Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.7% → 99.9%

Time: 3.3s

Alternatives: 11

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+17} \lor \neg \left(x \leq 1.8 \cdot 10^{+15}\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y} \cdot x, x - -1, \left(x - -1\right) \cdot x\right)}{\left(x - -1\right) \cdot \left(x - -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e+17) (not (<= x 1.8e+15)))
   (- 1.0 (/ (- x) y))
   (/
    (fma (* (/ x y) x) (- x -1.0) (* (- x -1.0) x))
    (* (- x -1.0) (- x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e+17) || !(x <= 1.8e+15)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = fma(((x / y) * x), (x - -1.0), ((x - -1.0) * x)) / ((x - -1.0) * (x - -1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1e+17) || !(x <= 1.8e+15))
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	else
		tmp = Float64(fma(Float64(Float64(x / y) * x), Float64(x - -1.0), Float64(Float64(x - -1.0) * x)) / Float64(Float64(x - -1.0) * Float64(x - -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1e+17], N[Not[LessEqual[x, 1.8e+15]], $MachinePrecision]], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * N[(x - -1.0), $MachinePrecision] + N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[(N[(x - -1.0), $MachinePrecision] * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e17 or 1.8e15 < x

   1. Initial program 76.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -1e17 < x < 1.8e15

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+17} \lor \neg \left(x \leq 1.8 \cdot 10^{+15}\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y} \cdot x, x - -1, \left(x - -1\right) \cdot x\right)}{\left(x - -1\right) \cdot \left(x - -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{-x}{y}\\ \end{array} \end{array} \]

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 -4e-11)
       (/ (* x x) (fma y x y))
       (if (<= t_0 0.01) (/ x (- x -1.0)) (- 1.0 (/ (- x) y)))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= -4e-11) {
		tmp = (x * x) / fma(y, x, y);
	} else if (t_0 <= 0.01) {
		tmp = x / (x - -1.0);
	} else {
		tmp = 1.0 - (-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 <= -4e-11)
		tmp = Float64(Float64(x * x) / fma(y, x, y));
	elseif (t_0 <= 0.01)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(1.0 - Float64(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, -4e-11], N[(N[(x * x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[((-x) / y), $MachinePrecision]), $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

   1. Initial program 56.0%

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

   3. Taylor expanded in x around inf

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

      1. lift-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -3.99999999999999976e-11

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -3.99999999999999976e-11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0100000000000000002

   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 90.2%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 85.5%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 90.3

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 90.6

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

   11. Applied rewrites 90.6%

       \[\leadsto 1 - \frac{-x}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 0.01:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{-x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (- (/ x y) -1.0)) (- x -1.0))))
   (if (<= t_0 -2e+22)
     (/ (- x 1.0) y)
     (if (<= t_0 0.01) x (if (<= t_0 2.0) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -2e+22) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 0.01) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) - (-1.0d0))) / (x - (-1.0d0))
    if (t_0 <= (-2d+22)) then
        tmp = (x - 1.0d0) / y
    else if (t_0 <= 0.01d0) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -2e+22) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 0.01) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) - -1.0)) / (x - -1.0)
	tmp = 0
	if t_0 <= -2e+22:
		tmp = (x - 1.0) / y
	elif t_0 <= 0.01:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) - -1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_0 <= -2e+22)
		tmp = Float64(Float64(x - 1.0) / y);
	elseif (t_0 <= 0.01)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	tmp = 0.0;
	if (t_0 <= -2e+22)
		tmp = (x - 1.0) / y;
	elseif (t_0 <= 0.01)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+22], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 0.01], x, If[LessEqual[t$95$0, 2.0], 1.0, 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))) < -2e22

   1. Initial program 73.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 84.4

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in x around 0

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

       1. sub-div N/A

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

       2. lower-/.f64 N/A

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

       3. lower--.f64 84.4

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

   11. Applied rewrites 84.4%

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

   if -2e22 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0100000000000000002

   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

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

   5. Applied rewrites 87.9%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 46.3

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 88.4%

      \[\leadsto 1 \]

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

   1. Initial program 76.4%

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

   3. Taylor expanded in x around inf

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

      1. lift-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 0.01:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (- (/ x y) -1.0)) (- x -1.0))))
   (if (<= t_0 -4e-11)
     (/ x y)
     (if (<= t_0 0.01) x (if (<= t_0 2.0) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -4e-11) {
		tmp = x / y;
	} else if (t_0 <= 0.01) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) - (-1.0d0))) / (x - (-1.0d0))
    if (t_0 <= (-4d-11)) then
        tmp = x / y
    else if (t_0 <= 0.01d0) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -4e-11) {
		tmp = x / y;
	} else if (t_0 <= 0.01) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) - -1.0)) / (x - -1.0)
	tmp = 0
	if t_0 <= -4e-11:
		tmp = x / y
	elif t_0 <= 0.01:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) - -1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_0 <= -4e-11)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.01)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	tmp = 0.0;
	if (t_0 <= -4e-11)
		tmp = x / y;
	elseif (t_0 <= 0.01)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-11], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.01], x, If[LessEqual[t$95$0, 2.0], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -3.99999999999999976e-11 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.3%

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

   3. Taylor expanded in x around inf

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

      1. lift-/.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -3.99999999999999976e-11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0100000000000000002

   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

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

   5. Applied rewrites 89.3%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 46.3

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 88.4%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 0.01:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (- (/ x y) -1.0)) (- x -1.0))))
   (if (<= t_0 -2e+22)
     (/ (- x 1.0) y)
     (if (<= t_0 2.0) (/ x (- x -1.0)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -2e+22) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) - (-1.0d0))) / (x - (-1.0d0))
    if (t_0 <= (-2d+22)) then
        tmp = (x - 1.0d0) / y
    else if (t_0 <= 2.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -2e+22) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) - -1.0)) / (x - -1.0)
	tmp = 0
	if t_0 <= -2e+22:
		tmp = (x - 1.0) / y
	elif t_0 <= 2.0:
		tmp = x / (x - -1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) - -1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_0 <= -2e+22)
		tmp = Float64(Float64(x - 1.0) / y);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	tmp = 0.0;
	if (t_0 <= -2e+22)
		tmp = (x - 1.0) / y;
	elseif (t_0 <= 2.0)
		tmp = x / (x - -1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+22], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 84.4

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in x around 0

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

       1. sub-div N/A

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

       2. lower-/.f64 N/A

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

       3. lower--.f64 84.4

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

   11. Applied rewrites 84.4%

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

   if -2e22 < (/.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 89.0%

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

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

   1. Initial program 76.4%

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

   3. Taylor expanded in x around inf

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

      1. lift-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+54} \lor \neg \left(x \leq 3.8 \cdot 10^{+15}\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7.2e+54) (not (<= x 3.8e+15)))
   (- 1.0 (/ (- x) y))
   (/ (fma (/ x y) x x) (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7.2e+54) || !(x <= 3.8e+15)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = fma((x / y), x, x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7.2e+54) || !(x <= 3.8e+15))
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	else
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7.2e+54], N[Not[LessEqual[x, 3.8e+15]], $MachinePrecision]], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2000000000000003e54 or 3.8e15 < x

   1. Initial program 75.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -7.2000000000000003e54 < x < 3.8e15

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+54} \lor \neg \left(x \leq 3.8 \cdot 10^{+15}\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{elif}\;x \leq 1.22:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (- 1.0 (/ (- x) y))
   (if (<= x 1.22) (/ (fma (/ x y) x x) 1.0) (- 1.0 (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 - (-x / y);
	} else if (x <= 1.22) {
		tmp = fma((x / y), x, x) / 1.0;
	} else {
		tmp = 1.0 - ((1.0 - x) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	elseif (x <= 1.22)
		tmp = Float64(fma(Float64(x / y), x, x) / 1.0);
	else
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / 1.0), $MachinePrecision], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 78.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.0

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 99.0

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

   11. Applied rewrites 99.0%

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

   if -1 < x < 1.21999999999999997

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.7%

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

   if 1.21999999999999997 < x

   1. Initial program 77.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.5

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

   8. Applied rewrites 99.5%

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

Alternative 8: 50.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 0.01:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (- (/ x y) -1.0)) (- x -1.0)) 0.01) x 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= 0.01) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * ((x / y) - (-1.0d0))) / (x - (-1.0d0))) <= 0.01d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= 0.01) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) - -1.0)) / (x - -1.0)) <= 0.01:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) - -1.0)) / Float64(x - -1.0)) <= 0.01)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= 0.01)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], 0.01], x, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0100000000000000002

   1. Initial program 92.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.6%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 85.5%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 37.2%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 0.01:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{elif}\;x \leq 24000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -175000000.0)
   (- 1.0 (/ (- x) y))
   (if (<= x 24000.0) (/ x (- x -1.0)) (- 1.0 (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -175000000.0) {
		tmp = 1.0 - (-x / y);
	} else if (x <= 24000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = 1.0 - ((1.0 - 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) :: tmp
    if (x <= (-175000000.0d0)) then
        tmp = 1.0d0 - (-x / y)
    else if (x <= 24000.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = 1.0d0 - ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -175000000.0) {
		tmp = 1.0 - (-x / y);
	} else if (x <= 24000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = 1.0 - ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -175000000.0:
		tmp = 1.0 - (-x / y)
	elif x <= 24000.0:
		tmp = x / (x - -1.0)
	else:
		tmp = 1.0 - ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	elseif (x <= 24000.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = 1.0 - (-x / y);
	elseif (x <= 24000.0)
		tmp = x / (x - -1.0);
	else
		tmp = 1.0 - ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -175000000.0], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 24000.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   1. Initial program 78.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -1.75e8 < x < 24000

   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 79.8%

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

   if 24000 < x

   1. Initial program 77.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.5

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

   8. Applied rewrites 99.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{elif}\;x \leq 24000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -175000000 \lor \neg \left(x \leq 90000\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -175000000.0) (not (<= x 90000.0)))
   (- 1.0 (/ (- x) y))
   (/ x (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -175000000.0) || !(x <= 90000.0)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-175000000.0d0)) .or. (.not. (x <= 90000.0d0))) then
        tmp = 1.0d0 - (-x / y)
    else
        tmp = x / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -175000000.0) || !(x <= 90000.0)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -175000000.0) or not (x <= 90000.0):
		tmp = 1.0 - (-x / y)
	else:
		tmp = x / (x - -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -175000000.0) || !(x <= 90000.0))
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -175000000.0) || ~((x <= 90000.0)))
		tmp = 1.0 - (-x / y);
	else
		tmp = x / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -175000000.0], N[Not[LessEqual[x, 90000.0]], $MachinePrecision]], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e8 or 9e4 < x

   1. Initial program 77.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. frac-add N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 99.4

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

   11. Applied rewrites 99.4%

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

   if -1.75e8 < x < 9e4

   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 79.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -175000000 \lor \neg \left(x \leq 90000\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.4% 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 90.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 44.5%

   \[\leadsto \color{blue}{x} \]
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 2025037 
(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)))