Main:z from

Percentage Accurate: 92.0% → 99.1%

Time: 13.8s

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: 92.0% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ t_5 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right) + t\_4\\ \mathbf{if}\;t\_5 \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, t\_2\right)\\ \mathbf{elif}\;t\_5 \leq 2.0001:\\ \;\;\;\;\left(\sqrt{1 + y} + t\_2\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t\_1 + 1\right) + t\_3\right) - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\right) + t\_4\\ \end{array} \end{array} \]

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 (+ y 1.0)))
        (t_2 (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_5
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y)))
           (- t_3 (sqrt z)))
          t_4)))
   (if (<= t_5 1.0001)
     (fma 0.5 (/ 1.0 (sqrt y)) t_2)
     (if (<= t_5 2.0001)
       (- (+ (sqrt (+ 1.0 y)) t_2) (sqrt y))
       (+ (- (- (+ (+ t_1 1.0) t_3) (sqrt x)) (+ (sqrt z) (sqrt y))) t_4)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(x) + sqrt((1.0 + x)))));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double t_5 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + t_4;
	double tmp;
	if (t_5 <= 1.0001) {
		tmp = fma(0.5, (1.0 / sqrt(y)), t_2);
	} else if (t_5 <= 2.0001) {
		tmp = (sqrt((1.0 + y)) + t_2) - sqrt(y);
	} else {
		tmp = ((((t_1 + 1.0) + t_3) - sqrt(x)) - (sqrt(z) + sqrt(y))) + t_4;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_5 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))) + t_4)
	tmp = 0.0
	if (t_5 <= 1.0001)
		tmp = fma(0.5, Float64(1.0 / sqrt(y)), t_2);
	elseif (t_5 <= 2.0001)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) + t_2) - sqrt(y));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_1 + 1.0) + t_3) - sqrt(x)) - Float64(sqrt(z) + sqrt(y))) + t_4);
	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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0001], N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 2.0001], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00009999999999999

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 5.6%

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

   10. Taylor expanded in z around inf

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

       1. lower-fma.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-sqrt.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. lift-+.f64 N/A

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

       15. lift-/.f64 34.7

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

   12. Applied rewrites 34.7%

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

   if 1.00009999999999999 < (+.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))) < 2.00010000000000021

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 46.1%

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

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 39.6%

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

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ t_2 := \sqrt{z + 1} - \sqrt{z}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_2\right) + t\_3\\ \mathbf{if}\;t\_4 \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, t\_1\right)\\ \mathbf{elif}\;t\_4 \leq 2.0001:\\ \;\;\;\;\left(\sqrt{1 + y} + t\_1\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + t\_2\right) + t\_3\\ \end{array} \end{array} \]

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 (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))))
        (t_2 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_2)
          t_3)))
   (if (<= t_4 1.0001)
     (fma 0.5 (/ 1.0 (sqrt y)) t_1)
     (if (<= t_4 2.0001)
       (- (+ (sqrt (+ 1.0 y)) t_1) (sqrt y))
       (+ (+ (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))) t_2) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(x) + sqrt((1.0 + x)))));
	double t_2 = sqrt((z + 1.0)) - sqrt(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_2) + t_3;
	double tmp;
	if (t_4 <= 1.0001) {
		tmp = fma(0.5, (1.0 / sqrt(y)), t_1);
	} else if (t_4 <= 2.0001) {
		tmp = (sqrt((1.0 + y)) + t_1) - sqrt(y);
	} else {
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + t_2) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))))
	t_2 = Float64(sqrt(Float64(z + 1.0)) - sqrt(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_2) + t_3)
	tmp = 0.0
	if (t_4 <= 1.0001)
		tmp = fma(0.5, Float64(1.0 / sqrt(y)), t_1);
	elseif (t_4 <= 2.0001)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) + t_1) - sqrt(y));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))) + t_2) + 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[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $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$2), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0001], N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 2.0001], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00009999999999999

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 5.6%

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

   10. Taylor expanded in z around inf

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

       1. lower-fma.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-sqrt.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. lift-+.f64 N/A

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

       15. lift-/.f64 34.7

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

   12. Applied rewrites 34.7%

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

   if 1.00009999999999999 < (+.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))) < 2.00010000000000021

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 46.1%

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

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 67.1

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

   7. Applied rewrites 67.1%

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

Alternative 3: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 13600000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 13600000000.0)
   (+
    (+
     (+
      (- (sqrt (+ x 1.0)) (sqrt x))
      (/ (- (+ y 1.0) y) (+ (sqrt (+ y 1.0)) (sqrt y))))
     (- (sqrt (+ z 1.0)) (sqrt z)))
    (- (sqrt (+ t 1.0)) (sqrt t)))
   (fma
    0.5
    (/ 1.0 (sqrt y))
    (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 13600000000.0) {
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (((y + 1.0) - y) / (sqrt((y + 1.0)) + sqrt(y)))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	} else {
		tmp = fma(0.5, (1.0 / sqrt(y)), fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(x) + sqrt((1.0 + x))))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 13600000000.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(Float64(Float64(y + 1.0) - y) / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	else
		tmp = fma(0.5, Float64(1.0 / sqrt(y)), fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x))))));
	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_] := If[LessEqual[y, 13600000000.0], N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y + 1.0), $MachinePrecision] - y), $MachinePrecision] / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $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], N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.36e10

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt 83.2

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt 92.6

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

   5. Applied rewrites 92.6%

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

   if 1.36e10 < y

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 5.6%

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

   10. Taylor expanded in z around inf

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

       1. lower-fma.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-sqrt.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. lift-+.f64 N/A

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

       15. lift-/.f64 34.7

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

   12. Applied rewrites 34.7%

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

Alternative 4: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 280000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\left(\frac{1}{y} + 1\right) \cdot y} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 280000000.0)
   (+
    (+
     (+
      (- (sqrt (+ x 1.0)) (sqrt x))
      (- (sqrt (* (+ (/ 1.0 y) 1.0) y)) (sqrt y)))
     (- (sqrt (+ z 1.0)) (sqrt z)))
    (- (sqrt (+ t 1.0)) (sqrt t)))
   (fma
    0.5
    (/ 1.0 (sqrt y))
    (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 280000000.0) {
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((((1.0 / y) + 1.0) * y)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	} else {
		tmp = fma(0.5, (1.0 / sqrt(y)), fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(x) + sqrt((1.0 + x))))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 280000000.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(Float64(Float64(1.0 / y) + 1.0) * y)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	else
		tmp = fma(0.5, Float64(1.0 / sqrt(y)), fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x))))));
	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_] := If[LessEqual[y, 280000000.0], N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(N[(N[(1.0 / y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $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], N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8e8

   1. Initial program 92.0%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 92.0

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

   4. Applied rewrites 92.0%

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

   if 2.8e8 < y

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 5.6%

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

   10. Taylor expanded in z around inf

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

       1. lower-fma.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-sqrt.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. lift-+.f64 N/A

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

       15. lift-/.f64 34.7

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

   12. Applied rewrites 34.7%

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

Alternative 5: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 280000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 280000000.0)
   (+
    (+
     (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
     (- (sqrt (+ z 1.0)) (sqrt z)))
    (- (sqrt (+ t 1.0)) (sqrt t)))
   (fma
    0.5
    (/ 1.0 (sqrt y))
    (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 280000000.0) {
		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));
	} else {
		tmp = fma(0.5, (1.0 / sqrt(y)), fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(x) + sqrt((1.0 + x))))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 280000000.0)
		tmp = 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)));
	else
		tmp = fma(0.5, Float64(1.0 / sqrt(y)), fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x))))));
	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_] := If[LessEqual[y, 280000000.0], 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] + 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], N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8e8

   1. Initial program 92.0%

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

   if 2.8e8 < y

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 5.6%

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

   10. Taylor expanded in z around inf

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

       1. lower-fma.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-sqrt.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. lift-+.f64 N/A

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

       15. lift-/.f64 34.7

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

   12. Applied rewrites 34.7%

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

Alternative 6: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;t\_2 \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 2.0001:\\ \;\;\;\;\left(t\_3 + t\_1\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 + \left(\sqrt{1 + z} + \frac{1}{1 + \sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \end{array} \end{array} \]

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 (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))))
        (t_2
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_3 (sqrt (+ 1.0 y))))
   (if (<= t_2 1.0001)
     (fma 0.5 (/ 1.0 (sqrt y)) t_1)
     (if (<= t_2 2.0001)
       (- (+ t_3 t_1) (sqrt y))
       (-
        (+ t_3 (+ (sqrt (+ 1.0 z)) (/ 1.0 (+ 1.0 (sqrt x)))))
        (+ (sqrt y) (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(x) + sqrt((1.0 + x)))));
	double t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_3 = sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 1.0001) {
		tmp = fma(0.5, (1.0 / sqrt(y)), t_1);
	} else if (t_2 <= 2.0001) {
		tmp = (t_3 + t_1) - sqrt(y);
	} else {
		tmp = (t_3 + (sqrt((1.0 + z)) + (1.0 / (1.0 + sqrt(x))))) - (sqrt(y) + sqrt(z));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))))
	t_2 = 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)))
	t_3 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_2 <= 1.0001)
		tmp = fma(0.5, Float64(1.0 / sqrt(y)), t_1);
	elseif (t_2 <= 2.0001)
		tmp = Float64(Float64(t_3 + t_1) - sqrt(y));
	else
		tmp = Float64(Float64(t_3 + Float64(sqrt(Float64(1.0 + z)) + Float64(1.0 / Float64(1.0 + sqrt(x))))) - Float64(sqrt(y) + sqrt(z)));
	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[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1.0001], N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2.0001], N[(N[(t$95$3 + t$95$1), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00009999999999999

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 5.6%

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

   10. Taylor expanded in z around inf

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

       1. lower-fma.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-sqrt.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. lift-+.f64 N/A

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

       15. lift-/.f64 34.7

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

   12. Applied rewrites 34.7%

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

   if 1.00009999999999999 < (+.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))) < 2.00010000000000021

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 46.1%

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

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 34.0

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

   9. Applied rewrites 34.0%

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

Alternative 7: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_1\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;t\_2 \leq 1.0001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\left(\left(\left(1 + t\_3\right) - \sqrt{x}\right) - \sqrt{y}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 + \left(\sqrt{1 + z} + \frac{1}{1 + \sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \end{array} \end{array} \]

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 (+ t 1.0)) (sqrt t)))
        (t_2
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          t_1))
        (t_3 (sqrt (+ 1.0 y))))
   (if (<= t_2 1.0001)
     (fma
      0.5
      (/ 1.0 (sqrt y))
      (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))))
     (if (<= t_2 2.0)
       (+ (- (- (+ 1.0 t_3) (sqrt x)) (sqrt y)) t_1)
       (-
        (+ t_3 (+ (sqrt (+ 1.0 z)) (/ 1.0 (+ 1.0 (sqrt x)))))
        (+ (sqrt y) (sqrt z)))))))

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 t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_1;
	double t_3 = sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 1.0001) {
		tmp = fma(0.5, (1.0 / sqrt(y)), fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(x) + sqrt((1.0 + x))))));
	} else if (t_2 <= 2.0) {
		tmp = (((1.0 + t_3) - sqrt(x)) - sqrt(y)) + t_1;
	} else {
		tmp = (t_3 + (sqrt((1.0 + z)) + (1.0 / (1.0 + sqrt(x))))) - (sqrt(y) + sqrt(z));
	}
	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))
	t_2 = 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)
	t_3 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_2 <= 1.0001)
		tmp = fma(0.5, Float64(1.0 / sqrt(y)), fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x))))));
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_3) - sqrt(x)) - sqrt(y)) + t_1);
	else
		tmp = Float64(Float64(t_3 + Float64(sqrt(Float64(1.0 + z)) + Float64(1.0 / Float64(1.0 + sqrt(x))))) - Float64(sqrt(y) + sqrt(z)));
	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[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1.0001], N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(N[(N[(1.0 + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(t$95$3 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00009999999999999

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 5.6%

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

   10. Taylor expanded in z around inf

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

       1. lower-fma.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. sqrt-div N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-sqrt.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. lift-+.f64 N/A

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

       15. lift-/.f64 34.7

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

   12. Applied rewrites 34.7%

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

   if 1.00009999999999999 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 92.0%

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

   2. Taylor expanded in 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

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 47.7

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

   7. Applied rewrites 47.7%

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

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

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 34.0

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

   9. Applied rewrites 34.0%

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

Alternative 8: 87.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_2\\ t_4 := \sqrt{1 + y}\\ \mathbf{if}\;t\_3 \leq 1.0000295:\\ \;\;\;\;\left(\left(t\_1 + 0.5 \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{x}\right) + t\_2\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\left(t\_4 + \frac{1}{\sqrt{x} + t\_1}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 + \left(\sqrt{1 + z} + \frac{1}{1 + \sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \end{array} \end{array} \]

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 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          t_2))
        (t_4 (sqrt (+ 1.0 y))))
   (if (<= t_3 1.0000295)
     (+ (- (+ t_1 (* 0.5 (/ 1.0 (sqrt y)))) (sqrt x)) t_2)
     (if (<= t_3 2.0)
       (- (+ t_4 (/ 1.0 (+ (sqrt x) t_1))) (sqrt y))
       (-
        (+ t_4 (+ (sqrt (+ 1.0 z)) (/ 1.0 (+ 1.0 (sqrt x)))))
        (+ (sqrt y) (sqrt z)))))))

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 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_2;
	double t_4 = sqrt((1.0 + y));
	double tmp;
	if (t_3 <= 1.0000295) {
		tmp = ((t_1 + (0.5 * (1.0 / sqrt(y)))) - sqrt(x)) + t_2;
	} else if (t_3 <= 2.0) {
		tmp = (t_4 + (1.0 / (sqrt(x) + t_1))) - sqrt(y);
	} else {
		tmp = (t_4 + (sqrt((1.0 + z)) + (1.0 / (1.0 + sqrt(x))))) - (sqrt(y) + sqrt(z));
	}
	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) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    t_3 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + t_2
    t_4 = sqrt((1.0d0 + y))
    if (t_3 <= 1.0000295d0) then
        tmp = ((t_1 + (0.5d0 * (1.0d0 / sqrt(y)))) - sqrt(x)) + t_2
    else if (t_3 <= 2.0d0) then
        tmp = (t_4 + (1.0d0 / (sqrt(x) + t_1))) - sqrt(y)
    else
        tmp = (t_4 + (sqrt((1.0d0 + z)) + (1.0d0 / (1.0d0 + sqrt(x))))) - (sqrt(y) + sqrt(z))
    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 = 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))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + t_2;
	double t_4 = Math.sqrt((1.0 + y));
	double tmp;
	if (t_3 <= 1.0000295) {
		tmp = ((t_1 + (0.5 * (1.0 / Math.sqrt(y)))) - Math.sqrt(x)) + t_2;
	} else if (t_3 <= 2.0) {
		tmp = (t_4 + (1.0 / (Math.sqrt(x) + t_1))) - Math.sqrt(y);
	} else {
		tmp = (t_4 + (Math.sqrt((1.0 + z)) + (1.0 / (1.0 + Math.sqrt(x))))) - (Math.sqrt(y) + Math.sqrt(z));
	}
	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 = 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))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + t_2
	t_4 = math.sqrt((1.0 + y))
	tmp = 0
	if t_3 <= 1.0000295:
		tmp = ((t_1 + (0.5 * (1.0 / math.sqrt(y)))) - math.sqrt(x)) + t_2
	elif t_3 <= 2.0:
		tmp = (t_4 + (1.0 / (math.sqrt(x) + t_1))) - math.sqrt(y)
	else:
		tmp = (t_4 + (math.sqrt((1.0 + z)) + (1.0 / (1.0 + math.sqrt(x))))) - (math.sqrt(y) + math.sqrt(z))
	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(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))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + t_2)
	t_4 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_3 <= 1.0000295)
		tmp = Float64(Float64(Float64(t_1 + Float64(0.5 * Float64(1.0 / sqrt(y)))) - sqrt(x)) + t_2);
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(t_4 + Float64(1.0 / Float64(sqrt(x) + t_1))) - sqrt(y));
	else
		tmp = Float64(Float64(t_4 + Float64(sqrt(Float64(1.0 + z)) + Float64(1.0 / Float64(1.0 + sqrt(x))))) - Float64(sqrt(y) + sqrt(z)));
	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 = sqrt((t + 1.0)) - sqrt(t);
	t_3 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_2;
	t_4 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t_3 <= 1.0000295)
		tmp = ((t_1 + (0.5 * (1.0 / sqrt(y)))) - sqrt(x)) + t_2;
	elseif (t_3 <= 2.0)
		tmp = (t_4 + (1.0 / (sqrt(x) + t_1))) - sqrt(y);
	else
		tmp = (t_4 + (sqrt((1.0 + z)) + (1.0 / (1.0 + sqrt(x))))) - (sqrt(y) + sqrt(z));
	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[(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] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1.0000295], N[(N[(N[(t$95$1 + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(t$95$4 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0000294999999999

   1. Initial program 92.0%

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

   2. Taylor expanded in 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

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 29.4

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

   7. Applied rewrites 29.4%

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

   if 1.0000294999999999 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 47.8

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

   9. Applied rewrites 47.8%

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

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

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 34.0

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

   9. Applied rewrites 34.0%

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

Alternative 9: 86.5% accurate, 0.4× speedup

Math

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

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 (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          (- (sqrt (+ z 1.0)) (sqrt z))))
        (t_2 (sqrt (+ 1.0 y))))
   (if (<= t_1 1.0)
     (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (* 0.5 (/ 1.0 (sqrt t))))
     (if (<= t_1 2.0)
       (+ (- (- (+ 1.0 t_2) (sqrt x)) (sqrt y)) (- (sqrt (+ t 1.0)) (sqrt t)))
       (-
        (+ t_2 (+ (sqrt (+ 1.0 z)) (/ 1.0 (+ 1.0 (sqrt x)))))
        (+ (sqrt y) (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z));
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (t_1 <= 1.0) {
		tmp = (sqrt((1.0 + x)) - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	} else if (t_1 <= 2.0) {
		tmp = (((1.0 + t_2) - sqrt(x)) - sqrt(y)) + (sqrt((t + 1.0)) - sqrt(t));
	} else {
		tmp = (t_2 + (sqrt((1.0 + z)) + (1.0 / (1.0 + sqrt(x))))) - (sqrt(y) + sqrt(z));
	}
	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((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))
    t_2 = sqrt((1.0d0 + y))
    if (t_1 <= 1.0d0) then
        tmp = (sqrt((1.0d0 + x)) - sqrt(x)) + (0.5d0 * (1.0d0 / sqrt(t)))
    else if (t_1 <= 2.0d0) then
        tmp = (((1.0d0 + t_2) - sqrt(x)) - sqrt(y)) + (sqrt((t + 1.0d0)) - sqrt(t))
    else
        tmp = (t_2 + (sqrt((1.0d0 + z)) + (1.0d0 / (1.0d0 + sqrt(x))))) - (sqrt(y) + sqrt(z))
    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((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z));
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (t_1 <= 1.0) {
		tmp = (Math.sqrt((1.0 + x)) - Math.sqrt(x)) + (0.5 * (1.0 / Math.sqrt(t)));
	} else if (t_1 <= 2.0) {
		tmp = (((1.0 + t_2) - Math.sqrt(x)) - Math.sqrt(y)) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	} else {
		tmp = (t_2 + (Math.sqrt((1.0 + z)) + (1.0 / (1.0 + Math.sqrt(x))))) - (Math.sqrt(y) + Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = ((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_2 = math.sqrt((1.0 + y))
	tmp = 0
	if t_1 <= 1.0:
		tmp = (math.sqrt((1.0 + x)) - math.sqrt(x)) + (0.5 * (1.0 / math.sqrt(t)))
	elif t_1 <= 2.0:
		tmp = (((1.0 + t_2) - math.sqrt(x)) - math.sqrt(y)) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	else:
		tmp = (t_2 + (math.sqrt((1.0 + z)) + (1.0 / (1.0 + math.sqrt(x))))) - (math.sqrt(y) + math.sqrt(z))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = 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_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_1 <= 1.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(0.5 * Float64(1.0 / sqrt(t))));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_2) - sqrt(x)) - sqrt(y)) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	else
		tmp = Float64(Float64(t_2 + Float64(sqrt(Float64(1.0 + z)) + Float64(1.0 / Float64(1.0 + sqrt(x))))) - Float64(sqrt(y) + sqrt(z)));
	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((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z));
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t_1 <= 1.0)
		tmp = (sqrt((1.0 + x)) - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	elseif (t_1 <= 2.0)
		tmp = (((1.0 + t_2) - sqrt(x)) - sqrt(y)) + (sqrt((t + 1.0)) - sqrt(t));
	else
		tmp = (t_2 + (sqrt((1.0 + z)) + (1.0 / (1.0 + sqrt(x))))) - (sqrt(y) + sqrt(z));
	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[(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]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 1.0], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(N[(N[(1.0 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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

   1. Initial program 92.0%

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

   2. Taylor expanded in 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

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 35.3

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

   7. Applied rewrites 35.3%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 34.9

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

   10. Applied rewrites 34.9%

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

   if 1 < (+.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))) < 2

   1. Initial program 92.0%

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

   2. Taylor expanded in 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

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 47.7

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

   7. Applied rewrites 47.7%

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

   if 2 < (+.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 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 34.0

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

   9. Applied rewrites 34.0%

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

Alternative 10: 86.5% accurate, 0.4× speedup

Math

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

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 (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          (- (sqrt (+ z 1.0)) (sqrt z))))
        (t_2 (sqrt (+ 1.0 x))))
   (if (<= t_1 1.0)
     (+ (- t_2 (sqrt x)) (* 0.5 (/ 1.0 (sqrt t))))
     (if (<= t_1 2.0)
       (+
        (- (- (+ 1.0 (sqrt (+ 1.0 y))) (sqrt x)) (sqrt y))
        (- (sqrt (+ t 1.0)) (sqrt t)))
       (-
        (+ 1.0 (+ (sqrt (+ 1.0 z)) (/ 1.0 (+ (sqrt x) t_2))))
        (+ (sqrt y) (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (t_1 <= 1.0) {
		tmp = (t_2 - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	} else if (t_1 <= 2.0) {
		tmp = (((1.0 + sqrt((1.0 + y))) - sqrt(x)) - sqrt(y)) + (sqrt((t + 1.0)) - sqrt(t));
	} else {
		tmp = (1.0 + (sqrt((1.0 + z)) + (1.0 / (sqrt(x) + t_2)))) - (sqrt(y) + sqrt(z));
	}
	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((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))
    t_2 = sqrt((1.0d0 + x))
    if (t_1 <= 1.0d0) then
        tmp = (t_2 - sqrt(x)) + (0.5d0 * (1.0d0 / sqrt(t)))
    else if (t_1 <= 2.0d0) then
        tmp = (((1.0d0 + sqrt((1.0d0 + y))) - sqrt(x)) - sqrt(y)) + (sqrt((t + 1.0d0)) - sqrt(t))
    else
        tmp = (1.0d0 + (sqrt((1.0d0 + z)) + (1.0d0 / (sqrt(x) + t_2)))) - (sqrt(y) + sqrt(z))
    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((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_1 <= 1.0) {
		tmp = (t_2 - Math.sqrt(x)) + (0.5 * (1.0 / Math.sqrt(t)));
	} else if (t_1 <= 2.0) {
		tmp = (((1.0 + Math.sqrt((1.0 + y))) - Math.sqrt(x)) - Math.sqrt(y)) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	} else {
		tmp = (1.0 + (Math.sqrt((1.0 + z)) + (1.0 / (Math.sqrt(x) + t_2)))) - (Math.sqrt(y) + Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = ((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_2 = math.sqrt((1.0 + x))
	tmp = 0
	if t_1 <= 1.0:
		tmp = (t_2 - math.sqrt(x)) + (0.5 * (1.0 / math.sqrt(t)))
	elif t_1 <= 2.0:
		tmp = (((1.0 + math.sqrt((1.0 + y))) - math.sqrt(x)) - math.sqrt(y)) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	else:
		tmp = (1.0 + (math.sqrt((1.0 + z)) + (1.0 / (math.sqrt(x) + t_2)))) - (math.sqrt(y) + math.sqrt(z))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = 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_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= 1.0)
		tmp = Float64(Float64(t_2 - sqrt(x)) + Float64(0.5 * Float64(1.0 / sqrt(t))));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + sqrt(Float64(1.0 + y))) - sqrt(x)) - sqrt(y)) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	else
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) + Float64(1.0 / Float64(sqrt(x) + t_2)))) - Float64(sqrt(y) + sqrt(z)));
	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((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_1 <= 1.0)
		tmp = (t_2 - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	elseif (t_1 <= 2.0)
		tmp = (((1.0 + sqrt((1.0 + y))) - sqrt(x)) - sqrt(y)) + (sqrt((t + 1.0)) - sqrt(t));
	else
		tmp = (1.0 + (sqrt((1.0 + z)) + (1.0 / (sqrt(x) + t_2)))) - (sqrt(y) + sqrt(z));
	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[(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]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 1.0], N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(N[(N[(1.0 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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

   1. Initial program 92.0%

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

   2. Taylor expanded in 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

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 35.3

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

   7. Applied rewrites 35.3%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 34.9

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

   10. Applied rewrites 34.9%

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

   if 1 < (+.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))) < 2

   1. Initial program 92.0%

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

   2. Taylor expanded in 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

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 47.7

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

   7. Applied rewrites 47.7%

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

   if 2 < (+.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 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 25.1

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

   9. Applied rewrites 25.1%

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

   10. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 33.3

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

   12. Applied rewrites 33.3%

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

Alternative 11: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_1\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;t\_2 \leq 1:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + 0.5 \cdot \frac{1}{\sqrt{t}}\\ \mathbf{elif}\;t\_2 \leq 2.4:\\ \;\;\;\;\left(\left(\left(1 + t\_3\right) - \sqrt{x}\right) - \sqrt{y}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 + \left(1 + \frac{1}{1 + \sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \end{array} \end{array} \]

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 (+ t 1.0)) (sqrt t)))
        (t_2
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          t_1))
        (t_3 (sqrt (+ 1.0 y))))
   (if (<= t_2 1.0)
     (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (* 0.5 (/ 1.0 (sqrt t))))
     (if (<= t_2 2.4)
       (+ (- (- (+ 1.0 t_3) (sqrt x)) (sqrt y)) t_1)
       (- (+ t_3 (+ 1.0 (/ 1.0 (+ 1.0 (sqrt x))))) (+ (sqrt y) (sqrt z)))))))

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 t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_1;
	double t_3 = sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 1.0) {
		tmp = (sqrt((1.0 + x)) - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	} else if (t_2 <= 2.4) {
		tmp = (((1.0 + t_3) - sqrt(x)) - sqrt(y)) + t_1;
	} else {
		tmp = (t_3 + (1.0 + (1.0 / (1.0 + sqrt(x))))) - (sqrt(y) + sqrt(z));
	}
	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((t + 1.0d0)) - sqrt(t)
    t_2 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + t_1
    t_3 = sqrt((1.0d0 + y))
    if (t_2 <= 1.0d0) then
        tmp = (sqrt((1.0d0 + x)) - sqrt(x)) + (0.5d0 * (1.0d0 / sqrt(t)))
    else if (t_2 <= 2.4d0) then
        tmp = (((1.0d0 + t_3) - sqrt(x)) - sqrt(y)) + t_1
    else
        tmp = (t_3 + (1.0d0 + (1.0d0 / (1.0d0 + sqrt(x))))) - (sqrt(y) + sqrt(z))
    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 t_2 = (((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;
	double t_3 = Math.sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 1.0) {
		tmp = (Math.sqrt((1.0 + x)) - Math.sqrt(x)) + (0.5 * (1.0 / Math.sqrt(t)));
	} else if (t_2 <= 2.4) {
		tmp = (((1.0 + t_3) - Math.sqrt(x)) - Math.sqrt(y)) + t_1;
	} else {
		tmp = (t_3 + (1.0 + (1.0 / (1.0 + Math.sqrt(x))))) - (Math.sqrt(y) + Math.sqrt(z));
	}
	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)
	t_2 = (((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
	t_3 = math.sqrt((1.0 + y))
	tmp = 0
	if t_2 <= 1.0:
		tmp = (math.sqrt((1.0 + x)) - math.sqrt(x)) + (0.5 * (1.0 / math.sqrt(t)))
	elif t_2 <= 2.4:
		tmp = (((1.0 + t_3) - math.sqrt(x)) - math.sqrt(y)) + t_1
	else:
		tmp = (t_3 + (1.0 + (1.0 / (1.0 + math.sqrt(x))))) - (math.sqrt(y) + math.sqrt(z))
	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))
	t_2 = 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)
	t_3 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_2 <= 1.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(0.5 * Float64(1.0 / sqrt(t))));
	elseif (t_2 <= 2.4)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_3) - sqrt(x)) - sqrt(y)) + t_1);
	else
		tmp = Float64(Float64(t_3 + Float64(1.0 + Float64(1.0 / Float64(1.0 + sqrt(x))))) - Float64(sqrt(y) + sqrt(z)));
	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);
	t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_1;
	t_3 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t_2 <= 1.0)
		tmp = (sqrt((1.0 + x)) - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	elseif (t_2 <= 2.4)
		tmp = (((1.0 + t_3) - sqrt(x)) - sqrt(y)) + t_1;
	else
		tmp = (t_3 + (1.0 + (1.0 / (1.0 + sqrt(x))))) - (sqrt(y) + sqrt(z));
	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]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1.0], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.4], N[(N[(N[(N[(1.0 + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(t$95$3 + N[(1.0 + N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 92.0%

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

   2. 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

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 35.3

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

   7. Applied rewrites 35.3%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 34.9

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

   10. Applied rewrites 34.9%

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

   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))) < 2.39999999999999991

   1. Initial program 92.0%

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

   2. Taylor expanded in 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

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 47.7

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

   7. Applied rewrites 47.7%

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

   if 2.39999999999999991 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 25.1

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

   9. Applied rewrites 25.1%

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

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lift-sqrt.f64 25.1

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

   12. Applied rewrites 25.1%

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

Alternative 12: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;t\_2 \leq 1:\\ \;\;\;\;\left(t\_1 - \sqrt{x}\right) + 0.5 \cdot \frac{1}{\sqrt{t}}\\ \mathbf{elif}\;t\_2 \leq 2.4:\\ \;\;\;\;\left(t\_3 + \frac{1}{\sqrt{x} + t\_1}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 + \left(1 + \frac{1}{1 + \sqrt{x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \end{array} \end{array} \]

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
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_3 (sqrt (+ 1.0 y))))
   (if (<= t_2 1.0)
     (+ (- t_1 (sqrt x)) (* 0.5 (/ 1.0 (sqrt t))))
     (if (<= t_2 2.4)
       (- (+ t_3 (/ 1.0 (+ (sqrt x) t_1))) (sqrt y))
       (- (+ t_3 (+ 1.0 (/ 1.0 (+ 1.0 (sqrt x))))) (+ (sqrt y) (sqrt z)))))))

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 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_3 = sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 1.0) {
		tmp = (t_1 - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	} else if (t_2 <= 2.4) {
		tmp = (t_3 + (1.0 / (sqrt(x) + t_1))) - sqrt(y);
	} else {
		tmp = (t_3 + (1.0 + (1.0 / (1.0 + sqrt(x))))) - (sqrt(y) + sqrt(z));
	}
	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((1.0d0 + x))
    t_2 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_3 = sqrt((1.0d0 + y))
    if (t_2 <= 1.0d0) then
        tmp = (t_1 - sqrt(x)) + (0.5d0 * (1.0d0 / sqrt(t)))
    else if (t_2 <= 2.4d0) then
        tmp = (t_3 + (1.0d0 / (sqrt(x) + t_1))) - sqrt(y)
    else
        tmp = (t_3 + (1.0d0 + (1.0d0 / (1.0d0 + sqrt(x))))) - (sqrt(y) + sqrt(z))
    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 = (((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));
	double t_3 = Math.sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 1.0) {
		tmp = (t_1 - Math.sqrt(x)) + (0.5 * (1.0 / Math.sqrt(t)));
	} else if (t_2 <= 2.4) {
		tmp = (t_3 + (1.0 / (Math.sqrt(x) + t_1))) - Math.sqrt(y);
	} else {
		tmp = (t_3 + (1.0 + (1.0 / (1.0 + Math.sqrt(x))))) - (Math.sqrt(y) + Math.sqrt(z));
	}
	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 = (((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))
	t_3 = math.sqrt((1.0 + y))
	tmp = 0
	if t_2 <= 1.0:
		tmp = (t_1 - math.sqrt(x)) + (0.5 * (1.0 / math.sqrt(t)))
	elif t_2 <= 2.4:
		tmp = (t_3 + (1.0 / (math.sqrt(x) + t_1))) - math.sqrt(y)
	else:
		tmp = (t_3 + (1.0 + (1.0 / (1.0 + math.sqrt(x))))) - (math.sqrt(y) + math.sqrt(z))
	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(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)))
	t_3 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_2 <= 1.0)
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(0.5 * Float64(1.0 / sqrt(t))));
	elseif (t_2 <= 2.4)
		tmp = Float64(Float64(t_3 + Float64(1.0 / Float64(sqrt(x) + t_1))) - sqrt(y));
	else
		tmp = Float64(Float64(t_3 + Float64(1.0 + Float64(1.0 / Float64(1.0 + sqrt(x))))) - Float64(sqrt(y) + sqrt(z)));
	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 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_3 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t_2 <= 1.0)
		tmp = (t_1 - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	elseif (t_2 <= 2.4)
		tmp = (t_3 + (1.0 / (sqrt(x) + t_1))) - sqrt(y);
	else
		tmp = (t_3 + (1.0 + (1.0 / (1.0 + sqrt(x))))) - (sqrt(y) + sqrt(z));
	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[(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]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1.0], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.4], N[(N[(t$95$3 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 + N[(1.0 + N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 92.0%

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

   2. 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

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 35.3

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

   7. Applied rewrites 35.3%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 34.9

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

   10. Applied rewrites 34.9%

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

   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))) < 2.39999999999999991

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 47.8

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

   9. Applied rewrites 47.8%

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

   if 2.39999999999999991 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 25.1

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

   9. Applied rewrites 25.1%

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

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lift-sqrt.f64 25.1

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

   12. Applied rewrites 25.1%

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

Alternative 13: 65.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 2.45 \cdot 10^{+16}:\\ \;\;\;\;\left(\sqrt{1 + y} + \frac{1}{\sqrt{x} + t\_1}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \sqrt{x}\right) + 0.5 \cdot \frac{1}{\sqrt{t}}\\ \end{array} \end{array} \]

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 2.45e+16)
     (- (+ (sqrt (+ 1.0 y)) (/ 1.0 (+ (sqrt x) t_1))) (sqrt y))
     (+ (- t_1 (sqrt x)) (* 0.5 (/ 1.0 (sqrt t)))))))

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 <= 2.45e+16) {
		tmp = (sqrt((1.0 + y)) + (1.0 / (sqrt(x) + t_1))) - sqrt(y);
	} else {
		tmp = (t_1 - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	}
	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 <= 2.45d+16) then
        tmp = (sqrt((1.0d0 + y)) + (1.0d0 / (sqrt(x) + t_1))) - sqrt(y)
    else
        tmp = (t_1 - sqrt(x)) + (0.5d0 * (1.0d0 / sqrt(t)))
    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 <= 2.45e+16) {
		tmp = (Math.sqrt((1.0 + y)) + (1.0 / (Math.sqrt(x) + t_1))) - Math.sqrt(y);
	} else {
		tmp = (t_1 - Math.sqrt(x)) + (0.5 * (1.0 / Math.sqrt(t)));
	}
	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 <= 2.45e+16:
		tmp = (math.sqrt((1.0 + y)) + (1.0 / (math.sqrt(x) + t_1))) - math.sqrt(y)
	else:
		tmp = (t_1 - math.sqrt(x)) + (0.5 * (1.0 / math.sqrt(t)))
	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 <= 2.45e+16)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) + Float64(1.0 / Float64(sqrt(x) + t_1))) - sqrt(y));
	else
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(0.5 * Float64(1.0 / sqrt(t))));
	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 <= 2.45e+16)
		tmp = (sqrt((1.0 + y)) + (1.0 / (sqrt(x) + t_1))) - sqrt(y);
	else
		tmp = (t_1 - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	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, 2.45e+16], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.45e16

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 34.0%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 47.8

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

   9. Applied rewrites 47.8%

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

   if 2.45e16 < y

   1. Initial program 92.0%

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

   2. 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

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 35.3

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

   7. Applied rewrites 35.3%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 34.9

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

   10. Applied rewrites 34.9%

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

Alternative 14: 35.3% accurate, 2.1× speedup

Math

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

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 x)) (sqrt x)) (- (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 + x)) - sqrt(x)) + (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 + x)) - sqrt(x)) + (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 + x)) - Math.sqrt(x)) + (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 + x)) - math.sqrt(x)) + (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 + x)) - sqrt(x)) + 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 + x)) - sqrt(x)) + (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 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.0%

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

2. 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

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

   3. lower--.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-sqrt.f64 47.9

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

4. Applied rewrites 47.9%

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

5. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 35.3

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

7. Applied rewrites 35.3%

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

Alternative 15: 34.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{1 + x} - \sqrt{x}\right) + 0.5 \cdot \frac{1}{\sqrt{t}} \end{array} \]

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 x)) (* 0.5 (/ 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 + x)) - sqrt(x)) + (0.5 * (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 + x)) - sqrt(x)) + (0.5d0 * (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 + x)) - Math.sqrt(x)) + (0.5 * (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 + x)) - math.sqrt(x)) + (0.5 * (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 + x)) - sqrt(x)) + Float64(0.5 * 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 = (sqrt((1.0 + x)) - sqrt(x)) + (0.5 * (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 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.0%

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

2. Taylor expanded in 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

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

   3. lower--.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-sqrt.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. lift-sqrt.f64 47.9

       \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

4. Applied rewrites 47.9%

   \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Taylor expanded in y around inf

   \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lift-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lift-sqrt.f64 35.3

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

7. Applied rewrites 35.3%

   \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

8. Taylor expanded in t around inf

   \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{t}}} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{t}}} \]

   2. sqrt-div N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{1}{2} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{t}}} \]

   3. metadata-eval N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{1}{2} \cdot \frac{1}{\sqrt{\color{blue}{t}}} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \frac{1}{2} \cdot \frac{1}{\sqrt{t}} \]

   5. lift-/.f64 34.9

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + 0.5 \cdot \frac{1}{\color{blue}{\sqrt{t}}} \]

10. Applied rewrites 34.9%

    \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \color{blue}{0.5 \cdot \frac{1}{\sqrt{t}}} \]
11. Add Preprocessing

Alternative 16: 5.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{1 + z} + 0.5 \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{z} \end{array} \]

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 z)) (* 0.5 (/ 1.0 (sqrt y)))) (sqrt z)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (sqrt((1.0 + z)) + (0.5 * (1.0 / sqrt(y)))) - sqrt(z);
}

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 + z)) + (0.5d0 * (1.0d0 / sqrt(y)))) - sqrt(z)
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 + z)) + (0.5 * (1.0 / Math.sqrt(y)))) - Math.sqrt(z);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt((1.0 + z)) + (0.5 * (1.0 / math.sqrt(y)))) - math.sqrt(z)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(sqrt(Float64(1.0 + z)) + Float64(0.5 * Float64(1.0 / sqrt(y)))) - sqrt(z))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (sqrt((1.0 + z)) + (0.5 * (1.0 / sqrt(y)))) - sqrt(z);
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 + z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\color{blue}{\sqrt{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. +-commutative N/A

      \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. flip-- N/A

      \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

3. Applied rewrites 92.1%

   \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

4. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
5. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)} \]

6. Applied rewrites 34.0%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

7. Taylor expanded in y around inf

   \[\leadsto \left(\sqrt{1 + z} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) - \color{blue}{\sqrt{z}} \]
8. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + z} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) - \sqrt{z} \]

9. Applied rewrites 5.6%

   \[\leadsto \left(\sqrt{1 + z} + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) - \color{blue}{\sqrt{z}} \]

10. Taylor expanded in x around inf

    \[\leadsto \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{z} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{z} \]

    2. sqrt-div N/A

       \[\leadsto \left(\sqrt{1 + z} + \frac{1}{2} \cdot \frac{\sqrt{1}}{\sqrt{y}}\right) - \sqrt{z} \]

    3. metadata-eval N/A

       \[\leadsto \left(\sqrt{1 + z} + \frac{1}{2} \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{z} \]

    4. lift-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + z} + \frac{1}{2} \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{z} \]

    5. lift-/.f64 5.2

       \[\leadsto \left(\sqrt{1 + z} + 0.5 \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{z} \]

12. Applied rewrites 5.2%

    \[\leadsto \left(\sqrt{1 + z} + 0.5 \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{z} \]
13. Add Preprocessing

Alternative 17: 2.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(1 - \sqrt{t}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- 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 + x)) - sqrt(x)) + (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 + x)) - sqrt(x)) + (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 + x)) - Math.sqrt(x)) + (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 + x)) - math.sqrt(x)) + (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 + x)) - sqrt(x)) + 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 = (sqrt((1.0 + x)) - sqrt(x)) + (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 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.0%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

2. 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

      \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right) - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lower--.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right) - \sqrt{\color{blue}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. +-commutative N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} + \sqrt{1 + y}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} + \sqrt{1 + y}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. +-commutative N/A

      \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. lift-+.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. lift-sqrt.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. lift-sqrt.f64 47.9

       \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

4. Applied rewrites 47.9%

   \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Taylor expanded in y around inf

   \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lift-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lift-sqrt.f64 35.3

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

7. Applied rewrites 35.3%

   \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

8. Taylor expanded in t around 0

   \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{t}\right)} \]
9. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(1 - \color{blue}{\sqrt{t}}\right) \]

   2. lift-sqrt.f64 2.3

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \left(1 - \sqrt{t}\right) \]

10. Applied rewrites 2.3%

    \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{t}\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(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))))