Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.8% → 99.3%

Time: 7.5s

Alternatives: 10

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))

C

double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

Python

def code(x, y):
	return (x * ((x / y) + 1.0)) / (x + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * ((x / y) + 1.0)) / (x + 1.0);
end

Wolfram

code[x_, y_] := N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))

C

double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

Python

def code(x, y):
	return (x * ((x / y) + 1.0)) / (x + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * ((x / y) + 1.0)) / (x + 1.0);
end

Wolfram

code[x_, y_] := N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+91} \lor \neg \left(x \leq 110000000\right):\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.1e+91) (not (<= x 110000000.0)))
   (- 1.0 (/ (- 1.0 x) y))
   (/ (fma (/ x y) x x) (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.1e+91) || !(x <= 110000000.0)) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else {
		tmp = fma((x / y), x, x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.1e+91) || !(x <= 110000000.0))
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.1e+91], N[Not[LessEqual[x, 110000000.0]], $MachinePrecision]], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.10000000000000013e91 or 1.1e8 < x

   1. Initial program 72.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1 - x}{y}} \]
   7. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - 1 \cdot \frac{1 - x}{y} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{-1}{-1} \cdot \frac{1 - x}{y} \]

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

         \[\leadsto 1 - \frac{1 - x}{y} \]

      8. lower--.f64 N/A

         \[\leadsto 1 - \frac{1 - x}{\color{blue}{y}} \]

      9. lower-/.f64 N/A

         \[\leadsto 1 - \frac{1 - x}{y} \]

      10. lower--.f64 100.0

          \[\leadsto 1 - \frac{1 - x}{y} \]

   8. Applied rewrites 100.0%

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   if -5.10000000000000013e91 < x < 1.1e8

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + 1 \cdot x}}{x + 1} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x + \color{blue}{x}}{x + 1} \]

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+91} \lor \neg \left(x \leq 110000000\right):\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ t_2 := \frac{x - 1}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+154}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -20000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+20}:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+172}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x x) (fma y x y)))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (- x -1.0)))
        (t_2 (/ (- x 1.0) y)))
   (if (<= t_1 -1e+154)
     t_2
     (if (<= t_1 -20000000000000.0)
       t_0
       (if (<= t_1 1e+20) (/ x (- x -1.0)) (if (<= t_1 4e+172) t_0 t_2))))))

C

double code(double x, double y) {
	double t_0 = (x * x) / fma(y, x, y);
	double t_1 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double t_2 = (x - 1.0) / y;
	double tmp;
	if (t_1 <= -1e+154) {
		tmp = t_2;
	} else if (t_1 <= -20000000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+20) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 4e+172) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) / fma(y, x, y))
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	t_2 = Float64(Float64(x - 1.0) / y)
	tmp = 0.0
	if (t_1 <= -1e+154)
		tmp = t_2;
	elseif (t_1 <= -20000000000000.0)
		tmp = t_0;
	elseif (t_1 <= 1e+20)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_1 <= 4e+172)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+154], t$95$2, If[LessEqual[t$95$1, -20000000000000.0], t$95$0, If[LessEqual[t$95$1, 1e+20], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+172], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1.00000000000000004e154 or 4.0000000000000003e172 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 65.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{1 - x}{y}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{1 - x}{y} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{1 - x}{y} \]

      4. lower--.f64 100.0

         \[\leadsto -\frac{1 - x}{y} \]

   8. Applied rewrites 100.0%

      \[\leadsto -\frac{1 - x}{y} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{x}{y} - \frac{1}{\color{blue}{y}} \]
   10. Step-by-step derivation

       1. sub-div N/A

          \[\leadsto \frac{x - 1}{y} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{x - 1}{y} \]

       3. lower--.f64 100.0

          \[\leadsto \frac{x - 1}{y} \]

   11. Applied rewrites 100.0%

       \[\leadsto \frac{x - 1}{y} \]

   if -1.00000000000000004e154 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e13 or 1e20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.0000000000000003e172

   1. Initial program 99.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{x}^{2}}{\color{blue}{y \cdot \left(1 + x\right)}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

         \[\leadsto \frac{x \cdot x}{y \cdot x + \color{blue}{y \cdot 1}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{x \cdot x}{y \cdot x + y} \]

      7. lower-fma.f64 83.9

         \[\leadsto \frac{x \cdot x}{\mathsf{fma}\left(y, \color{blue}{x}, y\right)} \]

   5. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}} \]

   if -2e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e20

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x}}{x + 1} \]
   4. Step-by-step derivation

   5. Applied rewrites 85.2%

      \[\leadsto \frac{\color{blue}{x}}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -20000000000000:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 10^{+20}:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 4 \cdot 10^{+172}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;t\_0 \leq -20000000000000 \lor \neg \left(t\_0 \leq 10^{+20}\right):\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (<= t_0 -1e+154)
     (/ (- x 1.0) y)
     (if (or (<= t_0 -20000000000000.0) (not (<= t_0 1e+20)))
       (* x (/ x (fma y x y)))
       (/ x (- x -1.0))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -1e+154) {
		tmp = (x - 1.0) / y;
	} else if ((t_0 <= -20000000000000.0) || !(t_0 <= 1e+20)) {
		tmp = x * (x / fma(y, x, y));
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_0 <= -1e+154)
		tmp = Float64(Float64(x - 1.0) / y);
	elseif ((t_0 <= -20000000000000.0) || !(t_0 <= 1e+20))
		tmp = Float64(x * Float64(x / fma(y, x, y)));
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+154], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], If[Or[LessEqual[t$95$0, -20000000000000.0], N[Not[LessEqual[t$95$0, 1e+20]], $MachinePrecision]], N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1.00000000000000004e154

   1. Initial program 59.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{1 - x}{y}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{1 - x}{y} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{1 - x}{y} \]

      4. lower--.f64 100.0

         \[\leadsto -\frac{1 - x}{y} \]

   8. Applied rewrites 100.0%

      \[\leadsto -\frac{1 - x}{y} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{x}{y} - \frac{1}{\color{blue}{y}} \]
   10. Step-by-step derivation

       1. sub-div N/A

          \[\leadsto \frac{x - 1}{y} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{x - 1}{y} \]

       3. lower--.f64 100.0

          \[\leadsto \frac{x - 1}{y} \]

   11. Applied rewrites 100.0%

       \[\leadsto \frac{x - 1}{y} \]

   if -1.00000000000000004e154 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e13 or 1e20 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. rgt-mult-inverse N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

          \[\leadsto \frac{\color{blue}{{x}^{2}} \cdot \left(\frac{1}{x} + \frac{1}{y}\right)}{x + 1} \]

      11. frac-add N/A

          \[\leadsto \frac{{x}^{2} \cdot \color{blue}{\frac{1 \cdot y + x \cdot 1}{x \cdot y}}}{x + 1} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot \left(1 \cdot y + x \cdot 1\right)}{x \cdot y}}}{x + 1} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot \left(1 \cdot y + x \cdot 1\right)}{x \cdot y}}}{x + 1} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot \left(1 \cdot y + x \cdot 1\right)}}{x \cdot y}}{x + 1} \]

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. *-lft-identity N/A

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

      18. *-rgt-identity N/A

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

      19. lower-+.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 66.7

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot x}{y + \color{blue}{x \cdot y}} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot x}{y + y \cdot \color{blue}{x}} \]

      6. +-commutative N/A

         \[\leadsto \frac{x \cdot x}{y \cdot x + \color{blue}{y}} \]

      7. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{x}{y \cdot x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{x}{y \cdot x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto x \cdot \frac{x}{\color{blue}{y \cdot x + y}} \]

      10. lift-fma.f64 90.8

          \[\leadsto x \cdot \frac{x}{\mathsf{fma}\left(y, \color{blue}{x}, y\right)} \]

   7. Applied rewrites 90.8%

      \[\leadsto \color{blue}{x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}} \]

   if -2e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e20

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x}}{x + 1} \]
   4. Step-by-step derivation

   5. Applied rewrites 85.2%

      \[\leadsto \frac{\color{blue}{x}}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -20000000000000 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 10^{+20}\right):\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ t_1 := \frac{x - 1}{y}\\ \mathbf{if}\;t\_0 \leq -20000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))) (t_1 (/ (- x 1.0) y)))
   (if (<= t_0 -20000000000000.0)
     t_1
     (if (<= t_0 0.001) x (if (<= t_0 2.0) 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double t_1 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -20000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.001) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x - (-1.0d0))
    t_1 = (x - 1.0d0) / y
    if (t_0 <= (-20000000000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.001d0) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double t_1 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -20000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.001) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0)
	t_1 = (x - 1.0) / y
	tmp = 0
	if t_0 <= -20000000000000.0:
		tmp = t_1
	elif t_0 <= 0.001:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	t_1 = Float64(Float64(x - 1.0) / y)
	tmp = 0.0
	if (t_0 <= -20000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 0.001)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	t_1 = (x - 1.0) / y;
	tmp = 0.0;
	if (t_0 <= -20000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 0.001)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -20000000000000.0], t$95$1, If[LessEqual[t$95$0, 0.001], x, If[LessEqual[t$95$0, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e13 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 77.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{1 - x}{y}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{1 - x}{y} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{1 - x}{y} \]

      4. lower--.f64 85.7

         \[\leadsto -\frac{1 - x}{y} \]

   8. Applied rewrites 85.7%

      \[\leadsto -\frac{1 - x}{y} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{x}{y} - \frac{1}{\color{blue}{y}} \]
   10. Step-by-step derivation

       1. sub-div N/A

          \[\leadsto \frac{x - 1}{y} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{x - 1}{y} \]

       3. lower--.f64 85.7

          \[\leadsto \frac{x - 1}{y} \]

   11. Applied rewrites 85.7%

       \[\leadsto \frac{x - 1}{y} \]

   if -2e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e-3

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{x} \]

   if 1e-3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 98.9%

      \[\leadsto 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -20000000000000:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 0.001:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -20000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (<= t_0 -20000000000000.0)
     (/ x y)
     (if (<= t_0 0.001) x (if (<= t_0 2.0) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -20000000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.001) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x - (-1.0d0))
    if (t_0 <= (-20000000000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 0.001d0) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -20000000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.001) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0)
	tmp = 0
	if t_0 <= -20000000000000.0:
		tmp = x / y
	elif t_0 <= 0.001:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_0 <= -20000000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.001)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	tmp = 0.0;
	if (t_0 <= -20000000000000.0)
		tmp = x / y;
	elseif (t_0 <= 0.001)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20000000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.001], x, If[LessEqual[t$95$0, 2.0], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e13 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 77.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   4. Step-by-step derivation

      1. lift-/.f64 85.0

         \[\leadsto \frac{x}{\color{blue}{y}} \]

   5. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if -2e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e-3

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{x} \]

   if 1e-3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 98.9%

      \[\leadsto 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -20000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 0.001:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -20000000000000 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (or (<= t_0 -20000000000000.0) (not (<= t_0 2.0)))
     (/ (- x 1.0) y)
     (/ x (- x -1.0)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if ((t_0 <= -20000000000000.0) || !(t_0 <= 2.0)) {
		tmp = (x - 1.0) / y;
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x - (-1.0d0))
    if ((t_0 <= (-20000000000000.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = (x - 1.0d0) / y
    else
        tmp = x / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if ((t_0 <= -20000000000000.0) || !(t_0 <= 2.0)) {
		tmp = (x - 1.0) / y;
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0)
	tmp = 0
	if (t_0 <= -20000000000000.0) or not (t_0 <= 2.0):
		tmp = (x - 1.0) / y
	else:
		tmp = x / (x - -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if ((t_0 <= -20000000000000.0) || !(t_0 <= 2.0))
		tmp = Float64(Float64(x - 1.0) / y);
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	tmp = 0.0;
	if ((t_0 <= -20000000000000.0) || ~((t_0 <= 2.0)))
		tmp = (x - 1.0) / y;
	else
		tmp = x / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -20000000000000.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e13 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 77.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{1 - x}{y}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{1 - x}{y} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{1 - x}{y} \]

      4. lower--.f64 85.7

         \[\leadsto -\frac{1 - x}{y} \]

   8. Applied rewrites 85.7%

      \[\leadsto -\frac{1 - x}{y} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{x}{y} - \frac{1}{\color{blue}{y}} \]
   10. Step-by-step derivation

       1. sub-div N/A

          \[\leadsto \frac{x - 1}{y} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{x - 1}{y} \]

       3. lower--.f64 85.7

          \[\leadsto \frac{x - 1}{y} \]

   11. Applied rewrites 85.7%

       \[\leadsto \frac{x - 1}{y} \]

   if -2e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x}}{x + 1} \]
   4. Step-by-step derivation

   5. Applied rewrites 85.8%

      \[\leadsto \frac{\color{blue}{x}}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -20000000000000 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 2\right):\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.25)))
   (- 1.0 (/ (- 1.0 x) y))
   (/ (fma (/ x y) x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.25)) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else {
		tmp = fma((x / y), x, x) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.25))
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(fma(Float64(x / y), x, x) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.25]], $MachinePrecision]], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.25 < x

   1. Initial program 78.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1 - x}{y}} \]
   7. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - 1 \cdot \frac{1 - x}{y} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{-1}{-1} \cdot \frac{1 - x}{y} \]

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

         \[\leadsto 1 - \frac{1 - x}{y} \]

      8. lower--.f64 N/A

         \[\leadsto 1 - \frac{1 - x}{\color{blue}{y}} \]

      9. lower-/.f64 N/A

         \[\leadsto 1 - \frac{1 - x}{y} \]

      10. lower--.f64 99.9

          \[\leadsto 1 - \frac{1 - x}{y} \]

   8. Applied rewrites 99.9%

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   if -1 < x < 1.25

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + 1 \cdot x}}{x + 1} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x + \color{blue}{x}}{x + 1} \]

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 0.001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (- x -1.0)) 0.001) x 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x - -1.0)) <= 0.001) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * ((x / y) + 1.0d0)) / (x - (-1.0d0))) <= 0.001d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x - -1.0)) <= 0.001) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x - -1.0)) <= 0.001:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0)) <= 0.001)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) + 1.0)) / (x - -1.0)) <= 0.001)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], 0.001], x, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e-3

   1. Initial program 89.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 50.4%

      \[\leadsto \color{blue}{x} \]

   if 1e-3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 31.3%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 0.001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -31000 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -31000.0) (not (<= x 1.15)))
   (- 1.0 (/ (- 1.0 x) y))
   (/ x (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -31000.0) || !(x <= 1.15)) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-31000.0d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = 1.0d0 - ((1.0d0 - x) / y)
    else
        tmp = x / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -31000.0) || !(x <= 1.15)) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -31000.0) or not (x <= 1.15):
		tmp = 1.0 - ((1.0 - x) / y)
	else:
		tmp = x / (x - -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -31000.0) || !(x <= 1.15))
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -31000.0) || ~((x <= 1.15)))
		tmp = 1.0 - ((1.0 - x) / y);
	else
		tmp = x / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -31000.0], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -31000 or 1.1499999999999999 < x

   1. Initial program 78.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1 - x}{y}} \]
   7. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - 1 \cdot \frac{1 - x}{y} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{-1}{-1} \cdot \frac{1 - x}{y} \]

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

         \[\leadsto 1 - \frac{1 - x}{y} \]

      8. lower--.f64 N/A

         \[\leadsto 1 - \frac{1 - x}{\color{blue}{y}} \]

      9. lower-/.f64 N/A

         \[\leadsto 1 - \frac{1 - x}{y} \]

      10. lower--.f64 99.3

          \[\leadsto 1 - \frac{1 - x}{y} \]

   8. Applied rewrites 99.3%

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   if -31000 < x < 1.1499999999999999

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x}}{x + 1} \]
   4. Step-by-step derivation

   5. Applied rewrites 72.3%

      \[\leadsto \frac{\color{blue}{x}}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -31000 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.2% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 89.0%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Applied rewrites 36.0%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0))))

C

double code(double x, double y) {
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / 1.0d0) * (((x / y) + 1.0d0) / (x + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
}

Python

def code(x, y):
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x / 1.0) * Float64(Float64(Float64(x / y) + 1.0) / Float64(x + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x / 1.0), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025051 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1))))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))