Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.4% → 99.9%

Time: 4.7s

Alternatives: 13

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: 88.4% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500000000000:\\ \;\;\;\;\frac{x - 1}{y} - -1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7500000000000.0)
   (- (/ (- x 1.0) y) -1.0)
   (if (<= x 3e+16) (/ (fma (/ x y) x x) (- x -1.0)) (- (/ x y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7500000000000.0) {
		tmp = ((x - 1.0) / y) - -1.0;
	} else if (x <= 3e+16) {
		tmp = fma((x / y), x, x) / (x - -1.0);
	} else {
		tmp = (x / y) - -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7500000000000.0)
		tmp = Float64(Float64(Float64(x - 1.0) / y) - -1.0);
	elseif (x <= 3e+16)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	else
		tmp = Float64(Float64(x / y) - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7500000000000.0], N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[x, 3e+16], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e12

   1. Initial program 76.4%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -7.5e12 < x < 3e16

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 99.8

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

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

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 99.8

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

   3. Applied rewrites 99.8%

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

   if 3e16 < x

   1. Initial program 76.0%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3400000000000:\\ \;\;\;\;\frac{x - 1}{y} - -1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{y + x}{\mathsf{fma}\left(y, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3400000000000.0)
   (- (/ (- x 1.0) y) -1.0)
   (if (<= x 6.5e+14) (* x (/ (+ y x) (fma y x y))) (- (/ x y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3400000000000.0) {
		tmp = ((x - 1.0) / y) - -1.0;
	} else if (x <= 6.5e+14) {
		tmp = x * ((y + x) / fma(y, x, y));
	} else {
		tmp = (x / y) - -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3400000000000.0)
		tmp = Float64(Float64(Float64(x - 1.0) / y) - -1.0);
	elseif (x <= 6.5e+14)
		tmp = Float64(x * Float64(Float64(y + x) / fma(y, x, y)));
	else
		tmp = Float64(Float64(x / y) - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3400000000000.0], N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[x, 6.5e+14], N[(x * N[(N[(y + x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e12

   1. Initial program 76.4%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -3.4e12 < x < 6.5e14

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   if 6.5e14 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

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

Alternative 3: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} - -1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ (- x 1.0) y) -1.0)))
   (if (<= x -1.0) t_0 (if (<= x 1.3) (/ (fma (/ x y) x x) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) - -1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.3) {
		tmp = fma((x / y), x, x) / 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) - -1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.3)
		tmp = Float64(fma(Float64(x / y), x, x) / 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.3], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.30000000000000004 < x

   1. Initial program 77.3%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if -1 < x < 1.30000000000000004

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

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

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 97.9%

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

Alternative 4: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{\mathsf{fma}\left(y, x, y\right)}\\ t_1 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+213}:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+206}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ x (fma y x y))))
        (t_1 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_1 -1e+213)
     (/ (- x 1.0) y)
     (if (<= t_1 -5e-64)
       t_0
       (if (<= t_1 2.0) (/ x (- x -1.0)) (if (<= t_1 5e+206) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x * (x / fma(y, x, y));
	double t_1 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_1 <= -1e+213) {
		tmp = (x - 1.0) / y;
	} else if (t_1 <= -5e-64) {
		tmp = t_0;
	} else if (t_1 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_1 <= 5e+206) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x / fma(y, x, y)))
	t_1 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+213)
		tmp = Float64(Float64(x - 1.0) / y);
	elseif (t_1 <= -5e-64)
		tmp = t_0;
	elseif (t_1 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_1 <= 5e+206)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+213], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, -5e-64], t$95$0, If[LessEqual[t$95$1, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+206], t$95$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))) < -9.99999999999999984e212

   1. Initial program 55.5%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in y around 0

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 97.7

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

   10. Applied rewrites 97.7%

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

   if -9.99999999999999984e212 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000033e-64 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000002e206

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 75.8

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

   4. Applied rewrites 75.8%

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

   if -5.00000000000000033e-64 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lft-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if 5.0000000000000002e206 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 52.1

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

   4. Applied rewrites 52.1%

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

Alternative 5: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} - -1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot x}{y}\\ \mathbf{elif}\;x \leq 220:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ (- x 1.0) y) -1.0)))
   (if (<= x -1.0)
     t_0
     (if (<= x -2.9e-63)
       (/ (* x x) y)
       (if (<= x 220.0) (/ x (- x -1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) - -1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -2.9e-63) {
		tmp = (x * x) / y;
	} else if (x <= 220.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - 1.0d0) / y) - (-1.0d0)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-2.9d-63)) then
        tmp = (x * x) / y
    else if (x <= 220.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) - -1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -2.9e-63) {
		tmp = (x * x) / y;
	} else if (x <= 220.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - 1.0) / y) - -1.0
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -2.9e-63:
		tmp = (x * x) / y
	elif x <= 220.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) - -1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -2.9e-63)
		tmp = Float64(Float64(x * x) / y);
	elseif (x <= 220.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - 1.0) / y) - -1.0;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -2.9e-63)
		tmp = (x * x) / y;
	elseif (x <= 220.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -2.9e-63], N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 220.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 220 < x

   1. Initial program 77.2%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if -1 < x < -2.89999999999999975e-63

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 46.0

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 41.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

         \[\leadsto \frac{{x}^{2}}{y} \]

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 41.4

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

   9. Applied rewrites 41.4%

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

   if -2.89999999999999975e-63 < x < 220

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lft-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 77.6

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

   4. Applied rewrites 77.6%

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

Alternative 6: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} - -1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot x}{y}\\ \mathbf{elif}\;x \leq 12500000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ x y) -1.0)))
   (if (<= x -1.0)
     t_0
     (if (<= x -2.9e-63)
       (/ (* x x) y)
       (if (<= x 12500000.0) (/ x (- x -1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x / y) - -1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -2.9e-63) {
		tmp = (x * x) / y;
	} else if (x <= 12500000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / y) - (-1.0d0)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-2.9d-63)) then
        tmp = (x * x) / y
    else if (x <= 12500000.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) - -1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -2.9e-63) {
		tmp = (x * x) / y;
	} else if (x <= 12500000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) - -1.0
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -2.9e-63:
		tmp = (x * x) / y
	elif x <= 12500000.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) - -1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -2.9e-63)
		tmp = Float64(Float64(x * x) / y);
	elseif (x <= 12500000.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) - -1.0;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -2.9e-63)
		tmp = (x * x) / y;
	elseif (x <= 12500000.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -2.9e-63], N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 12500000.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 1.25e7 < x

   1. Initial program 77.0%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.6%

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

   if -1 < x < -2.89999999999999975e-63

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 46.0

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 41.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

         \[\leadsto \frac{{x}^{2}}{y} \]

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 41.4

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

   9. Applied rewrites 41.4%

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

   if -2.89999999999999975e-63 < x < 1.25e7

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lft-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 77.3

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

   4. Applied rewrites 77.3%

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

Alternative 7: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} - -1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 12500000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ x y) -1.0)))
   (if (<= x -1.0)
     t_0
     (if (<= x -2.9e-63)
       (* x (/ x y))
       (if (<= x 12500000.0) (/ x (- x -1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x / y) - -1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -2.9e-63) {
		tmp = x * (x / y);
	} else if (x <= 12500000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / y) - (-1.0d0)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-2.9d-63)) then
        tmp = x * (x / y)
    else if (x <= 12500000.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) - -1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -2.9e-63) {
		tmp = x * (x / y);
	} else if (x <= 12500000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) - -1.0
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -2.9e-63:
		tmp = x * (x / y)
	elif x <= 12500000.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) - -1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -2.9e-63)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 12500000.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) - -1.0;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -2.9e-63)
		tmp = x * (x / y);
	elseif (x <= 12500000.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -2.9e-63], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 12500000.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 1.25e7 < x

   1. Initial program 77.0%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.6%

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

   if -1 < x < -2.89999999999999975e-63

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 46.0

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 41.4%

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

   if -2.89999999999999975e-63 < x < 1.25e7

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lft-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 77.3

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

   4. Applied rewrites 77.3%

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

Alternative 8: 86.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} - -1\\ \mathbf{if}\;x \leq -3950:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 12500000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ x y) -1.0)))
   (if (<= x -3950.0) t_0 (if (<= x 12500000.0) (/ x (- x -1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) - -1.0;
	double tmp;
	if (x <= -3950.0) {
		tmp = t_0;
	} else if (x <= 12500000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / y) - (-1.0d0)
    if (x <= (-3950.0d0)) then
        tmp = t_0
    else if (x <= 12500000.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) - -1.0;
	double tmp;
	if (x <= -3950.0) {
		tmp = t_0;
	} else if (x <= 12500000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) - -1.0
	tmp = 0
	if x <= -3950.0:
		tmp = t_0
	elif x <= 12500000.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) - -1.0)
	tmp = 0.0
	if (x <= -3950.0)
		tmp = t_0;
	elseif (x <= 12500000.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) - -1.0;
	tmp = 0.0;
	if (x <= -3950.0)
		tmp = t_0;
	elseif (x <= 12500000.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -3950.0], t$95$0, If[LessEqual[x, 12500000.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3950 or 1.25e7 < x

   1. Initial program 76.9%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.0%

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

   if -3950 < x < 1.25e7

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lft-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 74.9

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

   4. Applied rewrites 74.9%

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

Alternative 9: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{x}{y} - -1\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (- (/ x y) -1.0)))
   (if (<= t_0 -400.0) t_1 (if (<= t_0 0.1) x t_1))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = (x / y) - -1.0;
	double tmp;
	if (t_0 <= -400.0) {
		tmp = t_1;
	} else if (t_0 <= 0.1) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    t_1 = (x / y) - (-1.0d0)
    if (t_0 <= (-400.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.1d0) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = (x / y) - -1.0;
	double tmp;
	if (t_0 <= -400.0) {
		tmp = t_1;
	} else if (t_0 <= 0.1) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	t_1 = (x / y) - -1.0
	tmp = 0
	if t_0 <= -400.0:
		tmp = t_1
	elif t_0 <= 0.1:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64(Float64(x / y) - -1.0)
	tmp = 0.0
	if (t_0 <= -400.0)
		tmp = t_1;
	elseif (t_0 <= 0.1)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	t_1 = (x / y) - -1.0;
	tmp = 0.0;
	if (t_0 <= -400.0)
		tmp = t_1;
	elseif (t_0 <= 0.1)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -400.0], t$95$1, If[LessEqual[t$95$0, 0.1], x, t$95$1]]]]

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))) < -400 or 0.10000000000000001 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 79.8%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 87.2

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 87.4

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

   7. Applied rewrites 87.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 87.3%

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

   if -400 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 84.9%

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

Alternative 10: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 200:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -400.0)
     (/ (- x 1.0) y)
     (if (<= t_0 0.1) x (if (<= t_0 200.0) 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 <= -400.0) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 0.1) {
		tmp = x;
	} else if (t_0 <= 200.0) {
		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 <= (-400.0d0)) then
        tmp = (x - 1.0d0) / y
    else if (t_0 <= 0.1d0) then
        tmp = x
    else if (t_0 <= 200.0d0) 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 <= -400.0) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 0.1) {
		tmp = x;
	} else if (t_0 <= 200.0) {
		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 <= -400.0:
		tmp = (x - 1.0) / y
	elif t_0 <= 0.1:
		tmp = x
	elif t_0 <= 200.0:
		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 <= -400.0)
		tmp = Float64(Float64(x - 1.0) / y);
	elseif (t_0 <= 0.1)
		tmp = x;
	elseif (t_0 <= 200.0)
		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 <= -400.0)
		tmp = (x - 1.0) / y;
	elseif (t_0 <= 0.1)
		tmp = x;
	elseif (t_0 <= 200.0)
		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, -400.0], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 0.1], x, If[LessEqual[t$95$0, 200.0], 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))) < -400

   1. Initial program 73.6%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 84.6

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 84.8

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in y around 0

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 84.0

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

   10. Applied rewrites 84.0%

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

   if -400 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 84.9%

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

   if 0.10000000000000001 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 200

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 90.5%

      \[\leadsto 1 \]

   if 200 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.3%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 83.9

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

   4. Applied rewrites 83.9%

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

Alternative 11: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 200:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -400.0)
     (/ x y)
     (if (<= t_0 0.1) x (if (<= t_0 200.0) 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 <= -400.0) {
		tmp = x / y;
	} else if (t_0 <= 0.1) {
		tmp = x;
	} else if (t_0 <= 200.0) {
		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 <= (-400.0d0)) then
        tmp = x / y
    else if (t_0 <= 0.1d0) then
        tmp = x
    else if (t_0 <= 200.0d0) 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 <= -400.0) {
		tmp = x / y;
	} else if (t_0 <= 0.1) {
		tmp = x;
	} else if (t_0 <= 200.0) {
		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 <= -400.0:
		tmp = x / y
	elif t_0 <= 0.1:
		tmp = x
	elif t_0 <= 200.0:
		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 <= -400.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.1)
		tmp = x;
	elseif (t_0 <= 200.0)
		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 <= -400.0)
		tmp = x / y;
	elseif (t_0 <= 0.1)
		tmp = x;
	elseif (t_0 <= 200.0)
		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, -400.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.1], x, If[LessEqual[t$95$0, 200.0], 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))) < -400 or 200 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.5%

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

   2. Taylor expanded in x around inf

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

      1. lift-/.f64 83.9

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

   4. Applied rewrites 83.9%

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

   if -400 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 84.9%

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

   if 0.10000000000000001 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 200

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 90.5%

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

Alternative 12: 50.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 0.1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) 0.1) x 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.1) {
		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)) <= 0.1d0) 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)) <= 0.1) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= 0.1:
		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)) <= 0.1)
		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)) <= 0.1)
		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], 0.1], 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))) < 0.10000000000000001

   1. Initial program 90.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 57.1%

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

   if 0.10000000000000001 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 83.7%

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

   2. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 38.3%

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

Alternative 13: 38.6% accurate, 16.1× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 38.6%

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

Reproduce

herbie shell --seed 2025130 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64
  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))