t\_14\\ \mathbf{elif}\;t\_16 \leq 2:\\ \;\;\;\;\left(t\_17 + t\_12\right) - \left(t\_6 + t\_15\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(t\_12 + \sqrt{1 + t\_9}\right)\right) - \left(t\_6 + \left(t\_15 + t\_10\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmax (fmin x y) z))
        (t_2 (fmax (fmax x y) t_1))
        (t_3 (fmin (fmax x y) t_1))
        (t_4 (fmin (fmin x y) z))
        (t_5 (fmin t_4 t))
        (t_6 (sqrt t_5))
        (t_7 (fmax t_4 t))
        (t_8 (fmax t_3 t_7))
        (t_9 (fmin t_2 t_8))
        (t_10 (sqrt t_9))
        (t_11 (fmin t_3 t_7))
        (t_12 (sqrt (+ 1.0 t_11)))
        (t_13 (fmax t_2 t_8))
        (t_14 (- (sqrt (+ t_13 1.0)) (sqrt t_13)))
        (t_15 (sqrt t_11))
        (t_16
         (+
          (+
           (+ (- (sqrt (+ t_5 1.0)) t_6) (- (sqrt (+ t_11 1.0)) t_15))
           (- (sqrt (+ t_9 1.0)) t_10))
          t_14))
        (t_17 (sqrt (+ 1.0 t_5))))
   (if (<= t_16 1.0)
     (+ (/ 1.0 (+ t_6 t_17)) t_14)
     (if (<= t_16 2.0)
       (- (+ t_17 t_12) (+ t_6 t_15))
       (- (+ 1.0 (+ t_12 (sqrt (+ 1.0 t_9)))) (+ t_6 (+ t_15 t_10)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = sqrt((1.0 + t_11));
	double t_13 = fmax(t_2, t_8);
	double t_14 = sqrt((t_13 + 1.0)) - sqrt(t_13);
	double t_15 = sqrt(t_11);
	double t_16 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_15)) + (sqrt((t_9 + 1.0)) - t_10)) + t_14;
	double t_17 = sqrt((1.0 + t_5));
	double tmp;
	if (t_16 <= 1.0) {
		tmp = (1.0 / (t_6 + t_17)) + t_14;
	} else if (t_16 <= 2.0) {
		tmp = (t_17 + t_12) - (t_6 + t_15);
	} else {
		tmp = (1.0 + (t_12 + sqrt((1.0 + t_9)))) - (t_6 + (t_15 + t_10));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_10
    real(8) :: t_11
    real(8) :: t_12
    real(8) :: t_13
    real(8) :: t_14
    real(8) :: t_15
    real(8) :: t_16
    real(8) :: t_17
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    t_4 = fmin(fmin(x, y), z)
    t_5 = fmin(t_4, t)
    t_6 = sqrt(t_5)
    t_7 = fmax(t_4, t)
    t_8 = fmax(t_3, t_7)
    t_9 = fmin(t_2, t_8)
    t_10 = sqrt(t_9)
    t_11 = fmin(t_3, t_7)
    t_12 = sqrt((1.0d0 + t_11))
    t_13 = fmax(t_2, t_8)
    t_14 = sqrt((t_13 + 1.0d0)) - sqrt(t_13)
    t_15 = sqrt(t_11)
    t_16 = (((sqrt((t_5 + 1.0d0)) - t_6) + (sqrt((t_11 + 1.0d0)) - t_15)) + (sqrt((t_9 + 1.0d0)) - t_10)) + t_14
    t_17 = sqrt((1.0d0 + t_5))
    if (t_16 <= 1.0d0) then
        tmp = (1.0d0 / (t_6 + t_17)) + t_14
    else if (t_16 <= 2.0d0) then
        tmp = (t_17 + t_12) - (t_6 + t_15)
    else
        tmp = (1.0d0 + (t_12 + sqrt((1.0d0 + t_9)))) - (t_6 + (t_15 + t_10))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double t_4 = fmin(fmin(x, y), z);
	double t_5 = fmin(t_4, t);
	double t_6 = Math.sqrt(t_5);
	double t_7 = fmax(t_4, t);
	double t_8 = fmax(t_3, t_7);
	double t_9 = fmin(t_2, t_8);
	double t_10 = Math.sqrt(t_9);
	double t_11 = fmin(t_3, t_7);
	double t_12 = Math.sqrt((1.0 + t_11));
	double t_13 = fmax(t_2, t_8);
	double t_14 = Math.sqrt((t_13 + 1.0)) - Math.sqrt(t_13);
	double t_15 = Math.sqrt(t_11);
	double t_16 = (((Math.sqrt((t_5 + 1.0)) - t_6) + (Math.sqrt((t_11 + 1.0)) - t_15)) + (Math.sqrt((t_9 + 1.0)) - t_10)) + t_14;
	double t_17 = Math.sqrt((1.0 + t_5));
	double tmp;
	if (t_16 <= 1.0) {
		tmp = (1.0 / (t_6 + t_17)) + t_14;
	} else if (t_16 <= 2.0) {
		tmp = (t_17 + t_12) - (t_6 + t_15);
	} else {
		tmp = (1.0 + (t_12 + Math.sqrt((1.0 + t_9)))) - (t_6 + (t_15 + t_10));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = math.sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = math.sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = math.sqrt((1.0 + t_11))
	t_13 = fmax(t_2, t_8)
	t_14 = math.sqrt((t_13 + 1.0)) - math.sqrt(t_13)
	t_15 = math.sqrt(t_11)
	t_16 = (((math.sqrt((t_5 + 1.0)) - t_6) + (math.sqrt((t_11 + 1.0)) - t_15)) + (math.sqrt((t_9 + 1.0)) - t_10)) + t_14
	t_17 = math.sqrt((1.0 + t_5))
	tmp = 0
	if t_16 <= 1.0:
		tmp = (1.0 / (t_6 + t_17)) + t_14
	elif t_16 <= 2.0:
		tmp = (t_17 + t_12) - (t_6 + t_15)
	else:
		tmp = (1.0 + (t_12 + math.sqrt((1.0 + t_9)))) - (t_6 + (t_15 + t_10))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	t_4 = fmin(fmin(x, y), z)
	t_5 = fmin(t_4, t)
	t_6 = sqrt(t_5)
	t_7 = fmax(t_4, t)
	t_8 = fmax(t_3, t_7)
	t_9 = fmin(t_2, t_8)
	t_10 = sqrt(t_9)
	t_11 = fmin(t_3, t_7)
	t_12 = sqrt(Float64(1.0 + t_11))
	t_13 = fmax(t_2, t_8)
	t_14 = Float64(sqrt(Float64(t_13 + 1.0)) - sqrt(t_13))
	t_15 = sqrt(t_11)
	t_16 = Float64(Float64(Float64(Float64(sqrt(Float64(t_5 + 1.0)) - t_6) + Float64(sqrt(Float64(t_11 + 1.0)) - t_15)) + Float64(sqrt(Float64(t_9 + 1.0)) - t_10)) + t_14)
	t_17 = sqrt(Float64(1.0 + t_5))
	tmp = 0.0
	if (t_16 <= 1.0)
		tmp = Float64(Float64(1.0 / Float64(t_6 + t_17)) + t_14);
	elseif (t_16 <= 2.0)
		tmp = Float64(Float64(t_17 + t_12) - Float64(t_6 + t_15));
	else
		tmp = Float64(Float64(1.0 + Float64(t_12 + sqrt(Float64(1.0 + t_9)))) - Float64(t_6 + Float64(t_15 + t_10)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	t_4 = min(min(x, y), z);
	t_5 = min(t_4, t);
	t_6 = sqrt(t_5);
	t_7 = max(t_4, t);
	t_8 = max(t_3, t_7);
	t_9 = min(t_2, t_8);
	t_10 = sqrt(t_9);
	t_11 = min(t_3, t_7);
	t_12 = sqrt((1.0 + t_11));
	t_13 = max(t_2, t_8);
	t_14 = sqrt((t_13 + 1.0)) - sqrt(t_13);
	t_15 = sqrt(t_11);
	t_16 = (((sqrt((t_5 + 1.0)) - t_6) + (sqrt((t_11 + 1.0)) - t_15)) + (sqrt((t_9 + 1.0)) - t_10)) + t_14;
	t_17 = sqrt((1.0 + t_5));
	tmp = 0.0;
	if (t_16 <= 1.0)
		tmp = (1.0 / (t_6 + t_17)) + t_14;
	elseif (t_16 <= 2.0)
		tmp = (t_17 + t_12) - (t_6 + t_15);
	else
		tmp = (1.0 + (t_12 + sqrt((1.0 + t_9)))) - (t_6 + (t_15 + t_10));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$4, t], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Max[t$95$4, t], $MachinePrecision]}, Block[{t$95$8 = N[Max[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[Min[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sqrt[t$95$9], $MachinePrecision]}, Block[{t$95$11 = N[Min[t$95$3, t$95$7], $MachinePrecision]}, Block[{t$95$12 = N[Sqrt[N[(1.0 + t$95$11), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$13 = N[Max[t$95$2, t$95$8], $MachinePrecision]}, Block[{t$95$14 = N[(N[Sqrt[N[(t$95$13 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$13], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$15 = N[Sqrt[t$95$11], $MachinePrecision]}, Block[{t$95$16 = N[(N[(N[(N[(N[Sqrt[N[(t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$6), $MachinePrecision] + N[(N[Sqrt[N[(t$95$11 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$15), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$9 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$10), $MachinePrecision]), $MachinePrecision] + t$95$14), $MachinePrecision]}, Block[{t$95$17 = N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$16, 1.0], N[(N[(1.0 / N[(t$95$6 + t$95$17), $MachinePrecision]), $MachinePrecision] + t$95$14), $MachinePrecision], If[LessEqual[t$95$16, 2.0], N[(N[(t$95$17 + t$95$12), $MachinePrecision] - N[(t$95$6 + t$95$15), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$12 + N[Sqrt[N[(1.0 + t$95$9), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$6 + N[(t$95$15 + t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 31.3

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 31.3%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{\color{blue}{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   4. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{\color{blue}{y}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      9. lower-sqrt.f64 13.8

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

   7. Applied rewrites 13.8%

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{\color{blue}{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   4. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 10.8

         \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. Applied rewrites 10.8%

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(t\_1, t\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\right)\\ t_4 := \mathsf{max}\left(t\_3, t\_2\right)\\ t_5 := \mathsf{min}\left(t\_3, t\_2\right)\\ t_6 := \mathsf{min}\left(t\_1, t\right)\\ t_7 := \sqrt{1 + t\_6}\\ t_8 := \sqrt{t\_6}\\ \mathbf{if}\;t\_5 \leq 1600000000000:\\ \;\;\;\;\left(t\_7 + \sqrt{1 + t\_5}\right) - \left(t\_8 + \sqrt{t\_5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_8 + t\_7} + \left(\sqrt{t\_4 + 1} - \sqrt{t\_4}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmin (fmin x y) z))
        (t_2 (fmax t_1 t))
        (t_3 (fmin (fmax x y) (fmax (fmin x y) z)))
        (t_4 (fmax t_3 t_2))
        (t_5 (fmin t_3 t_2))
        (t_6 (fmin t_1 t))
        (t_7 (sqrt (+ 1.0 t_6)))
        (t_8 (sqrt t_6)))
   (if (<= t_5 1600000000000.0)
     (- (+ t_7 (sqrt (+ 1.0 t_5))) (+ t_8 (sqrt t_5)))
     (+ (/ 1.0 (+ t_8 t_7)) (- (sqrt (+ t_4 1.0)) (sqrt t_4))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmin(fmax(x, y), fmax(fmin(x, y), z));
	double t_4 = fmax(t_3, t_2);
	double t_5 = fmin(t_3, t_2);
	double t_6 = fmin(t_1, t);
	double t_7 = sqrt((1.0 + t_6));
	double t_8 = sqrt(t_6);
	double tmp;
	if (t_5 <= 1600000000000.0) {
		tmp = (t_7 + sqrt((1.0 + t_5))) - (t_8 + sqrt(t_5));
	} else {
		tmp = (1.0 / (t_8 + t_7)) + (sqrt((t_4 + 1.0)) - sqrt(t_4));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: tmp
    t_1 = fmin(fmin(x, y), z)
    t_2 = fmax(t_1, t)
    t_3 = fmin(fmax(x, y), fmax(fmin(x, y), z))
    t_4 = fmax(t_3, t_2)
    t_5 = fmin(t_3, t_2)
    t_6 = fmin(t_1, t)
    t_7 = sqrt((1.0d0 + t_6))
    t_8 = sqrt(t_6)
    if (t_5 <= 1600000000000.0d0) then
        tmp = (t_7 + sqrt((1.0d0 + t_5))) - (t_8 + sqrt(t_5))
    else
        tmp = (1.0d0 / (t_8 + t_7)) + (sqrt((t_4 + 1.0d0)) - sqrt(t_4))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(x, y), z);
	double t_2 = fmax(t_1, t);
	double t_3 = fmin(fmax(x, y), fmax(fmin(x, y), z));
	double t_4 = fmax(t_3, t_2);
	double t_5 = fmin(t_3, t_2);
	double t_6 = fmin(t_1, t);
	double t_7 = Math.sqrt((1.0 + t_6));
	double t_8 = Math.sqrt(t_6);
	double tmp;
	if (t_5 <= 1600000000000.0) {
		tmp = (t_7 + Math.sqrt((1.0 + t_5))) - (t_8 + Math.sqrt(t_5));
	} else {
		tmp = (1.0 / (t_8 + t_7)) + (Math.sqrt((t_4 + 1.0)) - Math.sqrt(t_4));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmin(fmax(x, y), fmax(fmin(x, y), z))
	t_4 = fmax(t_3, t_2)
	t_5 = fmin(t_3, t_2)
	t_6 = fmin(t_1, t)
	t_7 = math.sqrt((1.0 + t_6))
	t_8 = math.sqrt(t_6)
	tmp = 0
	if t_5 <= 1600000000000.0:
		tmp = (t_7 + math.sqrt((1.0 + t_5))) - (t_8 + math.sqrt(t_5))
	else:
		tmp = (1.0 / (t_8 + t_7)) + (math.sqrt((t_4 + 1.0)) - math.sqrt(t_4))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(x, y), z)
	t_2 = fmax(t_1, t)
	t_3 = fmin(fmax(x, y), fmax(fmin(x, y), z))
	t_4 = fmax(t_3, t_2)
	t_5 = fmin(t_3, t_2)
	t_6 = fmin(t_1, t)
	t_7 = sqrt(Float64(1.0 + t_6))
	t_8 = sqrt(t_6)
	tmp = 0.0
	if (t_5 <= 1600000000000.0)
		tmp = Float64(Float64(t_7 + sqrt(Float64(1.0 + t_5))) - Float64(t_8 + sqrt(t_5)));
	else
		tmp = Float64(Float64(1.0 / Float64(t_8 + t_7)) + Float64(sqrt(Float64(t_4 + 1.0)) - sqrt(t_4)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = min(min(x, y), z);
	t_2 = max(t_1, t);
	t_3 = min(max(x, y), max(min(x, y), z));
	t_4 = max(t_3, t_2);
	t_5 = min(t_3, t_2);
	t_6 = min(t_1, t);
	t_7 = sqrt((1.0 + t_6));
	t_8 = sqrt(t_6);
	tmp = 0.0;
	if (t_5 <= 1600000000000.0)
		tmp = (t_7 + sqrt((1.0 + t_5))) - (t_8 + sqrt(t_5));
	else
		tmp = (1.0 / (t_8 + t_7)) + (sqrt((t_4 + 1.0)) - sqrt(t_4));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[t$95$1, t], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Max[t$95$3, t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[Min[t$95$3, t$95$2], $MachinePrecision]}, Block[{t$95$6 = N[Min[t$95$1, t], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 + t$95$6), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[t$95$6], $MachinePrecision]}, If[LessEqual[t$95$5, 1600000000000.0], N[(N[(t$95$7 + N[Sqrt[N[(1.0 + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$8 + N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$8 + t$95$7), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6e12

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{\color{blue}{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   4. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{\color{blue}{y}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      9. lower-sqrt.f64 13.8

         \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

   7. Applied rewrites 13.8%

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

   if 1.6e12 < y

   1. Initial program 92.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. add-flip N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-unsound-+.f64 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. add-flip N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. metadata-eval 73.7

          \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. Applied rewrites 73.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. rem-square-sqrt 92.4

         \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 92.4%

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 40.6

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lower-+.f64 31.3

          \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 31.3%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 47.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{min}\left(y, \mathsf{max}\left(x, z\right)\right), \mathsf{max}\left(\mathsf{min}\left(x, z\right), t\right)\right)\\ t_2 := \mathsf{min}\left(\mathsf{min}\left(x, z\right), t\right)\\ \left(\sqrt{1 + t\_2} + \sqrt{1 + t\_1}\right) - \left(\sqrt{t\_2} + \sqrt{t\_1}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmin (fmin y (fmax x z)) (fmax (fmin x z) t)))
        (t_2 (fmin (fmin x z) t)))
   (- (+ (sqrt (+ 1.0 t_2)) (sqrt (+ 1.0 t_1))) (+ (sqrt t_2) (sqrt t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(y, fmax(x, z)), fmax(fmin(x, z), t));
	double t_2 = fmin(fmin(x, z), t);
	return (sqrt((1.0 + t_2)) + sqrt((1.0 + t_1))) - (sqrt(t_2) + sqrt(t_1));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = fmin(fmin(y, fmax(x, z)), fmax(fmin(x, z), t))
    t_2 = fmin(fmin(x, z), t)
    code = (sqrt((1.0d0 + t_2)) + sqrt((1.0d0 + t_1))) - (sqrt(t_2) + sqrt(t_1))
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmin(y, fmax(x, z)), fmax(fmin(x, z), t));
	double t_2 = fmin(fmin(x, z), t);
	return (Math.sqrt((1.0 + t_2)) + Math.sqrt((1.0 + t_1))) - (Math.sqrt(t_2) + Math.sqrt(t_1));
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmin(y, fmax(x, z)), fmax(fmin(x, z), t))
	t_2 = fmin(fmin(x, z), t)
	return (math.sqrt((1.0 + t_2)) + math.sqrt((1.0 + t_1))) - (math.sqrt(t_2) + math.sqrt(t_1))

Julia

function code(x, y, z, t)
	t_1 = fmin(fmin(y, fmax(x, z)), fmax(fmin(x, z), t))
	t_2 = fmin(fmin(x, z), t)
	return Float64(Float64(sqrt(Float64(1.0 + t_2)) + sqrt(Float64(1.0 + t_1))) - Float64(sqrt(t_2) + sqrt(t_1)))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = min(min(y, max(x, z)), max(min(x, z), t));
	t_2 = min(min(x, z), t);
	tmp = (sqrt((1.0 + t_2)) + sqrt((1.0 + t_1))) - (sqrt(t_2) + sqrt(t_1));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Min[y, N[Max[x, z], $MachinePrecision]], $MachinePrecision], N[Max[N[Min[x, z], $MachinePrecision], t], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Min[x, z], $MachinePrecision], t], $MachinePrecision]}, N[(N[(N[Sqrt[N[(1.0 + t$95$2), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t$95$2], $MachinePrecision] + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 92.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

2. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{\color{blue}{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

4. Applied rewrites 12.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

5. Taylor expanded in z around inf

   \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{\color{blue}{y}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

   9. lower-sqrt.f64 13.8

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

7. Applied rewrites 13.8%

   \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
8. Add Preprocessing

Alternative 16: 40.6% accurate, 1.6× speedup

Math

\[\left(\sqrt{1 + \mathsf{min}\left(y, z\right)} - \sqrt{\mathsf{min}\left(y, z\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (- (sqrt (+ 1.0 (fmin y z))) (sqrt (fmin y z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (sqrt((1.0 + fmin(y, z))) - sqrt(fmin(y, z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (sqrt((1.0d0 + fmin(y, z))) - sqrt(fmin(y, z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (Math.sqrt((1.0 + fmin(y, z))) - Math.sqrt(fmin(y, z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (math.sqrt((1.0 + fmin(y, z))) - math.sqrt(fmin(y, z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(sqrt(Float64(1.0 + fmin(y, z))) - sqrt(fmin(y, z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (sqrt((1.0 + min(y, z))) - sqrt(min(y, z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(1.0 + N[Min[y, z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[Min[y, z], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. flip-- N/A

      \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lower-unsound--.f64 N/A

      \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. lower-unsound-*.f32 N/A

      \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. rem-square-sqrt N/A

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. add-flip N/A

       \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. lower--.f64 N/A

       \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   14. lower-unsound-*.f64 N/A

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   15. lower-unsound-+.f64 73.7

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   16. lift-+.f64 N/A

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   17. add-flip N/A

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   18. lower--.f64 N/A

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   19. metadata-eval 73.7

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

3. Applied rewrites 73.7%

   \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. rem-square-sqrt 92.4

      \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Applied rewrites 92.4%

   \[\leadsto \left(\left(\frac{\color{blue}{\left(x - -1\right) - x}}{\sqrt{x - -1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. lower-sqrt.f64 40.6

       \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

8. Applied rewrites 40.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

9. Taylor expanded in x around inf

   \[\leadsto \left(\sqrt{1 + y} - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
10. Step-by-step derivation

    1. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

    2. lower-+.f64 29.1

       \[\leadsto \left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

11. Applied rewrites 29.1%

    \[\leadsto \left(\sqrt{1 + y} - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
12. Add Preprocessing

Alternative 17: 7.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\mathsf{max}\left(y, z\right), \mathsf{max}\left(\mathsf{min}\left(y, z\right), \mathsf{max}\left(x, t\right)\right)\right)\\ 0.5 \cdot \left(t\_1 \cdot \sqrt{\frac{1}{t\_1}}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fmin (fmax y z) (fmax (fmin y z) (fmax x t)))))
   (* 0.5 (* t_1 (sqrt (/ 1.0 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmax(y, z), fmax(fmin(y, z), fmax(x, t)));
	return 0.5 * (t_1 * sqrt((1.0 / t_1)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = fmin(fmax(y, z), fmax(fmin(y, z), fmax(x, t)))
    code = 0.5d0 * (t_1 * sqrt((1.0d0 / t_1)))
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = fmin(fmax(y, z), fmax(fmin(y, z), fmax(x, t)));
	return 0.5 * (t_1 * Math.sqrt((1.0 / t_1)));
}

Python

def code(x, y, z, t):
	t_1 = fmin(fmax(y, z), fmax(fmin(y, z), fmax(x, t)))
	return 0.5 * (t_1 * math.sqrt((1.0 / t_1)))

Julia

function code(x, y, z, t)
	t_1 = fmin(fmax(y, z), fmax(fmin(y, z), fmax(x, t)))
	return Float64(0.5 * Float64(t_1 * sqrt(Float64(1.0 / t_1))))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = min(max(y, z), max(min(y, z), max(x, t)));
	tmp = 0.5 * (t_1 * sqrt((1.0 / t_1)));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Min[N[Max[y, z], $MachinePrecision], N[Max[N[Min[y, z], $MachinePrecision], N[Max[x, t], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(0.5 * N[(t$95$1 * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

2. Taylor expanded in y around inf

   \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{\color{blue}{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lower-sqrt.f64 50.8

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

4. Applied rewrites 50.8%

   \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
5. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. flip-- N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lower-unsound--.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. lower-unsound-*.f32 N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{z + 1} \cdot \sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{z + 1} \cdot \sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. rem-square-sqrt N/A

      \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. add-flip N/A

       \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\color{blue}{\left(z - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\left(z - \color{blue}{-1}\right) - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. lower--.f64 N/A

       \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\color{blue}{\left(z - -1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   14. lower-unsound-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\left(z - -1\right) - \color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   15. lower-unsound-+.f64 41.2

       \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\left(z - -1\right) - \sqrt{z} \cdot \sqrt{z}}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   16. lift-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\left(z - -1\right) - \sqrt{z} \cdot \sqrt{z}}{\sqrt{\color{blue}{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   17. add-flip N/A

       \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\left(z - -1\right) - \sqrt{z} \cdot \sqrt{z}}{\sqrt{\color{blue}{z - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   18. metadata-eval N/A

       \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\left(z - -1\right) - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z - \color{blue}{-1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   19. lower--.f64 41.2

       \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{\left(z - -1\right) - \sqrt{z} \cdot \sqrt{z}}{\sqrt{\color{blue}{z - -1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Applied rewrites 41.2%

   \[\leadsto \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \color{blue}{\frac{\left(z - -1\right) - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z - -1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

7. Taylor expanded in z around -inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(z \cdot \sqrt{\frac{1}{z}}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot \sqrt{\frac{1}{z}}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(z \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(z \cdot \sqrt{\frac{1}{z}}\right) \]

   4. lower-/.f64 6.9

      \[\leadsto 0.5 \cdot \left(z \cdot \sqrt{\frac{1}{z}}\right) \]

9. Applied rewrites 6.9%

   \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot \sqrt{\frac{1}{z}}\right)} \]
10. Add Preprocessing

Alternative 18: 6.9% accurate, 5.1× speedup

Math

\[\sqrt{\mathsf{min}\left(x, t\right)} \cdot 0.5 \]

FPCore

(FPCore (x y z t) :precision binary64 (* (sqrt (fmin x t)) 0.5))

C

double code(double x, double y, double z, double t) {
	return sqrt(fmin(x, t)) * 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt(fmin(x, t)) * 0.5d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.sqrt(fmin(x, t)) * 0.5;
}

Python

def code(x, y, z, t):
	return math.sqrt(fmin(x, t)) * 0.5

Julia

function code(x, y, z, t)
	return Float64(sqrt(fmin(x, t)) * 0.5)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = sqrt(min(x, t)) * 0.5;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[Min[x, t], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 92.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. flip-- N/A

      \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lower-unsound--.f64 N/A

      \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. lower-unsound-*.f32 N/A

      \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. rem-square-sqrt N/A

      \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. add-flip N/A

       \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. lower--.f64 N/A

       \[\leadsto \left(\left(\frac{\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\frac{\left(x - \color{blue}{-1}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   14. lower-unsound-*.f64 N/A

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   15. lower-unsound-+.f64 73.7

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   16. lift-+.f64 N/A

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   17. add-flip N/A

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   18. lower--.f64 N/A

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   19. metadata-eval 73.7

       \[\leadsto \left(\left(\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - \color{blue}{-1}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

3. Applied rewrites 73.7%

   \[\leadsto \left(\left(\color{blue}{\frac{\left(x - -1\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x - -1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

4. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot \sqrt{\frac{1}{x}}\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \sqrt{\frac{1}{x}}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \sqrt{\frac{1}{x}}\right) \]

   4. lower-/.f64 6.9

      \[\leadsto 0.5 \cdot \left(x \cdot \sqrt{\frac{1}{x}}\right) \]

6. Applied rewrites 6.9%

   \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \sqrt{\frac{1}{x}}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \sqrt{\frac{1}{x}}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{\frac{1}{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{1}{2} \]

   4. rem-square-sqrt N/A

      \[\leadsto \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{1}{2} \]

   5. sqrt-unprod N/A

      \[\leadsto \left(\sqrt{x \cdot x} \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{1}{2} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{x \cdot x} \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{1}{2} \]

   7. sqrt-unprod N/A

      \[\leadsto \sqrt{\left(x \cdot x\right) \cdot \frac{1}{x}} \cdot \frac{1}{2} \]

   8. pow2 N/A

      \[\leadsto \sqrt{{x}^{2} \cdot \frac{1}{x}} \cdot \frac{1}{2} \]

   9. lift-/.f64 N/A

      \[\leadsto \sqrt{{x}^{2} \cdot \frac{1}{x}} \cdot \frac{1}{2} \]

   10. inv-pow N/A

       \[\leadsto \sqrt{{x}^{2} \cdot {x}^{-1}} \cdot \frac{1}{2} \]

   11. pow-prod-up N/A

       \[\leadsto \sqrt{{x}^{\left(2 + -1\right)}} \cdot \frac{1}{2} \]

   12. metadata-eval N/A

       \[\leadsto \sqrt{{x}^{1}} \cdot \frac{1}{2} \]

   13. unpow1 N/A

       \[\leadsto \sqrt{x} \cdot \frac{1}{2} \]

   14. lift-sqrt.f64 N/A

       \[\leadsto \sqrt{x} \cdot \frac{1}{2} \]

   15. lower-*.f64 6.9

       \[\leadsto \sqrt{x} \cdot \color{blue}{0.5} \]

8. Applied rewrites 6.9%

   \[\leadsto \sqrt{x} \cdot \color{blue}{0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025170 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64
  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))