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

Percentage Accurate: 65.5% → 99.8%

Time: 10.2s

Alternatives: 15

Speedup: 1.2×

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: 65.5% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3400000:\\ \;\;\;\;\frac{1 - x}{y} + \left(\frac{-1}{y \cdot y} + x\right)\\ \mathbf{elif}\;y \leq 170000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{y}{x}, -1, y\right)}{y - -1}, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3400000.0)
   (+ (/ (- 1.0 x) y) (+ (/ -1.0 (* y y)) x))
   (if (<= y 170000000.0)
     (fma (/ (fma (/ y x) -1.0 y) (- y -1.0)) x 1.0)
     (- x (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3400000.0) {
		tmp = ((1.0 - x) / y) + ((-1.0 / (y * y)) + x);
	} else if (y <= 170000000.0) {
		tmp = fma((fma((y / x), -1.0, y) / (y - -1.0)), x, 1.0);
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3400000.0)
		tmp = Float64(Float64(Float64(1.0 - x) / y) + Float64(Float64(-1.0 / Float64(y * y)) + x));
	elseif (y <= 170000000.0)
		tmp = fma(Float64(fma(Float64(y / x), -1.0, y) / Float64(y - -1.0)), x, 1.0);
	else
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3400000.0], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 170000000.0], N[(N[(N[(N[(y / x), $MachinePrecision] * -1.0 + y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.4e6

   1. Initial program 28.3%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

   if -3.4e6 < y < 1.7e8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 3.7

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 100.0%

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

   if 1.7e8 < y

   1. Initial program 37.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 63.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))))
   (if (or (<= t_0 1e-12) (not (<= t_0 2.0)))
     (- 1.0 (- 1.0 x))
     (fma (- y 1.0) y 1.0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	double tmp;
	if ((t_0 <= 1e-12) || !(t_0 <= 2.0)) {
		tmp = 1.0 - (1.0 - x);
	} else {
		tmp = fma((y - 1.0), y, 1.0);
	}
	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 <= 1e-12) || !(t_0 <= 2.0))
		tmp = Float64(1.0 - Float64(1.0 - x));
	else
		tmp = fma(Float64(y - 1.0), y, 1.0);
	end
	return 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[Or[LessEqual[t$95$0, 1e-12], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(1.0 - N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(y - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 9.9999999999999998e-13 or 2 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 46.6%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 39.6

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

   5. Applied rewrites 39.6%

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

   if 9.9999999999999998e-13 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 10^{-12} \lor \neg \left(1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 2\right):\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - 1, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{if}\;t\_0 \leq 10^{-12} \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))))
   (if (or (<= t_0 1e-12) (not (<= t_0 2.0))) (- 1.0 (- 1.0 x)) 1.0)))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	double tmp;
	if ((t_0 <= 1e-12) || !(t_0 <= 2.0)) {
		tmp = 1.0 - (1.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (((1.0d0 - x) * y) / (y - (-1.0d0)))
    if ((t_0 <= 1d-12) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = 1.0d0 - (1.0d0 - x)
    else
        tmp = 1.0d0
    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 <= 1e-12) || !(t_0 <= 2.0)) {
		tmp = 1.0 - (1.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0))
	tmp = 0
	if (t_0 <= 1e-12) or not (t_0 <= 2.0):
		tmp = 1.0 - (1.0 - x)
	else:
		tmp = 1.0
	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 <= 1e-12) || !(t_0 <= 2.0))
		tmp = Float64(1.0 - Float64(1.0 - x));
	else
		tmp = 1.0;
	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 <= 1e-12) || ~((t_0 <= 2.0)))
		tmp = 1.0 - (1.0 - x);
	else
		tmp = 1.0;
	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[Or[LessEqual[t$95$0, 1e-12], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(1.0 - N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 9.9999999999999998e-13 or 2 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 46.6%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 39.6

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

   5. Applied rewrites 39.6%

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

   if 9.9999999999999998e-13 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.9%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 97.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 10^{-12} \lor \neg \left(1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 2\right):\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-15} \lor \neg \left(t\_0 \leq 400\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))))
   (if (or (<= t_0 -5e-15) (not (<= t_0 400.0))) (* y x) 1.0)))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	double tmp;
	if ((t_0 <= -5e-15) || !(t_0 <= 400.0)) {
		tmp = y * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (((1.0d0 - x) * y) / (y - (-1.0d0)))
    if ((t_0 <= (-5d-15)) .or. (.not. (t_0 <= 400.0d0))) then
        tmp = y * x
    else
        tmp = 1.0d0
    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 <= -5e-15) || !(t_0 <= 400.0)) {
		tmp = y * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0))
	tmp = 0
	if (t_0 <= -5e-15) or not (t_0 <= 400.0):
		tmp = y * x
	else:
		tmp = 1.0
	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 <= -5e-15) || !(t_0 <= 400.0))
		tmp = Float64(y * x);
	else
		tmp = 1.0;
	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 <= -5e-15) || ~((t_0 <= 400.0)))
		tmp = y * x;
	else
		tmp = 1.0;
	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[Or[LessEqual[t$95$0, -5e-15], N[Not[LessEqual[t$95$0, 400.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < -4.99999999999999999e-15 or 400 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 76.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 32.7

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

   5. Applied rewrites 32.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 32.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 34.8%

       \[\leadsto y \cdot x \]

   if -4.99999999999999999e-15 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 400

   1. Initial program 66.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

   8. Applied rewrites 66.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 65.7%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 63.7%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq -5 \cdot 10^{-15} \lor \neg \left(1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 400\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3700000:\\ \;\;\;\;\frac{1 - x}{y} + \left(\frac{-1}{y \cdot y} + x\right)\\ \mathbf{elif}\;y \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{1 - x}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3700000.0)
   (+ (/ (- 1.0 x) y) (+ (/ -1.0 (* y y)) x))
   (if (<= y 200000000.0)
     (fma (- y) (/ (- 1.0 x) (- y -1.0)) 1.0)
     (- x (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3700000.0) {
		tmp = ((1.0 - x) / y) + ((-1.0 / (y * y)) + x);
	} else if (y <= 200000000.0) {
		tmp = fma(-y, ((1.0 - x) / (y - -1.0)), 1.0);
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3700000.0)
		tmp = Float64(Float64(Float64(1.0 - x) / y) + Float64(Float64(-1.0 / Float64(y * y)) + x));
	elseif (y <= 200000000.0)
		tmp = fma(Float64(-y), Float64(Float64(1.0 - x) / Float64(y - -1.0)), 1.0);
	else
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3700000.0], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 200000000.0], N[((-y) * N[(N[(1.0 - x), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7e6

   1. Initial program 28.3%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

   if -3.7e6 < y < 2e8

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

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

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

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

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 2e8 < y

   1. Initial program 37.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3700000:\\ \;\;\;\;\frac{1 - x}{y} + \left(\frac{-1}{y \cdot y} + x\right)\\ \mathbf{elif}\;y \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{1 - x}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -235000:\\ \;\;\;\;\frac{\left(1 - x\right) + \frac{-1 + x}{y}}{y} + x\\ \mathbf{elif}\;y \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{1 - x}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -235000.0)
   (+ (/ (+ (- 1.0 x) (/ (+ -1.0 x) y)) y) x)
   (if (<= y 200000000.0)
     (fma (- y) (/ (- 1.0 x) (- y -1.0)) 1.0)
     (- x (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -235000.0) {
		tmp = (((1.0 - x) + ((-1.0 + x) / y)) / y) + x;
	} else if (y <= 200000000.0) {
		tmp = fma(-y, ((1.0 - x) / (y - -1.0)), 1.0);
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -235000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 - x) + Float64(Float64(-1.0 + x) / y)) / y) + x);
	elseif (y <= 200000000.0)
		tmp = fma(Float64(-y), Float64(Float64(1.0 - x) / Float64(y - -1.0)), 1.0);
	else
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -235000.0], N[(N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 200000000.0], N[((-y) * N[(N[(1.0 - x), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -235000

   1. Initial program 28.3%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 99.9%

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

   if -235000 < y < 2e8

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

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

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

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

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 2e8 < y

   1. Initial program 37.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -235000:\\ \;\;\;\;\frac{\left(1 - x\right) + \frac{-1 + x}{y}}{y} + x\\ \mathbf{elif}\;y \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{1 - x}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -180000000 \lor \neg \left(y \leq 200000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{1 - x}{y - -1}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -180000000.0) (not (<= y 200000000.0)))
   (- x (/ (- x 1.0) y))
   (fma (- y) (/ (- 1.0 x) (- y -1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -180000000.0) || !(y <= 200000000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = fma(-y, ((1.0 - x) / (y - -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -180000000.0) || !(y <= 200000000.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = fma(Float64(-y), Float64(Float64(1.0 - x) / Float64(y - -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -180000000.0], N[Not[LessEqual[y, 200000000.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(N[(1.0 - x), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e8 or 2e8 < y

   1. Initial program 33.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -1.8e8 < y < 2e8

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

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

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

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

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -180000000 \lor \neg \left(y \leq 200000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{1 - x}{y - -1}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -260000000 \lor \neg \left(y \leq 190000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y - -1}, -1 + x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -260000000.0) (not (<= y 190000000.0)))
   (- x (/ (- x 1.0) y))
   (fma (/ y (- y -1.0)) (+ -1.0 x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -260000000.0) || !(y <= 190000000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = fma((y / (y - -1.0)), (-1.0 + x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -260000000.0) || !(y <= 190000000.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = fma(Float64(y / Float64(y - -1.0)), Float64(-1.0 + x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -260000000.0], N[Not[LessEqual[y, 190000000.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6e8 or 1.9e8 < y

   1. Initial program 33.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -2.6e8 < y < 1.9e8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 3.7

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260000000 \lor \neg \left(y \leq 190000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y - -1}, -1 + x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -3.2e+80)
     t_0
     (if (<= y -1.0)
       (/ (- 1.0 x) y)
       (if (<= y 1.15) (fma (- x 1.0) y 1.0) t_0)))))

C

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -3.2e+80) {
		tmp = t_0;
	} else if (y <= -1.0) {
		tmp = (1.0 - x) / y;
	} else if (y <= 1.15) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(x / y))
	tmp = 0.0
	if (y <= -3.2e+80)
		tmp = t_0;
	elseif (y <= -1.0)
		tmp = Float64(Float64(1.0 - x) / y);
	elseif (y <= 1.15)
		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[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+80], t$95$0, If[LessEqual[y, -1.0], N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.15], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1999999999999999e80 or 1.1499999999999999 < y

   1. Initial program 33.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. rgt-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

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

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

      11. *-lft-identity N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lft-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 77.1%

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

   if -3.1999999999999999e80 < y < -1

   1. Initial program 43.1%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 96.3%

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 63.2%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 61.8%

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

   if -1 < y < 1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 10: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 34.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 98.8%

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

Alternative 11: 98.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 34.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.6%

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

Alternative 12: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.15\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.15))) (- x (/ x y)) (fma (- x 1.0) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.15)) {
		tmp = x - (x / y);
	} else {
		tmp = fma((x - 1.0), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.15))
		tmp = Float64(x - Float64(x / y));
	else
		tmp = fma(Float64(x - 1.0), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.15]], $MachinePrecision]], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.1499999999999999 < y

   1. Initial program 34.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. rgt-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

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

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

      11. *-lft-identity N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lft-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 71.6%

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

   if -1 < y < 1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.15\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.4% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (- 1.0 (- 1.0 x))
   (fma (- x 1.0) y 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 34.6%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 47.7

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

   5. Applied rewrites 47.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 74.8%

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

Alternative 14: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.35 \cdot 10^{-59}\right):\\ \;\;\;\;1 - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.35e-59))) (- 1.0 (- x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.35e-59)) {
		tmp = 1.0 - -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.35d-59))) then
        tmp = 1.0d0 - -x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.35e-59)) {
		tmp = 1.0 - -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.35e-59):
		tmp = 1.0 - -x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.35e-59))
		tmp = Float64(1.0 - Float64(-x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.35e-59)))
		tmp = 1.0 - -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.35e-59]], $MachinePrecision]], N[(1.0 - (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.3499999999999999e-59 < y

   1. Initial program 41.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 43.3

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.6%

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

   if -1 < y < 1.3499999999999999e-59

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 82.2%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.35 \cdot 10^{-59}\right):\\ \;\;\;\;1 - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.6% accurate, 26.0× 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 69.3%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 24.1

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

5. Applied rewrites 24.1%

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

6. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

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

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

   6. +-commutative N/A

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

8. Applied rewrites 76.7%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 76.2%

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

12. Taylor expanded in y around 0

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

14. Applied rewrites 43.5%

    \[\leadsto 1 \]
15. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 y) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / y) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / y) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / y) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / y) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024350 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -36938482788297247/10000000000000) (- (/ 1 y) (- (/ x y) x)) (if (< y 679931050341891/100000) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x)))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))