Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Percentage Accurate: 65.7% → 99.9%

Time: 3.4s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + 1\right) \cdot 2\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 3400000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x - y, 2, t\_0\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y 1.0) 2.0)))
   (if (<= y -8.8e+15)
     (- x (/ -1.0 y))
     (if (<= y 3400000000000.0)
       (/ (fma (- (* y x) y) 2.0 t_0) t_0)
       (- x (/ (- x 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = (y + 1.0) * 2.0;
	double tmp;
	if (y <= -8.8e+15) {
		tmp = x - (-1.0 / y);
	} else if (y <= 3400000000000.0) {
		tmp = fma(((y * x) - y), 2.0, t_0) / t_0;
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + 1.0) * 2.0)
	tmp = 0.0
	if (y <= -8.8e+15)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 3400000000000.0)
		tmp = Float64(fma(Float64(Float64(y * x) - y), 2.0, t_0) / t_0);
	else
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + 1.0), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[y, -8.8e+15], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3400000000000.0], N[(N[(N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision] * 2.0 + t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.8e15

   1. Initial program 28.4%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

   if -8.8e15 < y < 3.4e12

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot x - y, 2, \left(y + 1\right) \cdot 2\right)}{\color{blue}{\left(y + 1\right)} \cdot 2} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot x - y, 2, \left(y + 1\right) \cdot 2\right)}{\left(\color{blue}{y} + 1\right) \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot x - y, 2, \left(y + 1\right) \cdot 2\right)}{\left(y + 1\right) \cdot 2} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot x - y, 2, \left(y + 1\right) \cdot 2\right)}{\left(y + 1\right) \cdot 2} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot x - y, 2, \left(y + 1\right) \cdot 2\right)}{\left(y + 1\right) \cdot 2} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot x - y, 2, \left(y + 1\right) \cdot 2\right)}{\left(y + 1\right) \cdot \color{blue}{2}} \]

      16. lift-+.f64 99.9

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot x - y, 2, \left(y + 1\right) \cdot 2\right)}{\left(y + 1\right) \cdot 2} \]

   6. Applied rewrites 99.9%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot x - y, 2, \left(y + 1\right) \cdot 2\right)}{\color{blue}{\left(y + 1\right) \cdot 2}} \]

   if 3.4e12 < y

   1. Initial program 30.0%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 340000:\\ \;\;\;\;1 - \frac{\frac{y}{x} - y}{y + 1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\left(\left(-\frac{x - 1}{y}\right) + x\right) - 1}{y}\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.1e+19)
   (- x (/ -1.0 y))
   (if (<= y 340000.0)
     (- 1.0 (* (/ (- (/ y x) y) (+ y 1.0)) x))
     (+ (- (/ (- (+ (- (/ (- x 1.0) y)) x) 1.0) y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+19) {
		tmp = x - (-1.0 / y);
	} else if (y <= 340000.0) {
		tmp = 1.0 - ((((y / x) - y) / (y + 1.0)) * x);
	} else {
		tmp = -(((-((x - 1.0) / y) + x) - 1.0) / y) + x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.1d+19)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 340000.0d0) then
        tmp = 1.0d0 - ((((y / x) - y) / (y + 1.0d0)) * x)
    else
        tmp = -(((-((x - 1.0d0) / y) + x) - 1.0d0) / y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+19) {
		tmp = x - (-1.0 / y);
	} else if (y <= 340000.0) {
		tmp = 1.0 - ((((y / x) - y) / (y + 1.0)) * x);
	} else {
		tmp = -(((-((x - 1.0) / y) + x) - 1.0) / y) + x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.1e+19:
		tmp = x - (-1.0 / y)
	elif y <= 340000.0:
		tmp = 1.0 - ((((y / x) - y) / (y + 1.0)) * x)
	else:
		tmp = -(((-((x - 1.0) / y) + x) - 1.0) / y) + x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.1e+19)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 340000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(y / x) - y) / Float64(y + 1.0)) * x));
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(x - 1.0) / y)) + x) - 1.0) / y)) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.1e+19)
		tmp = x - (-1.0 / y);
	elseif (y <= 340000.0)
		tmp = 1.0 - ((((y / x) - y) / (y + 1.0)) * x);
	else
		tmp = -(((-((x - 1.0) / y) + x) - 1.0) / y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.1e+19], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 340000.0], N[(1.0 - N[(N[(N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(N[((-N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1e19

   1. Initial program 28.1%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

   if -3.1e19 < y < 3.4e5

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf

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

      1. lift--.f64 4.6

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

   4. Applied rewrites 4.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. div-add N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-+.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 99.3

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

   10. Applied rewrites 99.3%

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

   if 3.4e5 < y

   1. Initial program 31.0%

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

   2. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 31.0

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 3: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+15}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 108000000:\\ \;\;\;\;1 + \frac{y \cdot x - y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e+15)
   (- x (/ -1.0 y))
   (if (<= y 108000000.0)
     (+ 1.0 (/ (- (* y x) y) (+ y 1.0)))
     (- x (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+15) {
		tmp = x - (-1.0 / y);
	} else if (y <= 108000000.0) {
		tmp = 1.0 + (((y * x) - y) / (y + 1.0));
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.1d+15)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 108000000.0d0) then
        tmp = 1.0d0 + (((y * x) - y) / (y + 1.0d0))
    else
        tmp = x - ((x - 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+15) {
		tmp = x - (-1.0 / y);
	} else if (y <= 108000000.0) {
		tmp = 1.0 + (((y * x) - y) / (y + 1.0));
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e+15:
		tmp = x - (-1.0 / y)
	elif y <= 108000000.0:
		tmp = 1.0 + (((y * x) - y) / (y + 1.0))
	else:
		tmp = x - ((x - 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e+15)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 108000000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(y * x) - y) / Float64(y + 1.0)));
	else
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.1e+15)
		tmp = x - (-1.0 / y);
	elseif (y <= 108000000.0)
		tmp = 1.0 + (((y * x) - y) / (y + 1.0));
	else
		tmp = x - ((x - 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e+15], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 108000000.0], N[(1.0 + N[(N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e15

   1. Initial program 28.4%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

   if -1.1e15 < y < 1.08e8

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 1.08e8 < y

   1. Initial program 30.7%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 4: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+15}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 108000000:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e+15)
   (- x (/ -1.0 y))
   (if (<= y 108000000.0)
     (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0)))
     (- x (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+15) {
		tmp = x - (-1.0 / y);
	} else if (y <= 108000000.0) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.1d+15)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 108000000.0d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    else
        tmp = x - ((x - 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+15) {
		tmp = x - (-1.0 / y);
	} else if (y <= 108000000.0) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e+15:
		tmp = x - (-1.0 / y)
	elif y <= 108000000.0:
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	else:
		tmp = x - ((x - 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e+15)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 108000000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)));
	else
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.1e+15)
		tmp = x - (-1.0 / y);
	elseif (y <= 108000000.0)
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	else
		tmp = x - ((x - 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e+15], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 108000000.0], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e15

   1. Initial program 28.4%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

   if -1.1e15 < y < 1.08e8

   1. Initial program 99.5%

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

   if 1.08e8 < y

   1. Initial program 30.7%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 5: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+15}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 112000:\\ \;\;\;\;1 - \frac{\left(-x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e+15)
   (- x (/ -1.0 y))
   (if (<= y 112000.0)
     (- 1.0 (/ (* (- x) y) (+ y 1.0)))
     (- x (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+15) {
		tmp = x - (-1.0 / y);
	} else if (y <= 112000.0) {
		tmp = 1.0 - ((-x * y) / (y + 1.0));
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.1d+15)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 112000.0d0) then
        tmp = 1.0d0 - ((-x * y) / (y + 1.0d0))
    else
        tmp = x - ((x - 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+15) {
		tmp = x - (-1.0 / y);
	} else if (y <= 112000.0) {
		tmp = 1.0 - ((-x * y) / (y + 1.0));
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e+15:
		tmp = x - (-1.0 / y)
	elif y <= 112000.0:
		tmp = 1.0 - ((-x * y) / (y + 1.0))
	else:
		tmp = x - ((x - 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e+15)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 112000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(-x) * y) / Float64(y + 1.0)));
	else
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.1e+15)
		tmp = x - (-1.0 / y);
	elseif (y <= 112000.0)
		tmp = 1.0 - ((-x * y) / (y + 1.0));
	else
		tmp = x - ((x - 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e+15], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 112000.0], N[(1.0 - N[(N[((-x) * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e15

   1. Initial program 28.4%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

   if -1.1e15 < y < 112000

   1. Initial program 99.5%

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

   2. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.5

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

   4. Applied rewrites 97.5%

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

   if 112000 < y

   1. Initial program 31.1%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.5

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

   4. Applied rewrites 99.5%

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

Alternative 6: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 31.7%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 98.4

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

   4. Applied rewrites 98.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 7: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 31.7%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 98.4

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

   4. Applied rewrites 98.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 8: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.78:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 0.78) (fma (- x 1.0) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.78) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.78)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.78], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.78000000000000003 < y

   1. Initial program 31.7%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.8%

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

   if -1 < y < 0.78000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 9: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 31.7%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 98.0%

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

Alternative 10: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.5:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 5.5) (fma x y 1.0) (if (<= y 7.2e+97) (/ 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 5.5) {
		tmp = fma(x, y, 1.0);
	} else if (y <= 7.2e+97) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 5.5)
		tmp = fma(x, y, 1.0);
	elseif (y <= 7.2e+97)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 5.5], N[(x * y + 1.0), $MachinePrecision], If[LessEqual[y, 7.2e+97], N[(1.0 / y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 7.19999999999999932e97 < y

   1. Initial program 27.9%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 77.8%

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

   if -1 < y < 5.5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 98.0%

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

   if 5.5 < y < 7.19999999999999932e97

   1. Initial program 51.4%

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

   2. Taylor expanded in y around -inf

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

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 39.1

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

   7. Applied rewrites 39.1%

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

Alternative 11: 74.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{if}\;t\_0 \leq 0.6624206840315668:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\left(-y\right) + 1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+55}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0)))))
   (if (<= t_0 0.6624206840315668)
     x
     (if (<= t_0 4.0) (+ (- y) 1.0) (if (<= t_0 2e+55) (* x y) x)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	double tmp;
	if (t_0 <= 0.6624206840315668) {
		tmp = x;
	} else if (t_0 <= 4.0) {
		tmp = -y + 1.0;
	} else if (t_0 <= 2e+55) {
		tmp = x * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    if (t_0 <= 0.6624206840315668d0) then
        tmp = x
    else if (t_0 <= 4.0d0) then
        tmp = -y + 1.0d0
    else if (t_0 <= 2d+55) then
        tmp = x * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	double tmp;
	if (t_0 <= 0.6624206840315668) {
		tmp = x;
	} else if (t_0 <= 4.0) {
		tmp = -y + 1.0;
	} else if (t_0 <= 2e+55) {
		tmp = x * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	tmp = 0
	if t_0 <= 0.6624206840315668:
		tmp = x
	elif t_0 <= 4.0:
		tmp = -y + 1.0
	elif t_0 <= 2e+55:
		tmp = x * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)))
	tmp = 0.0
	if (t_0 <= 0.6624206840315668)
		tmp = x;
	elseif (t_0 <= 4.0)
		tmp = Float64(Float64(-y) + 1.0);
	elseif (t_0 <= 2e+55)
		tmp = Float64(x * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	tmp = 0.0;
	if (t_0 <= 0.6624206840315668)
		tmp = x;
	elseif (t_0 <= 4.0)
		tmp = -y + 1.0;
	elseif (t_0 <= 2e+55)
		tmp = x * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.6624206840315668], x, If[LessEqual[t$95$0, 4.0], N[((-y) + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+55], N[(x * y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 0.662420684031566842 or 2.00000000000000002e55 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 41.2%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 61.2%

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

   if 0.662420684031566842 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(y\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto 1 + \left(-y\right) \]

      3. +-commutative N/A

         \[\leadsto \left(-y\right) + 1 \]

      4. lower-+.f64 98.1

         \[\leadsto \left(-y\right) + 1 \]

   7. Applied rewrites 98.1%

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

   if 4 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2.00000000000000002e55

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. lift-+.f64 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 47.2%

      \[\leadsto x \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{if}\;t\_0 \leq 0.6624206840315668:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+55}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0)))))
   (if (<= t_0 0.6624206840315668)
     x
     (if (<= t_0 4.0) 1.0 (if (<= t_0 2e+55) (* x y) x)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	double tmp;
	if (t_0 <= 0.6624206840315668) {
		tmp = x;
	} else if (t_0 <= 4.0) {
		tmp = 1.0;
	} else if (t_0 <= 2e+55) {
		tmp = x * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    if (t_0 <= 0.6624206840315668d0) then
        tmp = x
    else if (t_0 <= 4.0d0) then
        tmp = 1.0d0
    else if (t_0 <= 2d+55) then
        tmp = x * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	double tmp;
	if (t_0 <= 0.6624206840315668) {
		tmp = x;
	} else if (t_0 <= 4.0) {
		tmp = 1.0;
	} else if (t_0 <= 2e+55) {
		tmp = x * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	tmp = 0
	if t_0 <= 0.6624206840315668:
		tmp = x
	elif t_0 <= 4.0:
		tmp = 1.0
	elif t_0 <= 2e+55:
		tmp = x * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)))
	tmp = 0.0
	if (t_0 <= 0.6624206840315668)
		tmp = x;
	elseif (t_0 <= 4.0)
		tmp = 1.0;
	elseif (t_0 <= 2e+55)
		tmp = Float64(x * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	tmp = 0.0;
	if (t_0 <= 0.6624206840315668)
		tmp = x;
	elseif (t_0 <= 4.0)
		tmp = 1.0;
	elseif (t_0 <= 2e+55)
		tmp = x * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.6624206840315668], x, If[LessEqual[t$95$0, 4.0], 1.0, If[LessEqual[t$95$0, 2e+55], N[(x * y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 0.662420684031566842 or 2.00000000000000002e55 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 41.2%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 61.2%

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

   if 0.662420684031566842 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 97.4%

      \[\leadsto \color{blue}{1} \]

   if 4 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2.00000000000000002e55

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. lift-+.f64 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 47.2%

      \[\leadsto x \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 1.8e-29) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.8e-29) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 1.8d-29) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.8e-29) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.8e-29:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.8e-29)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.8e-29)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.8e-29], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.79999999999999987e-29 < y

   1. Initial program 34.6%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 71.5%

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

   if -1 < y < 1.79999999999999987e-29

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 76.0%

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

Alternative 14: 38.9% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 65.7%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 38.9%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64
  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))