Main:z from

Percentage Accurate: 91.6% → 98.2%

Time: 10.2s

Alternatives: 17

Speedup: 0.9×

Specification

Math

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

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: 91.6% accurate, 1.0× speedup

Math

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

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: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_3\\ t_5 := \sqrt{1 + x}\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \frac{\sqrt{x}}{x} + t\_1\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 1.0005:\\ \;\;\;\;\left(t\_5 + \mathsf{fma}\left(0.5, \frac{1}{y \cdot \frac{1}{\sqrt{y}}}, t\_2\right)\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.3:\\ \;\;\;\;\left(t\_5 + \left(\sqrt{1 + y} + t\_2\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\sqrt{y - -1} + 1\right) - \sqrt{x}\right) - \sqrt{y}\right) + t\_1\right) + t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_1)
          t_3))
        (t_5 (sqrt (+ 1.0 x))))
   (if (<= t_4 5e-5)
     (+ (+ (* 0.5 (/ (sqrt x) x)) t_1) t_3)
     (if (<= t_4 1.0005)
       (- (+ t_5 (fma 0.5 (/ 1.0 (* y (/ 1.0 (sqrt y)))) t_2)) (sqrt x))
       (if (<= t_4 2.3)
         (- (+ t_5 (+ (sqrt (+ 1.0 y)) t_2)) (+ (sqrt x) (sqrt y)))
         (+
          (+ (- (- (+ (sqrt (- y -1.0)) 1.0) (sqrt x)) (sqrt y)) t_1)
          t_3))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3;
	double t_5 = sqrt((1.0 + x));
	double tmp;
	if (t_4 <= 5e-5) {
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_3;
	} else if (t_4 <= 1.0005) {
		tmp = (t_5 + fma(0.5, (1.0 / (y * (1.0 / sqrt(y)))), t_2)) - sqrt(x);
	} else if (t_4 <= 2.3) {
		tmp = (t_5 + (sqrt((1.0 + y)) + t_2)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = ((((sqrt((y - -1.0)) + 1.0) - sqrt(x)) - sqrt(y)) + t_1) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_3)
	t_5 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_4 <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(x) / x)) + t_1) + t_3);
	elseif (t_4 <= 1.0005)
		tmp = Float64(Float64(t_5 + fma(0.5, Float64(1.0 / Float64(y * Float64(1.0 / sqrt(y)))), t_2)) - sqrt(x));
	elseif (t_4 <= 2.3)
		tmp = Float64(Float64(t_5 + Float64(sqrt(Float64(1.0 + y)) + t_2)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(y - -1.0)) + 1.0) - sqrt(x)) - sqrt(y)) + t_1) + t_3);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 5e-5], N[(N[(N[(0.5 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 1.0005], N[(N[(t$95$5 + N[(0.5 * N[(1.0 / N[(y * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.3], N[(N[(t$95$5 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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))) < 5.00000000000000024e-5

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 5.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 10.6

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

   7. Applied rewrites 10.6%

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

   if 5.00000000000000024e-5 < (+.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))) < 1.00049999999999994

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 29.7%

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

   if 1.00049999999999994 < (+.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))) < 2.2999999999999998

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   if 2.2999999999999998 < (+.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 91.6%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 2: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;\left(t\_2 + t\_1\right) + t\_3 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(0.5 \cdot \frac{1}{y \cdot \frac{1}{\sqrt{y}}} - -0.5 \cdot \frac{\sqrt{x}}{x}\right) + t\_1\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \frac{\left(z - -1\right) - z}{\sqrt{z - -1} + \sqrt{z}}\right) + t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= (+ (+ t_2 t_1) t_3) 5e-5)
     (+
      (+
       (- (* 0.5 (/ 1.0 (* y (/ 1.0 (sqrt y))))) (* -0.5 (/ (sqrt x) x)))
       t_1)
      t_3)
     (+ (+ t_2 (/ (- (- z -1.0) z) (+ (sqrt (- z -1.0)) (sqrt z)))) t_3))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (((t_2 + t_1) + t_3) <= 5e-5) {
		tmp = (((0.5 * (1.0 / (y * (1.0 / sqrt(y))))) - (-0.5 * (sqrt(x) / x))) + t_1) + t_3;
	} else {
		tmp = (t_2 + (((z - -1.0) - z) / (sqrt((z - -1.0)) + sqrt(z)))) + t_3;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = (sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    if (((t_2 + t_1) + t_3) <= 5d-5) then
        tmp = (((0.5d0 * (1.0d0 / (y * (1.0d0 / sqrt(y))))) - ((-0.5d0) * (sqrt(x) / x))) + t_1) + t_3
    else
        tmp = (t_2 + (((z - (-1.0d0)) - z) / (sqrt((z - (-1.0d0))) + sqrt(z)))) + t_3
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if (((t_2 + t_1) + t_3) <= 5e-5) {
		tmp = (((0.5 * (1.0 / (y * (1.0 / Math.sqrt(y))))) - (-0.5 * (Math.sqrt(x) / x))) + t_1) + t_3;
	} else {
		tmp = (t_2 + (((z - -1.0) - z) / (Math.sqrt((z - -1.0)) + Math.sqrt(z)))) + t_3;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = (math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if ((t_2 + t_1) + t_3) <= 5e-5:
		tmp = (((0.5 * (1.0 / (y * (1.0 / math.sqrt(y))))) - (-0.5 * (math.sqrt(x) / x))) + t_1) + t_3
	else:
		tmp = (t_2 + (((z - -1.0) - z) / (math.sqrt((z - -1.0)) + math.sqrt(z)))) + t_3
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (Float64(Float64(t_2 + t_1) + t_3) <= 5e-5)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(1.0 / Float64(y * Float64(1.0 / sqrt(y))))) - Float64(-0.5 * Float64(sqrt(x) / x))) + t_1) + t_3);
	else
		tmp = Float64(Float64(t_2 + Float64(Float64(Float64(z - -1.0) - z) / Float64(sqrt(Float64(z - -1.0)) + sqrt(z)))) + t_3);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if (((t_2 + t_1) + t_3) <= 5e-5)
		tmp = (((0.5 * (1.0 / (y * (1.0 / sqrt(y))))) - (-0.5 * (sqrt(x) / x))) + t_1) + t_3;
	else
		tmp = (t_2 + (((z - -1.0) - z) / (sqrt((z - -1.0)) + sqrt(z)))) + t_3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], 5e-5], N[(N[(N[(N[(0.5 * N[(1.0 / N[(y * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(t$95$2 + N[(N[(N[(z - -1.0), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $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))) < 5.00000000000000024e-5

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 5.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 11.4

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

   7. Applied rewrites 11.4%

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

   if 5.00000000000000024e-5 < (+.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 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

Alternative 3: 97.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_3\\ t_5 := \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \frac{\sqrt{x}}{x} + t\_1\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\left(t\_2 + t\_5\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.3:\\ \;\;\;\;\left(t\_2 + \left(\sqrt{1 + y} + t\_5\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\sqrt{y - -1} + 1\right) - \sqrt{x}\right) - \sqrt{y}\right) + t\_1\right) + t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_1)
          t_3))
        (t_5 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z))))))
   (if (<= t_4 5e-5)
     (+ (+ (* 0.5 (/ (sqrt x) x)) t_1) t_3)
     (if (<= t_4 1.0)
       (- (+ t_2 t_5) (sqrt x))
       (if (<= t_4 2.3)
         (- (+ t_2 (+ (sqrt (+ 1.0 y)) t_5)) (+ (sqrt x) (sqrt y)))
         (+
          (+ (- (- (+ (sqrt (- y -1.0)) 1.0) (sqrt x)) (sqrt y)) t_1)
          t_3))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3;
	double t_5 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	double tmp;
	if (t_4 <= 5e-5) {
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_3;
	} else if (t_4 <= 1.0) {
		tmp = (t_2 + t_5) - sqrt(x);
	} else if (t_4 <= 2.3) {
		tmp = (t_2 + (sqrt((1.0 + y)) + t_5)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = ((((sqrt((y - -1.0)) + 1.0) - sqrt(x)) - sqrt(y)) + t_1) + t_3;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((1.0d0 + x))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_3
    t_5 = 1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))
    if (t_4 <= 5d-5) then
        tmp = ((0.5d0 * (sqrt(x) / x)) + t_1) + t_3
    else if (t_4 <= 1.0d0) then
        tmp = (t_2 + t_5) - sqrt(x)
    else if (t_4 <= 2.3d0) then
        tmp = (t_2 + (sqrt((1.0d0 + y)) + t_5)) - (sqrt(x) + sqrt(y))
    else
        tmp = ((((sqrt((y - (-1.0d0))) + 1.0d0) - sqrt(x)) - sqrt(y)) + t_1) + t_3
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) + t_3;
	double t_5 = 1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)));
	double tmp;
	if (t_4 <= 5e-5) {
		tmp = ((0.5 * (Math.sqrt(x) / x)) + t_1) + t_3;
	} else if (t_4 <= 1.0) {
		tmp = (t_2 + t_5) - Math.sqrt(x);
	} else if (t_4 <= 2.3) {
		tmp = (t_2 + (Math.sqrt((1.0 + y)) + t_5)) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = ((((Math.sqrt((y - -1.0)) + 1.0) - Math.sqrt(x)) - Math.sqrt(y)) + t_1) + t_3;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) + t_3
	t_5 = 1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))
	tmp = 0
	if t_4 <= 5e-5:
		tmp = ((0.5 * (math.sqrt(x) / x)) + t_1) + t_3
	elif t_4 <= 1.0:
		tmp = (t_2 + t_5) - math.sqrt(x)
	elif t_4 <= 2.3:
		tmp = (t_2 + (math.sqrt((1.0 + y)) + t_5)) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = ((((math.sqrt((y - -1.0)) + 1.0) - math.sqrt(x)) - math.sqrt(y)) + t_1) + t_3
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_3)
	t_5 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))
	tmp = 0.0
	if (t_4 <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(x) / x)) + t_1) + t_3);
	elseif (t_4 <= 1.0)
		tmp = Float64(Float64(t_2 + t_5) - sqrt(x));
	elseif (t_4 <= 2.3)
		tmp = Float64(Float64(t_2 + Float64(sqrt(Float64(1.0 + y)) + t_5)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(y - -1.0)) + 1.0) - sqrt(x)) - sqrt(y)) + t_1) + t_3);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3;
	t_5 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	tmp = 0.0;
	if (t_4 <= 5e-5)
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_3;
	elseif (t_4 <= 1.0)
		tmp = (t_2 + t_5) - sqrt(x);
	elseif (t_4 <= 2.3)
		tmp = (t_2 + (sqrt((1.0 + y)) + t_5)) - (sqrt(x) + sqrt(y));
	else
		tmp = ((((sqrt((y - -1.0)) + 1.0) - sqrt(x)) - sqrt(y)) + t_1) + t_3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-5], N[(N[(N[(0.5 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 1.0], N[(N[(t$95$2 + t$95$5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.3], N[(N[(t$95$2 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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))) < 5.00000000000000024e-5

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 5.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 10.6

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

   7. Applied rewrites 10.6%

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

   if 5.00000000000000024e-5 < (+.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 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-sqrt.f64 36.0

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

   11. Applied rewrites 36.0%

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

   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))) < 2.2999999999999998

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   if 2.2999999999999998 < (+.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 91.6%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 4: 97.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_3\\ t_5 := \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \frac{\sqrt{x}}{x} + t\_1\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\left(t\_2 + t\_5\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.5:\\ \;\;\;\;\left(t\_2 + \left(\sqrt{1 + y} + t\_5\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right) + t\_1\right) + t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_1)
          t_3))
        (t_5 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z))))))
   (if (<= t_4 5e-5)
     (+ (+ (* 0.5 (/ (sqrt x) x)) t_1) t_3)
     (if (<= t_4 1.0)
       (- (+ t_2 t_5) (sqrt x))
       (if (<= t_4 2.5)
         (- (+ t_2 (+ (sqrt (+ 1.0 y)) t_5)) (+ (sqrt x) (sqrt y)))
         (+ (+ (- (- 2.0 (sqrt x)) (sqrt y)) t_1) t_3))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3;
	double t_5 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	double tmp;
	if (t_4 <= 5e-5) {
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_3;
	} else if (t_4 <= 1.0) {
		tmp = (t_2 + t_5) - sqrt(x);
	} else if (t_4 <= 2.5) {
		tmp = (t_2 + (sqrt((1.0 + y)) + t_5)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((2.0 - sqrt(x)) - sqrt(y)) + t_1) + t_3;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((1.0d0 + x))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_3
    t_5 = 1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))
    if (t_4 <= 5d-5) then
        tmp = ((0.5d0 * (sqrt(x) / x)) + t_1) + t_3
    else if (t_4 <= 1.0d0) then
        tmp = (t_2 + t_5) - sqrt(x)
    else if (t_4 <= 2.5d0) then
        tmp = (t_2 + (sqrt((1.0d0 + y)) + t_5)) - (sqrt(x) + sqrt(y))
    else
        tmp = (((2.0d0 - sqrt(x)) - sqrt(y)) + t_1) + t_3
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) + t_3;
	double t_5 = 1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)));
	double tmp;
	if (t_4 <= 5e-5) {
		tmp = ((0.5 * (Math.sqrt(x) / x)) + t_1) + t_3;
	} else if (t_4 <= 1.0) {
		tmp = (t_2 + t_5) - Math.sqrt(x);
	} else if (t_4 <= 2.5) {
		tmp = (t_2 + (Math.sqrt((1.0 + y)) + t_5)) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (((2.0 - Math.sqrt(x)) - Math.sqrt(y)) + t_1) + t_3;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) + t_3
	t_5 = 1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))
	tmp = 0
	if t_4 <= 5e-5:
		tmp = ((0.5 * (math.sqrt(x) / x)) + t_1) + t_3
	elif t_4 <= 1.0:
		tmp = (t_2 + t_5) - math.sqrt(x)
	elif t_4 <= 2.5:
		tmp = (t_2 + (math.sqrt((1.0 + y)) + t_5)) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (((2.0 - math.sqrt(x)) - math.sqrt(y)) + t_1) + t_3
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_3)
	t_5 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))
	tmp = 0.0
	if (t_4 <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(x) / x)) + t_1) + t_3);
	elseif (t_4 <= 1.0)
		tmp = Float64(Float64(t_2 + t_5) - sqrt(x));
	elseif (t_4 <= 2.5)
		tmp = Float64(Float64(t_2 + Float64(sqrt(Float64(1.0 + y)) + t_5)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 - sqrt(x)) - sqrt(y)) + t_1) + t_3);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3;
	t_5 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	tmp = 0.0;
	if (t_4 <= 5e-5)
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_3;
	elseif (t_4 <= 1.0)
		tmp = (t_2 + t_5) - sqrt(x);
	elseif (t_4 <= 2.5)
		tmp = (t_2 + (sqrt((1.0 + y)) + t_5)) - (sqrt(x) + sqrt(y));
	else
		tmp = (((2.0 - sqrt(x)) - sqrt(y)) + t_1) + t_3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-5], N[(N[(N[(0.5 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 1.0], N[(N[(t$95$2 + t$95$5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.5], N[(N[(t$95$2 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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))) < 5.00000000000000024e-5

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 5.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 10.6

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

   7. Applied rewrites 10.6%

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

   if 5.00000000000000024e-5 < (+.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 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-sqrt.f64 36.0

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

   11. Applied rewrites 36.0%

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

   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))) < 2.5

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   if 2.5 < (+.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 91.6%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 66.6%

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

Alternative 5: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(0.5 \cdot \frac{1}{y \cdot \frac{1}{\sqrt{y}}} - -0.5 \cdot \frac{\sqrt{x}}{x}\right) + t\_1\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3 + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          t_1)))
   (if (<= t_3 5e-5)
     (+
      (+
       (- (* 0.5 (/ 1.0 (* y (/ 1.0 (sqrt y))))) (* -0.5 (/ (sqrt x) x)))
       t_1)
      t_2)
     (+ t_3 t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1;
	double tmp;
	if (t_3 <= 5e-5) {
		tmp = (((0.5 * (1.0 / (y * (1.0 / sqrt(y))))) - (-0.5 * (sqrt(x) / x))) + t_1) + t_2;
	} else {
		tmp = t_3 + t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    t_3 = ((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1
    if (t_3 <= 5d-5) then
        tmp = (((0.5d0 * (1.0d0 / (y * (1.0d0 / sqrt(y))))) - ((-0.5d0) * (sqrt(x) / x))) + t_1) + t_2
    else
        tmp = t_3 + t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_3 = ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1;
	double tmp;
	if (t_3 <= 5e-5) {
		tmp = (((0.5 * (1.0 / (y * (1.0 / Math.sqrt(y))))) - (-0.5 * (Math.sqrt(x) / x))) + t_1) + t_2;
	} else {
		tmp = t_3 + t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_3 = ((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1
	tmp = 0
	if t_3 <= 5e-5:
		tmp = (((0.5 * (1.0 / (y * (1.0 / math.sqrt(y))))) - (-0.5 * (math.sqrt(x) / x))) + t_1) + t_2
	else:
		tmp = t_3 + t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1)
	tmp = 0.0
	if (t_3 <= 5e-5)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(1.0 / Float64(y * Float64(1.0 / sqrt(y))))) - Float64(-0.5 * Float64(sqrt(x) / x))) + t_1) + t_2);
	else
		tmp = Float64(t_3 + t_2);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	t_3 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1;
	tmp = 0.0;
	if (t_3 <= 5e-5)
		tmp = (((0.5 * (1.0 / (y * (1.0 / sqrt(y))))) - (-0.5 * (sqrt(x) / x))) + t_1) + t_2;
	else
		tmp = t_3 + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-5], N[(N[(N[(N[(0.5 * N[(1.0 / N[(y * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], N[(t$95$3 + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))) < 5.00000000000000024e-5

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 5.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 11.4

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

   7. Applied rewrites 11.4%

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

   if 5.00000000000000024e-5 < (+.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)))

   1. Initial program 91.6%

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

Alternative 6: 95.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;x \leq 225000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\sqrt{x}}{x} + t\_1\right) + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= x 225000000.0)
     (+
      (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) t_1)
      t_2)
     (+ (+ (* 0.5 (/ (sqrt x) x)) t_1) t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (x <= 225000000.0) {
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_2;
	} else {
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    if (x <= 225000000.0d0) then
        tmp = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_2
    else
        tmp = ((0.5d0 * (sqrt(x) / x)) + t_1) + t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if (x <= 225000000.0) {
		tmp = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) + t_2;
	} else {
		tmp = ((0.5 * (Math.sqrt(x) / x)) + t_1) + t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if x <= 225000000.0:
		tmp = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) + t_2
	else:
		tmp = ((0.5 * (math.sqrt(x) / x)) + t_1) + t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (x <= 225000000.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_2);
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(x) / x)) + t_1) + t_2);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if (x <= 225000000.0)
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_2;
	else
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 225000000.0], 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] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(0.5 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25e8

   1. Initial program 91.6%

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

   if 2.25e8 < x

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 5.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 10.6

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

   7. Applied rewrites 10.6%

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

Alternative 7: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_3\\ t_5 := \sqrt{1 + x}\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \frac{\sqrt{x}}{x} + t\_1\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 1.2:\\ \;\;\;\;\left(t\_5 + t\_2\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.9999999:\\ \;\;\;\;\left(t\_5 + \left(1 + t\_2\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(2 + \sqrt{y - -1}\right) - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right) + t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_1)
          t_3))
        (t_5 (sqrt (+ 1.0 x))))
   (if (<= t_4 5e-5)
     (+ (+ (* 0.5 (/ (sqrt x) x)) t_1) t_3)
     (if (<= t_4 1.2)
       (- (+ t_5 t_2) (sqrt x))
       (if (<= t_4 2.9999999)
         (- (+ t_5 (+ 1.0 t_2)) (+ (sqrt x) (sqrt y)))
         (+
          (- (- (+ 2.0 (sqrt (- y -1.0))) (sqrt x)) (+ (sqrt z) (sqrt y)))
          t_3))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3;
	double t_5 = sqrt((1.0 + x));
	double tmp;
	if (t_4 <= 5e-5) {
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_3;
	} else if (t_4 <= 1.2) {
		tmp = (t_5 + t_2) - sqrt(x);
	} else if (t_4 <= 2.9999999) {
		tmp = (t_5 + (1.0 + t_2)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((2.0 + sqrt((y - -1.0))) - sqrt(x)) - (sqrt(z) + sqrt(y))) + t_3;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = 1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_3
    t_5 = sqrt((1.0d0 + x))
    if (t_4 <= 5d-5) then
        tmp = ((0.5d0 * (sqrt(x) / x)) + t_1) + t_3
    else if (t_4 <= 1.2d0) then
        tmp = (t_5 + t_2) - sqrt(x)
    else if (t_4 <= 2.9999999d0) then
        tmp = (t_5 + (1.0d0 + t_2)) - (sqrt(x) + sqrt(y))
    else
        tmp = (((2.0d0 + sqrt((y - (-1.0d0)))) - sqrt(x)) - (sqrt(z) + sqrt(y))) + t_3
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = 1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) + t_3;
	double t_5 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_4 <= 5e-5) {
		tmp = ((0.5 * (Math.sqrt(x) / x)) + t_1) + t_3;
	} else if (t_4 <= 1.2) {
		tmp = (t_5 + t_2) - Math.sqrt(x);
	} else if (t_4 <= 2.9999999) {
		tmp = (t_5 + (1.0 + t_2)) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (((2.0 + Math.sqrt((y - -1.0))) - Math.sqrt(x)) - (Math.sqrt(z) + Math.sqrt(y))) + t_3;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = 1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) + t_3
	t_5 = math.sqrt((1.0 + x))
	tmp = 0
	if t_4 <= 5e-5:
		tmp = ((0.5 * (math.sqrt(x) / x)) + t_1) + t_3
	elif t_4 <= 1.2:
		tmp = (t_5 + t_2) - math.sqrt(x)
	elif t_4 <= 2.9999999:
		tmp = (t_5 + (1.0 + t_2)) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (((2.0 + math.sqrt((y - -1.0))) - math.sqrt(x)) - (math.sqrt(z) + math.sqrt(y))) + t_3
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_3)
	t_5 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_4 <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(x) / x)) + t_1) + t_3);
	elseif (t_4 <= 1.2)
		tmp = Float64(Float64(t_5 + t_2) - sqrt(x));
	elseif (t_4 <= 2.9999999)
		tmp = Float64(Float64(t_5 + Float64(1.0 + t_2)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + sqrt(Float64(y - -1.0))) - sqrt(x)) - Float64(sqrt(z) + sqrt(y))) + t_3);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3;
	t_5 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_4 <= 5e-5)
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_3;
	elseif (t_4 <= 1.2)
		tmp = (t_5 + t_2) - sqrt(x);
	elseif (t_4 <= 2.9999999)
		tmp = (t_5 + (1.0 + t_2)) - (sqrt(x) + sqrt(y));
	else
		tmp = (((2.0 + sqrt((y - -1.0))) - sqrt(x)) - (sqrt(z) + sqrt(y))) + t_3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 5e-5], N[(N[(N[(0.5 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 1.2], N[(N[(t$95$5 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.9999999], N[(N[(t$95$5 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 + N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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))) < 5.00000000000000024e-5

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 5.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 10.6

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

   7. Applied rewrites 10.6%

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

   if 5.00000000000000024e-5 < (+.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))) < 1.19999999999999996

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-sqrt.f64 36.0

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

   11. Applied rewrites 36.0%

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

   if 1.19999999999999996 < (+.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))) < 2.99999990000000016

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-+.f64 N/A

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

       6. lift-/.f64 62.1

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

   11. Applied rewrites 62.1%

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

   if 2.99999990000000016 < (+.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 91.6%

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

   2. Taylor expanded in z around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 31.0%

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

Alternative 8: 93.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_2\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \frac{\sqrt{x}}{x} + t\_1\right) + t\_2\\ \mathbf{elif}\;t\_3 \leq 1.2:\\ \;\;\;\;\left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_1)
          t_2)))
   (if (<= t_3 5e-5)
     (+ (+ (* 0.5 (/ (sqrt x) x)) t_1) t_2)
     (if (<= t_3 1.2)
       (- (+ (sqrt (+ 1.0 x)) (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z))))) (sqrt x))
       (+ (+ (- (- 2.0 (sqrt x)) (sqrt y)) t_1) t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_2;
	double tmp;
	if (t_3 <= 5e-5) {
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_2;
	} else if (t_3 <= 1.2) {
		tmp = (sqrt((1.0 + x)) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) - sqrt(x);
	} else {
		tmp = (((2.0 - sqrt(x)) - sqrt(y)) + t_1) + t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    t_3 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_2
    if (t_3 <= 5d-5) then
        tmp = ((0.5d0 * (sqrt(x) / x)) + t_1) + t_2
    else if (t_3 <= 1.2d0) then
        tmp = (sqrt((1.0d0 + x)) + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))) - sqrt(x)
    else
        tmp = (((2.0d0 - sqrt(x)) - sqrt(y)) + t_1) + t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_3 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) + t_2;
	double tmp;
	if (t_3 <= 5e-5) {
		tmp = ((0.5 * (Math.sqrt(x) / x)) + t_1) + t_2;
	} else if (t_3 <= 1.2) {
		tmp = (Math.sqrt((1.0 + x)) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))) - Math.sqrt(x);
	} else {
		tmp = (((2.0 - Math.sqrt(x)) - Math.sqrt(y)) + t_1) + t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_3 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) + t_2
	tmp = 0
	if t_3 <= 5e-5:
		tmp = ((0.5 * (math.sqrt(x) / x)) + t_1) + t_2
	elif t_3 <= 1.2:
		tmp = (math.sqrt((1.0 + x)) + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))) - math.sqrt(x)
	else:
		tmp = (((2.0 - math.sqrt(x)) - math.sqrt(y)) + t_1) + t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_2)
	tmp = 0.0
	if (t_3 <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(x) / x)) + t_1) + t_2);
	elseif (t_3 <= 1.2)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))) - sqrt(x));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 - sqrt(x)) - sqrt(y)) + t_1) + t_2);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	t_3 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_2;
	tmp = 0.0;
	if (t_3 <= 5e-5)
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_2;
	elseif (t_3 <= 1.2)
		tmp = (sqrt((1.0 + x)) + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) - sqrt(x);
	else
		tmp = (((2.0 - sqrt(x)) - sqrt(y)) + t_1) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-5], N[(N[(N[(0.5 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 1.2], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $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))) < 5.00000000000000024e-5

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 5.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 10.6

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

   7. Applied rewrites 10.6%

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

   if 5.00000000000000024e-5 < (+.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))) < 1.19999999999999996

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-sqrt.f64 36.0

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

   11. Applied rewrites 36.0%

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

   if 1.19999999999999996 < (+.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 91.6%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 66.6%

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

Alternative 9: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_3\\ t_5 := \sqrt{1 + x}\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \frac{\sqrt{x}}{x} + t\_1\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 1.2:\\ \;\;\;\;\left(t\_5 + t\_2\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_5 + \left(1 + t\_2\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_1)
          t_3))
        (t_5 (sqrt (+ 1.0 x))))
   (if (<= t_4 5e-5)
     (+ (+ (* 0.5 (/ (sqrt x) x)) t_1) t_3)
     (if (<= t_4 1.2)
       (- (+ t_5 t_2) (sqrt x))
       (- (+ t_5 (+ 1.0 t_2)) (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3;
	double t_5 = sqrt((1.0 + x));
	double tmp;
	if (t_4 <= 5e-5) {
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_3;
	} else if (t_4 <= 1.2) {
		tmp = (t_5 + t_2) - sqrt(x);
	} else {
		tmp = (t_5 + (1.0 + t_2)) - (sqrt(x) + sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = 1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_3
    t_5 = sqrt((1.0d0 + x))
    if (t_4 <= 5d-5) then
        tmp = ((0.5d0 * (sqrt(x) / x)) + t_1) + t_3
    else if (t_4 <= 1.2d0) then
        tmp = (t_5 + t_2) - sqrt(x)
    else
        tmp = (t_5 + (1.0d0 + t_2)) - (sqrt(x) + sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = 1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) + t_3;
	double t_5 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_4 <= 5e-5) {
		tmp = ((0.5 * (Math.sqrt(x) / x)) + t_1) + t_3;
	} else if (t_4 <= 1.2) {
		tmp = (t_5 + t_2) - Math.sqrt(x);
	} else {
		tmp = (t_5 + (1.0 + t_2)) - (Math.sqrt(x) + Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = 1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) + t_3
	t_5 = math.sqrt((1.0 + x))
	tmp = 0
	if t_4 <= 5e-5:
		tmp = ((0.5 * (math.sqrt(x) / x)) + t_1) + t_3
	elif t_4 <= 1.2:
		tmp = (t_5 + t_2) - math.sqrt(x)
	else:
		tmp = (t_5 + (1.0 + t_2)) - (math.sqrt(x) + math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_3)
	t_5 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_4 <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 * Float64(sqrt(x) / x)) + t_1) + t_3);
	elseif (t_4 <= 1.2)
		tmp = Float64(Float64(t_5 + t_2) - sqrt(x));
	else
		tmp = Float64(Float64(t_5 + Float64(1.0 + t_2)) - Float64(sqrt(x) + sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3;
	t_5 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_4 <= 5e-5)
		tmp = ((0.5 * (sqrt(x) / x)) + t_1) + t_3;
	elseif (t_4 <= 1.2)
		tmp = (t_5 + t_2) - sqrt(x);
	else
		tmp = (t_5 + (1.0 + t_2)) - (sqrt(x) + sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 5e-5], N[(N[(N[(0.5 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 1.2], N[(N[(t$95$5 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$5 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $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))) < 5.00000000000000024e-5

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 5.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 10.6

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

   7. Applied rewrites 10.6%

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

   if 5.00000000000000024e-5 < (+.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))) < 1.19999999999999996

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-sqrt.f64 36.0

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

   11. Applied rewrites 36.0%

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

   if 1.19999999999999996 < (+.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 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-+.f64 N/A

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

       6. lift-/.f64 62.1

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

   11. Applied rewrites 62.1%

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

Alternative 10: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 1.2:\\ \;\;\;\;\left(t\_1 + t\_2\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \left(1 + t\_2\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z))))))
   (if (<=
        (+
         (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
         (- (sqrt (+ z 1.0)) (sqrt z)))
        1.2)
     (- (+ t_1 t_2) (sqrt x))
     (- (+ t_1 (+ 1.0 t_2)) (+ (sqrt x) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	double tmp;
	if ((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) <= 1.2) {
		tmp = (t_1 + t_2) - sqrt(x);
	} else {
		tmp = (t_1 + (1.0 + t_2)) - (sqrt(x) + sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = 1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))
    if ((((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) <= 1.2d0) then
        tmp = (t_1 + t_2) - sqrt(x)
    else
        tmp = (t_1 + (1.0d0 + t_2)) - (sqrt(x) + sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = 1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)));
	double tmp;
	if ((((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) <= 1.2) {
		tmp = (t_1 + t_2) - Math.sqrt(x);
	} else {
		tmp = (t_1 + (1.0 + t_2)) - (Math.sqrt(x) + Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = 1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))
	tmp = 0
	if (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) <= 1.2:
		tmp = (t_1 + t_2) - math.sqrt(x)
	else:
		tmp = (t_1 + (1.0 + t_2)) - (math.sqrt(x) + math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))
	tmp = 0.0
	if (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))) <= 1.2)
		tmp = Float64(Float64(t_1 + t_2) - sqrt(x));
	else
		tmp = Float64(Float64(t_1 + Float64(1.0 + t_2)) - Float64(sqrt(x) + sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = 1.0 / (sqrt(z) + sqrt((1.0 + z)));
	tmp = 0.0;
	if ((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) <= 1.2)
		tmp = (t_1 + t_2) - sqrt(x);
	else
		tmp = (t_1 + (1.0 + t_2)) - (sqrt(x) + sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 1.2], N[(N[(t$95$1 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))) < 1.19999999999999996

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-sqrt.f64 36.0

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

   11. Applied rewrites 36.0%

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

   if 1.19999999999999996 < (+.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)))

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-+.f64 N/A

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

       6. lift-/.f64 62.1

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

   11. Applied rewrites 62.1%

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

Alternative 11: 64.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.75 \cdot 10^{+21}:\\ \;\;\;\;\left(t\_1 + \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 1.75e+21)
     (- (+ t_1 (sqrt (+ 1.0 y))) (+ (sqrt y) (sqrt x)))
     (- (+ t_1 (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z))))) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 1.75e+21) {
		tmp = (t_1 + sqrt((1.0 + y))) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (t_1 + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 1.75d+21) then
        tmp = (t_1 + sqrt((1.0d0 + y))) - (sqrt(y) + sqrt(x))
    else
        tmp = (t_1 + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z))))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 1.75e+21) {
		tmp = (t_1 + Math.sqrt((1.0 + y))) - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = (t_1 + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z))))) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 1.75e+21:
		tmp = (t_1 + math.sqrt((1.0 + y))) - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = (t_1 + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z))))) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 1.75e+21)
		tmp = Float64(Float64(t_1 + sqrt(Float64(1.0 + y))) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z))))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 1.75e+21)
		tmp = (t_1 + sqrt((1.0 + y))) - (sqrt(y) + sqrt(x));
	else
		tmp = (t_1 + (1.0 / (sqrt(z) + sqrt((1.0 + z))))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.75e+21], N[(N[(t$95$1 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.75e21

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 3.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 47.2

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

   7. Applied rewrites 47.2%

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

   if 1.75e21 < y

   1. Initial program 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift--.f64 66.1

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-sqrt.f64 36.0

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

   11. Applied rewrites 36.0%

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

Alternative 12: 47.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (- (+ (sqrt (+ 1.0 x)) (sqrt (+ 1.0 y))) (+ (sqrt y) (sqrt x))))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 + x)) + sqrt((1.0d0 + y))) - (sqrt(y) + sqrt(x))
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (sqrt((1.0 + x)) + sqrt((1.0 + y))) - (sqrt(y) + sqrt(x));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

2. Taylor expanded in z around inf

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

   1. associate--r+ N/A

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

   2. lower--.f64 N/A

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

4. Applied rewrites 3.6%

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

5. Taylor expanded in t around inf

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

   1. lower-+.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-+.f64 47.2

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

7. Applied rewrites 47.2%

   \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) \]
8. Add Preprocessing

Alternative 13: 14.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- (sqrt (+ t 1.0)) (sqrt t))))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 + y)) - sqrt(y)) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (sqrt((1.0 + y)) - sqrt(y)) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

2. Taylor expanded in z around inf

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

   1. associate--r+ N/A

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

   2. lower--.f64 N/A

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

4. Applied rewrites 47.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \left(\sqrt{1 + y} - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
6. 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 14.1

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

7. Applied rewrites 14.1%

   \[\leadsto \left(\sqrt{1 + y} - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
8. Add Preprocessing

Alternative 14: 11.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.3:\\ \;\;\;\;0.5 \cdot \frac{1}{\sqrt{z}} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(2 - \left(\sqrt{x} + \sqrt{z}\right)\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= (- (sqrt (+ z 1.0)) (sqrt z)) 0.3)
     (+ (* 0.5 (/ 1.0 (sqrt z))) t_1)
     (+ (- 2.0 (+ (sqrt x) (sqrt z))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.3) {
		tmp = (0.5 * (1.0 / sqrt(z))) + t_1;
	} else {
		tmp = (2.0 - (sqrt(x) + sqrt(z))) + t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((t + 1.0d0)) - sqrt(t)
    if ((sqrt((z + 1.0d0)) - sqrt(z)) <= 0.3d0) then
        tmp = (0.5d0 * (1.0d0 / sqrt(z))) + t_1
    else
        tmp = (2.0d0 - (sqrt(x) + sqrt(z))) + t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) <= 0.3) {
		tmp = (0.5 * (1.0 / Math.sqrt(z))) + t_1;
	} else {
		tmp = (2.0 - (Math.sqrt(x) + Math.sqrt(z))) + t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if (math.sqrt((z + 1.0)) - math.sqrt(z)) <= 0.3:
		tmp = (0.5 * (1.0 / math.sqrt(z))) + t_1
	else:
		tmp = (2.0 - (math.sqrt(x) + math.sqrt(z))) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) <= 0.3)
		tmp = Float64(Float64(0.5 * Float64(1.0 / sqrt(z))) + t_1);
	else
		tmp = Float64(Float64(2.0 - Float64(sqrt(x) + sqrt(z))) + t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.3)
		tmp = (0.5 * (1.0 / sqrt(z))) + t_1;
	else
		tmp = (2.0 - (sqrt(x) + sqrt(z))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.3], N[(N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(2.0 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.299999999999999989

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. mult-flip-rev N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 45.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 7.8

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

   7. Applied rewrites 7.8%

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

   8. Taylor expanded in z around inf

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 7.8

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

   10. Applied rewrites 7.8%

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

   if 0.299999999999999989 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 91.6%

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

   2. Taylor expanded in z around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 7.4%

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

Alternative 15: 10.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_1 \leq 2.5:\\ \;\;\;\;0.5 \cdot \frac{1}{\sqrt{z}} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \sqrt{z}\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<=
        (+
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          (- (sqrt (+ z 1.0)) (sqrt z)))
         t_1)
        2.5)
     (+ (* 0.5 (/ 1.0 (sqrt z))) t_1)
     (+ (- 1.0 (sqrt z)) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_1) <= 2.5) {
		tmp = (0.5 * (1.0 / sqrt(z))) + t_1;
	} else {
		tmp = (1.0 - sqrt(z)) + t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) :: tmp
    t_1 = sqrt((t + 1.0d0)) - sqrt(t)
    if (((((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + t_1) <= 2.5d0) then
        tmp = (0.5d0 * (1.0d0 / sqrt(z))) + t_1
    else
        tmp = (1.0d0 - sqrt(z)) + t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if (((((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + t_1) <= 2.5) {
		tmp = (0.5 * (1.0 / Math.sqrt(z))) + t_1;
	} else {
		tmp = (1.0 - Math.sqrt(z)) + t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if ((((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + t_1) <= 2.5:
		tmp = (0.5 * (1.0 / math.sqrt(z))) + t_1
	else:
		tmp = (1.0 - math.sqrt(z)) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (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))) + t_1) <= 2.5)
		tmp = Float64(Float64(0.5 * Float64(1.0 / sqrt(z))) + t_1);
	else
		tmp = Float64(Float64(1.0 - sqrt(z)) + t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_1) <= 2.5)
		tmp = (0.5 * (1.0 / sqrt(z))) + t_1;
	else
		tmp = (1.0 - sqrt(z)) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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] + t$95$1), $MachinePrecision], 2.5], N[(N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$1), $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))) < 2.5

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. mult-flip-rev N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 45.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 7.8

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

   7. Applied rewrites 7.8%

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

   8. Taylor expanded in z around inf

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 7.8

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

   10. Applied rewrites 7.8%

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

   if 2.5 < (+.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 91.6%

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

   2. Taylor expanded in z around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 6.6

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

   10. Applied rewrites 6.6%

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

Alternative 16: 6.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(1 - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- 1.0 (sqrt z)) (- (sqrt (+ t 1.0)) (sqrt t))))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 = (1.0d0 - sqrt(z)) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (1.0 - sqrt(z)) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

2. Taylor expanded in z around 0

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

   1. associate--r+ N/A

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

   2. lower--.f64 N/A

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

4. Applied rewrites 31.0%

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

5. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 7.4

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

7. Applied rewrites 7.4%

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

8. Taylor expanded in x around inf

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

   1. lower--.f64 N/A

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

   2. lift-sqrt.f64 6.6

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

10. Applied rewrites 6.6%

    \[\leadsto \left(1 - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
11. Add Preprocessing

Alternative 17: 1.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot \frac{1}{\sqrt{z}} + \left(1 - \sqrt{t}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (* 0.5 (/ 1.0 (sqrt z))) (- 1.0 (sqrt t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (0.5 * (1.0 / sqrt(z))) + (1.0 - sqrt(t));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 = (0.5d0 * (1.0d0 / sqrt(z))) + (1.0d0 - sqrt(t))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return (0.5 * (1.0 / Math.sqrt(z))) + (1.0 - Math.sqrt(t));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (0.5 * (1.0 / math.sqrt(z))) + (1.0 - math.sqrt(t))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(0.5 * Float64(1.0 / sqrt(z))) + Float64(1.0 - sqrt(t)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (0.5 * (1.0 / sqrt(z))) + (1.0 - sqrt(t));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

2. Taylor expanded in z around inf

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

   1. associate--r+ N/A

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

   2. mult-flip-rev N/A

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

   3. lower--.f64 N/A

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

4. Applied rewrites 45.7%

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

5. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lift-sqrt.f64 7.8

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

7. Applied rewrites 7.8%

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

8. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. lift-sqrt.f64 1.9

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

10. Applied rewrites 1.9%

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

11. Taylor expanded in z around inf

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

    1. sqrt-div N/A

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

    2. metadata-eval N/A

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

    3. lower-/.f64 N/A

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

    4. lift-sqrt.f64 1.9

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

13. Applied rewrites 1.9%

    \[\leadsto 0.5 \cdot \frac{1}{\sqrt{z}} + \left(1 - \sqrt{t}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025136 
(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))))