Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.0% → 99.9%

Time: 5.5s

Alternatives: 15

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: 88.0% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+179} \lor \neg \left(t\_0 \leq 10^{+15}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{x - -1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -4e+179) (not (<= t_0 1e+15)))
     (* (/ (/ x y) (- x -1.0)) x)
     (/ (fma (/ x y) x 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 <= -4e+179) || !(t_0 <= 1e+15)) {
		tmp = ((x / y) / (x - -1.0)) * x;
	} else {
		tmp = fma((x / y), x, 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 <= -4e+179) || !(t_0 <= 1e+15))
		tmp = Float64(Float64(Float64(x / y) / Float64(x - -1.0)) * x);
	else
		tmp = Float64(fma(Float64(x / y), x, 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[Or[LessEqual[t$95$0, -4e+179], N[Not[LessEqual[t$95$0, 1e+15]], $MachinePrecision]], N[(N[(N[(x / y), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] * x), $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 (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -3.99999999999999992e179 or 1e15 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.9%

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

   if -3.99999999999999992e179 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e15

   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. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 99.9

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -4 \cdot 10^{+179} \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 10^{+15}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{x - -1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{t\_1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (fma (/ x y) x x)))
   (if (<= t_0 (- INFINITY))
     (/ x y)
     (if (<= t_0 -2e+15)
       (/ (* (+ y x) x) (fma y x y))
       (if (<= t_0 2e-19) t_1 (if (<= t_0 2e+22) (/ t_1 x) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = fma((x / y), x, x);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= -2e+15) {
		tmp = ((y + x) * x) / fma(y, x, y);
	} else if (t_0 <= 2e-19) {
		tmp = t_1;
	} else if (t_0 <= 2e+22) {
		tmp = t_1 / x;
	} 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))
	t_1 = fma(Float64(x / y), x, x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= -2e+15)
		tmp = Float64(Float64(Float64(y + x) * x) / fma(y, x, y));
	elseif (t_0 <= 2e-19)
		tmp = t_1;
	elseif (t_0 <= 2e+22)
		tmp = Float64(t_1 / x);
	else
		tmp = Float64(x / y);
	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]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -2e+15], N[(N[(N[(y + x), $MachinePrecision] * x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-19], t$95$1, If[LessEqual[t$95$0, 2e+22], N[(t$95$1 / x), $MachinePrecision], N[(x / y), $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))) < -inf.0 or 2e22 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 61.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 94.5%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e15

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.8

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.6%

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

   if -2e15 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e-19

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 99.9%

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

   if 2e-19 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e22

   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

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

   5. Applied rewrites 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 91.4

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

   7. Applied rewrites 91.4%

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

Alternative 3: 93.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 (- INFINITY))
     (/ x y)
     (if (<= t_0 -2e+15)
       (/ (* (+ y x) x) (fma y x y))
       (if (<= t_0 2e-19)
         (fma (/ x y) x x)
         (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 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= -2e+15) {
		tmp = ((y + x) * x) / fma(y, x, y);
	} else if (t_0 <= 2e-19) {
		tmp = fma((x / y), x, x);
	} else if (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 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= -2e+15)
		tmp = Float64(Float64(Float64(y + x) * x) / fma(y, x, y));
	elseif (t_0 <= 2e-19)
		tmp = fma(Float64(x / y), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	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, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -2e+15], N[(N[(N[(y + x), $MachinePrecision] * x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-19], N[(N[(x / y), $MachinePrecision] * x + x), $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 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 91.8%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e15

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.8

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.6%

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

   if -2e15 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e-19

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 99.9%

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

   if 2e-19 < (/.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 0

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

   5. Applied rewrites 97.2%

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

Alternative 4: 93.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 (- INFINITY))
     (/ x y)
     (if (<= t_0 -2e+15)
       (/ (* x x) (fma y x y))
       (if (<= t_0 2e-19)
         (fma (/ x y) x x)
         (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 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= -2e+15) {
		tmp = (x * x) / fma(y, x, y);
	} else if (t_0 <= 2e-19) {
		tmp = fma((x / y), x, x);
	} else if (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 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= -2e+15)
		tmp = Float64(Float64(x * x) / fma(y, x, y));
	elseif (t_0 <= 2e-19)
		tmp = fma(Float64(x / y), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	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, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -2e+15], N[(N[(x * x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-19], N[(N[(x / y), $MachinePrecision] * x + x), $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 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 91.8%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2e15

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.8

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.4%

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

   if -2e15 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e-19

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 99.9%

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

   if 2e-19 < (/.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 0

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

   5. Applied rewrites 97.2%

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

Alternative 5: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{y}}{x - -1} \cdot x\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_2 := \mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+15}:\\ \;\;\;\;\frac{t\_2}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (/ x y) (- x -1.0)) x))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))
        (t_2 (fma (/ x y) x x)))
   (if (<= t_1 -2e+15)
     t_0
     (if (<= t_1 2e-19) t_2 (if (<= t_1 1e+15) (/ t_2 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = ((x / y) / (x - -1.0)) * x;
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_2 = fma((x / y), x, x);
	double tmp;
	if (t_1 <= -2e+15) {
		tmp = t_0;
	} else if (t_1 <= 2e-19) {
		tmp = t_2;
	} else if (t_1 <= 1e+15) {
		tmp = t_2 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x / y) / Float64(x - -1.0)) * x)
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_2 = fma(Float64(x / y), x, x)
	tmp = 0.0
	if (t_1 <= -2e+15)
		tmp = t_0;
	elseif (t_1 <= 2e-19)
		tmp = t_2;
	elseif (t_1 <= 1e+15)
		tmp = Float64(t_2 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x / y), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] * x), $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 / y), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+15], t$95$0, If[LessEqual[t$95$1, 2e-19], t$95$2, If[LessEqual[t$95$1, 1e+15], N[(t$95$2 / x), $MachinePrecision], t$95$0]]]]]]

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))) < -2e15 or 1e15 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 70.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.9%

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

   if -2e15 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e-19

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 99.9%

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

   if 2e-19 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1e15

   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

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

   5. Applied rewrites 97.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 97.5

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

   7. Applied rewrites 97.5%

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

Alternative 6: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -4e+78)
     (/ x y)
     (if (<= t_0 2e-19)
       (fma (- (/ x y) x) x x)
       (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 <= -4e+78) {
		tmp = x / y;
	} else if (t_0 <= 2e-19) {
		tmp = fma(((x / y) - x), x, x);
	} else if (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 <= -4e+78)
		tmp = Float64(x / y);
	elseif (t_0 <= 2e-19)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	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, -4e+78], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2e-19], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $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 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000003e78 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 68.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 86.9%

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

   if -4.00000000000000003e78 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e-19

   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

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.8%

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

   if 2e-19 < (/.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 0

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

   5. Applied rewrites 97.2%

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

Alternative 7: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -4e+78)
     (/ x y)
     (if (<= t_0 2e-19)
       (fma (/ x y) x x)
       (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 <= -4e+78) {
		tmp = x / y;
	} else if (t_0 <= 2e-19) {
		tmp = fma((x / y), x, x);
	} else if (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 <= -4e+78)
		tmp = Float64(x / y);
	elseif (t_0 <= 2e-19)
		tmp = fma(Float64(x / y), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	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, -4e+78], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2e-19], N[(N[(x / y), $MachinePrecision] * x + x), $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 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000003e78 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 68.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 86.9%

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

   if -4.00000000000000003e78 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e-19

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 73.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.8%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 97.8%

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

   if 2e-19 < (/.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 0

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

   5. Applied rewrites 97.2%

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

Alternative 8: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y} \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -4e+78)
     (/ x y)
     (if (<= t_0 -5e-33)
       (* (/ x y) x)
       (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 <= -4e+78) {
		tmp = x / y;
	} else if (t_0 <= -5e-33) {
		tmp = (x / y) * x;
	} 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 <= (-4d+78)) then
        tmp = x / y
    else if (t_0 <= (-5d-33)) then
        tmp = (x / y) * x
    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 <= -4e+78) {
		tmp = x / y;
	} else if (t_0 <= -5e-33) {
		tmp = (x / y) * x;
	} 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 <= -4e+78:
		tmp = x / y
	elif t_0 <= -5e-33:
		tmp = (x / y) * x
	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 <= -4e+78)
		tmp = Float64(x / y);
	elseif (t_0 <= -5e-33)
		tmp = Float64(Float64(x / y) * x);
	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 <= -4e+78)
		tmp = x / y;
	elseif (t_0 <= -5e-33)
		tmp = (x / y) * x;
	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, -4e+78], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, -5e-33], N[(N[(x / y), $MachinePrecision] * x), $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 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -4.00000000000000003e78 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 68.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 86.9%

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

   if -4.00000000000000003e78 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000028e-33

   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

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 76.0%

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

   if -5.00000000000000028e-33 < (/.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 89.2%

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

Alternative 9: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -2e-6)
     (/ x y)
     (if (<= t_0 5e-22) x (if (<= t_0 2.0) (/ x x) (/ 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 <= -2e-6) {
		tmp = x / y;
	} else if (t_0 <= 5e-22) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = x / x;
	} 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 <= (-2d-6)) then
        tmp = x / y
    else if (t_0 <= 5d-22) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = x / x
    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 <= -2e-6) {
		tmp = x / y;
	} else if (t_0 <= 5e-22) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = x / x;
	} 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 <= -2e-6:
		tmp = x / y
	elif t_0 <= 5e-22:
		tmp = x
	elif t_0 <= 2.0:
		tmp = x / x
	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 <= -2e-6)
		tmp = Float64(x / y);
	elseif (t_0 <= 5e-22)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = Float64(x / x);
	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 <= -2e-6)
		tmp = x / y;
	elseif (t_0 <= 5e-22)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = x / x;
	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, -2e-6], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e-22], x, If[LessEqual[t$95$0, 2.0], N[(x / x), $MachinePrecision], 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))) < -1.99999999999999991e-6 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 81.0%

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

   if -1.99999999999999991e-6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.99999999999999954e-22

   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 87.7%

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

   if 4.99999999999999954e-22 < (/.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 93.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 90.9%

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

Alternative 10: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-6} \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -2e-6) (not (<= t_0 2.0))) (/ 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 <= -2e-6) || !(t_0 <= 2.0)) {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if ((t_0 <= (-2d-6)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = x / 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 <= -2e-6) || !(t_0 <= 2.0)) {
		tmp = x / 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 <= -2e-6) or not (t_0 <= 2.0):
		tmp = 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 <= -2e-6) || !(t_0 <= 2.0))
		tmp = Float64(x / 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 <= -2e-6) || ~((t_0 <= 2.0)))
		tmp = x / 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, -2e-6], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x / 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))) < -1.99999999999999991e-6 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 81.0%

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

   if -1.99999999999999991e-6 < (/.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 88.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -2 \cdot 10^{-6} \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-6} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -2e-6) (not (<= t_0 1.0))) (/ x y) x)))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -2e-6) || !(t_0 <= 1.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	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 <= (-2d-6)) .or. (.not. (t_0 <= 1.0d0))) then
        tmp = x / y
    else
        tmp = x
    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 <= -2e-6) || !(t_0 <= 1.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if (t_0 <= -2e-6) or not (t_0 <= 1.0):
		tmp = x / y
	else:
		tmp = x
	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-6) || !(t_0 <= 1.0))
		tmp = Float64(x / y);
	else
		tmp = x;
	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 <= -2e-6) || ~((t_0 <= 1.0)))
		tmp = x / y;
	else
		tmp = x;
	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, -2e-6], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]]

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))) < -1.99999999999999991e-6 or 1 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 72.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 79.7%

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

   if -1.99999999999999991e-6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1

   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 74.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -2 \cdot 10^{-6} \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 1\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) -2e-6) (* (- x) x) x))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -2e-6) {
		tmp = -x * x;
	} else {
		tmp = x;
	}
	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)) <= (-2d-6)) then
        tmp = -x * x
    else
        tmp = x
    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)) <= -2e-6) {
		tmp = -x * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= -2e-6:
		tmp = -x * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0)) <= -2e-6)
		tmp = Float64(Float64(-x) * x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -2e-6)
		tmp = -x * x;
	else
		tmp = x;
	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], -2e-6], N[((-x) * x), $MachinePrecision], x]

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))) < -1.99999999999999991e-6

   1. Initial program 71.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 23.1%

      \[\leadsto \left(1 - x\right) \cdot x \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 23.5%

       \[\leadsto \left(-x\right) \cdot x \]

   if -1.99999999999999991e-6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 92.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.3%

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

Alternative 13: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (((x / y) - -1.0) / (x - -1.0)) * 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 / y) - (-1.0d0)) / (x - (-1.0d0))) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

   7. lift-+.f64 N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval 99.9

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

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

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower--.f64 N/A

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

   22. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 14: 43.6% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (- 1.0 x) x))

C

double code(double x, double y) {
	return (1.0 - x) * 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 = (1.0d0 - x) * x
end function

Java

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

Python

def code(x, y):
	return (1.0 - x) * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 64.6%

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

6. Taylor expanded in y around inf

   \[\leadsto \left(1 + -1 \cdot x\right) \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 48.2%

   \[\leadsto \left(1 - x\right) \cdot x \]
9. Add Preprocessing

Alternative 15: 39.3% 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 88.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 44.4%

   \[\leadsto \color{blue}{x} \]
6. 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 2025019 
(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)))