Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.8% → 99.7%

Time: 2.9s

Alternatives: 9

Speedup: 1.0×

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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{1 - x}{y}, -1, 1\right)\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ (- 1.0 x) y) -1.0 1.0)))
   (if (<= x -4.3e+70)
     t_0
     (if (<= x 2e+16) (/ (fma (/ x y) x x) (- x -1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(((1.0 - x) / y), -1.0, 1.0);
	double tmp;
	if (x <= -4.3e+70) {
		tmp = t_0;
	} else if (x <= 2e+16) {
		tmp = fma((x / y), x, x) / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(1.0 - x) / y), -1.0, 1.0)
	tmp = 0.0
	if (x <= -4.3e+70)
		tmp = t_0;
	elseif (x <= 2e+16)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]}, If[LessEqual[x, -4.3e+70], t$95$0, If[LessEqual[x, 2e+16], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3000000000000001e70 or 2e16 < x

   1. Initial program 72.5%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{x \cdot y} - \left(\frac{1}{x} + \frac{1}{y}\right)\right)\right)} \]
   3. 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 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -4.3000000000000001e70 < x < 2e16

   1. Initial program 99.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. 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 99.4

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

      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 99.4

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

   3. Applied rewrites 99.4%

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

Alternative 2: 87.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, -1, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -2e+182)
     (/ x y)
     (if (<= t_0 -5e-12)
       (* x (/ x (fma y x y)))
       (if (<= t_0 0.5) (/ x (+ x 1.0)) (fma (/ (- 1.0 x) y) -1.0 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 <= -2e+182) {
		tmp = x / y;
	} else if (t_0 <= -5e-12) {
		tmp = x * (x / fma(y, x, y));
	} else if (t_0 <= 0.5) {
		tmp = x / (x + 1.0);
	} else {
		tmp = fma(((1.0 - x) / y), -1.0, 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 <= -2e+182)
		tmp = Float64(x / y);
	elseif (t_0 <= -5e-12)
		tmp = Float64(x * Float64(x / fma(y, x, y)));
	elseif (t_0 <= 0.5)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = fma(Float64(Float64(1.0 - x) / y), -1.0, 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, -2e+182], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -5e-12], N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2.0000000000000001e182

   1. Initial program 55.7%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if -2.0000000000000001e182 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.9999999999999997e-12

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

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

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

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 80.0

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

   6. Applied rewrites 80.0%

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

   if -4.9999999999999997e-12 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 85.4%

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

   if 0.5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.8%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{x \cdot y} - \left(\frac{1}{x} + \frac{1}{y}\right)\right)\right)} \]
   3. 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 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 89.1

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

   7. Applied rewrites 89.1%

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

Alternative 3: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -1e+16)
     (/ x y)
     (if (<= t_0 0.5) 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 <= -1e+16) {
		tmp = x / y;
	} else if (t_0 <= 0.5) {
		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 <= (-1d+16)) then
        tmp = x / y
    else if (t_0 <= 0.5d0) 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 <= -1e+16) {
		tmp = x / y;
	} else if (t_0 <= 0.5) {
		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 <= -1e+16:
		tmp = x / y
	elif t_0 <= 0.5:
		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 <= -1e+16)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.5)
		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 <= -1e+16)
		tmp = x / y;
	elseif (t_0 <= 0.5)
		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, -1e+16], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.5], 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))) < -1e16 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.9%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 85.5

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

   4. Applied rewrites 85.5%

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

   if -1e16 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 82.5%

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

   if 0.5 < (/.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. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{x \cdot y} - \left(\frac{1}{x} + \frac{1}{y}\right)\right)\right)} \]
   3. 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 46.9

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 91.4%

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

Alternative 4: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -1e+16) (/ x y) (if (<= t_0 2.0) (/ x (+ x 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 <= -1e+16) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 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 <= (-1d+16)) then
        tmp = x / y
    else if (t_0 <= 2.0d0) then
        tmp = x / (x + 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 <= -1e+16) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 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 <= -1e+16:
		tmp = x / y
	elif t_0 <= 2.0:
		tmp = x / (x + 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 <= -1e+16)
		tmp = Float64(x / y);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 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 <= -1e+16)
		tmp = x / y;
	elseif (t_0 <= 2.0)
		tmp = x / (x + 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, -1e+16], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $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))) < -1e16 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.9%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 85.5

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

   4. Applied rewrites 85.5%

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

   if -1e16 < (/.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. Taylor expanded in x around 0

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

   4. Applied rewrites 85.9%

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

Alternative 5: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{1 - x}{y}, -1, 1\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ (- 1.0 x) y) -1.0 1.0)))
   (if (<= x -1.0) t_0 (if (<= x 1.25) (/ (fma (/ x y) x x) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(((1.0 - x) / y), -1.0, 1.0);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.25) {
		tmp = fma((x / y), x, x) / 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(1.0 - x) / y), -1.0, 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.25)
		tmp = Float64(fma(Float64(x / y), x, x) / 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.25], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.25 < x

   1. Initial program 75.9%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{x \cdot y} - \left(\frac{1}{x} + \frac{1}{y}\right)\right)\right)} \]
   3. 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 72.8

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 98.2

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

   7. Applied rewrites 98.2%

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

   if -1 < x < 1.25

   1. Initial program 99.9%

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

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 98.2%

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

Alternative 6: 49.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 0.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) 0.5) x 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.5) {
		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)) <= 0.5d0) 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)) <= 0.5) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.5:
		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)) <= 0.5)
		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)) <= 0.5)
		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], 0.5], 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))) < 0.5

   1. Initial program 90.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 57.4%

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

   if 0.5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 82.8%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{x \cdot y} - \left(\frac{1}{x} + \frac{1}{y}\right)\right)\right)} \]
   3. 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 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 35.2%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

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(x * Float64(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[(x * N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.8%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. +-commutative N/A

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

   17. metadata-eval N/A

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

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

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower--.f64 99.8

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

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} - -1}{x - -1}} \]
4. Add Preprocessing

Alternative 8: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{1 - x}{y}, -1, 1\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ (- 1.0 x) y) -1.0 1.0)))
   (if (<= x -1e+29) t_0 (if (<= x 1.55e+16) (/ x (+ x 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(((1.0 - x) / y), -1.0, 1.0);
	double tmp;
	if (x <= -1e+29) {
		tmp = t_0;
	} else if (x <= 1.55e+16) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(1.0 - x) / y), -1.0, 1.0)
	tmp = 0.0
	if (x <= -1e+29)
		tmp = t_0;
	elseif (x <= 1.55e+16)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]}, If[LessEqual[x, -1e+29], t$95$0, If[LessEqual[x, 1.55e+16], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999914e28 or 1.55e16 < x

   1. Initial program 74.2%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{x \cdot y} - \left(\frac{1}{x} + \frac{1}{y}\right)\right)\right)} \]
   3. 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 72.7

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

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -9.99999999999999914e28 < x < 1.55e16

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 73.3%

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

Alternative 9: 38.7% 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 87.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 38.7%

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

Developer Target 1: 99.8% 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 2025095 
(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)))