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

Percentage Accurate: 65.4% → 99.9%

Time: 5.5s

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: 65.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -280000 \lor \neg \left(y \leq 280000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, \frac{-1}{y} - -1, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(y - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -280000.0) (not (<= y 280000.0)))
   (fma (/ (- 1.0 x) y) (- (/ -1.0 y) -1.0) x)
   (- 1.0 (* (/ (* y (- 1.0 x)) (fma y y -1.0)) (- y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -280000.0) || !(y <= 280000.0)) {
		tmp = fma(((1.0 - x) / y), ((-1.0 / y) - -1.0), x);
	} else {
		tmp = 1.0 - (((y * (1.0 - x)) / fma(y, y, -1.0)) * (y - 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -280000.0) || !(y <= 280000.0))
		tmp = fma(Float64(Float64(1.0 - x) / y), Float64(Float64(-1.0 / y) - -1.0), x);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(y * Float64(1.0 - x)) / fma(y, y, -1.0)) * Float64(y - 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -280000.0], N[Not[LessEqual[y, 280000.0]], $MachinePrecision]], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision] + x), $MachinePrecision], N[(1.0 - N[(N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(y - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e5 or 2.8e5 < y

   1. Initial program 27.6%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

   if -2.8e5 < y < 2.8e5

   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

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -280000 \lor \neg \left(y \leq 280000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, \frac{-1}{y} - -1, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(y - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+36} \lor \neg \left(t\_0 \leq 1.0000000000002\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (- y -1.0))))
   (if (or (<= t_0 -2e+36) (not (<= t_0 1.0000000000002))) (* y x) 1.0)))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (y - -1.0);
	double tmp;
	if ((t_0 <= -2e+36) || !(t_0 <= 1.0000000000002)) {
		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 - x) * y) / (y - (-1.0d0))
    if ((t_0 <= (-2d+36)) .or. (.not. (t_0 <= 1.0000000000002d0))) 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 - x) * y) / (y - -1.0);
	double tmp;
	if ((t_0 <= -2e+36) || !(t_0 <= 1.0000000000002)) {
		tmp = y * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (y - -1.0)
	tmp = 0
	if (t_0 <= -2e+36) or not (t_0 <= 1.0000000000002):
		tmp = y * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0))
	tmp = 0.0
	if ((t_0 <= -2e+36) || !(t_0 <= 1.0000000000002))
		tmp = Float64(y * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (y - -1.0);
	tmp = 0.0;
	if ((t_0 <= -2e+36) || ~((t_0 <= 1.0000000000002)))
		tmp = y * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+36], N[Not[LessEqual[t$95$0, 1.0000000000002]], $MachinePrecision]], N[(y * x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -2.00000000000000008e36 or 1.00000000000020006 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 66.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 34.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 35.4%

       \[\leadsto y \cdot x \]

   if -2.00000000000000008e36 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.00000000000020006

   1. Initial program 52.2%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 75.1%

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

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 46.6%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq -2 \cdot 10^{+36} \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{y - -1} \leq 1.0000000000002\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -290000 \lor \neg \left(y \leq 290000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, \frac{-1}{y} - -1, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -290000.0) (not (<= y 290000.0)))
   (fma (/ (- 1.0 x) y) (- (/ -1.0 y) -1.0) x)
   (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -290000.0) || !(y <= 290000.0)) {
		tmp = fma(((1.0 - x) / y), ((-1.0 / y) - -1.0), x);
	} else {
		tmp = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -290000.0) || !(y <= 290000.0))
		tmp = fma(Float64(Float64(1.0 - x) / y), Float64(Float64(-1.0 / y) - -1.0), x);
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -290000.0], N[Not[LessEqual[y, 290000.0]], $MachinePrecision]], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision] + x), $MachinePrecision], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e5 or 2.9e5 < y

   1. Initial program 27.6%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

   if -2.9e5 < y < 2.9e5

   1. Initial program 99.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -290000 \lor \neg \left(y \leq 290000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, \frac{-1}{y} - -1, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -240000000 \lor \neg \left(y \leq 210000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -240000000.0) (not (<= y 210000000.0)))
   (- x (/ (- x 1.0) y))
   (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -240000000.0) || !(y <= 210000000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = 1.0 - (((1.0 - x) * y) / (y - -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 <= (-240000000.0d0)) .or. (.not. (y <= 210000000.0d0))) then
        tmp = x - ((x - 1.0d0) / y)
    else
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -240000000.0) || !(y <= 210000000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -240000000.0) or not (y <= 210000000.0):
		tmp = x - ((x - 1.0) / y)
	else:
		tmp = 1.0 - (((1.0 - x) * y) / (y - -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -240000000.0) || !(y <= 210000000.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -240000000.0) || ~((y <= 210000000.0)))
		tmp = x - ((x - 1.0) / y);
	else
		tmp = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -240000000.0], N[Not[LessEqual[y, 210000000.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e8 or 2.1e8 < y

   1. Initial program 27.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(x + \frac{1}{y}\right) - \frac{x}{y}} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   if -2.4e8 < y < 2.1e8

   1. Initial program 99.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -240000000 \lor \neg \left(y \leq 210000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -24000 \lor \neg \left(y \leq 310000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(-x\right) \cdot y}{y - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -24000.0) (not (<= y 310000.0)))
   (- x (/ (- x 1.0) y))
   (- 1.0 (/ (* (- x) y) (- y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -24000.0) || !(y <= 310000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = 1.0 - ((-x * y) / (y - -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 <= (-24000.0d0)) .or. (.not. (y <= 310000.0d0))) then
        tmp = x - ((x - 1.0d0) / y)
    else
        tmp = 1.0d0 - ((-x * y) / (y - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -24000.0) || !(y <= 310000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = 1.0 - ((-x * y) / (y - -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -24000.0) or not (y <= 310000.0):
		tmp = x - ((x - 1.0) / y)
	else:
		tmp = 1.0 - ((-x * y) / (y - -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -24000.0) || !(y <= 310000.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(-x) * y) / Float64(y - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -24000.0) || ~((y <= 310000.0)))
		tmp = x - ((x - 1.0) / y);
	else
		tmp = 1.0 - ((-x * y) / (y - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -24000.0], N[Not[LessEqual[y, 310000.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[((-x) * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -24000 or 3.1e5 < y

   1. Initial program 27.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(x + \frac{1}{y}\right) - \frac{x}{y}} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   if -24000 < y < 3.1e5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -24000 \lor \neg \left(y \leq 310000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(-x\right) \cdot y}{y - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 2.1:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* -1.0 (- x))))
   (if (<= y -2.5e+64)
     t_0
     (if (<= y -1.0)
       (/ 1.0 y)
       (if (<= y 2.1)
         (fma (- x 1.0) y 1.0)
         (if (<= y 3.5e+60) (/ 1.0 y) t_0))))))

C

double code(double x, double y) {
	double t_0 = -1.0 * -x;
	double tmp;
	if (y <= -2.5e+64) {
		tmp = t_0;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 2.1) {
		tmp = fma((x - 1.0), y, 1.0);
	} else if (y <= 3.5e+60) {
		tmp = 1.0 / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 * Float64(-x))
	tmp = 0.0
	if (y <= -2.5e+64)
		tmp = t_0;
	elseif (y <= -1.0)
		tmp = Float64(1.0 / y);
	elseif (y <= 2.1)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	elseif (y <= 3.5e+60)
		tmp = Float64(1.0 / y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 * (-x)), $MachinePrecision]}, If[LessEqual[y, -2.5e+64], t$95$0, If[LessEqual[y, -1.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 2.1], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 3.5e+60], N[(1.0 / y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5e64 or 3.5000000000000002e60 < y

   1. Initial program 25.0%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 81.8%

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

   if -2.5e64 < y < -1 or 2.10000000000000009 < y < 3.5000000000000002e60

   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

      \[\leadsto \color{blue}{\left(x + \frac{1}{y}\right) - \frac{x}{y}} \]
   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 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.3%

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

   if -1 < y < 2.10000000000000009

   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

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

   5. Applied rewrites 97.8%

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

Alternative 7: 98.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (- x (/ (- x 1.0) y))
   (fma (- x (fma (- x 1.0) y 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 - fma((x - 1.0), y, 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 - fma(Float64(x - 1.0), y, 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 - N[(N[(x - 1.0), $MachinePrecision] * y + 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 28.5%

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

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

   5. Applied rewrites 98.3%

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

4. Final simplification 98.4%

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

Alternative 8: 98.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (- x (/ (- x 1.0) y))
   (fma (* (- 1.0 y) x) 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(((1.0 - y) * x), 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(Float64(1.0 - y) * x), 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 - y), $MachinePrecision] * x), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 28.5%

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

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.0%

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

4. Final simplification 98.3%

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

Alternative 9: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -66000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 55:\\ \;\;\;\;\mathsf{fma}\left(y - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* -1.0 (- x))))
   (if (<= y -66000000000.0)
     t_0
     (if (<= y -3.9e-36) (* y x) (if (<= y 55.0) (fma (- y 1.0) y 1.0) t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 * -x;
	double tmp;
	if (y <= -66000000000.0) {
		tmp = t_0;
	} else if (y <= -3.9e-36) {
		tmp = y * x;
	} else if (y <= 55.0) {
		tmp = fma((y - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 * Float64(-x))
	tmp = 0.0
	if (y <= -66000000000.0)
		tmp = t_0;
	elseif (y <= -3.9e-36)
		tmp = Float64(y * x);
	elseif (y <= 55.0)
		tmp = fma(Float64(y - 1.0), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 * (-x)), $MachinePrecision]}, If[LessEqual[y, -66000000000.0], t$95$0, If[LessEqual[y, -3.9e-36], N[(y * x), $MachinePrecision], If[LessEqual[y, 55.0], N[(N[(y - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.6e10 or 55 < y

   1. Initial program 27.5%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 68.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 73.4%

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

   if -6.6e10 < y < -3.9000000000000001e-36

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 70.0%

       \[\leadsto y \cdot x \]

   if -3.9000000000000001e-36 < y < 55

   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

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.7%

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

Alternative 10: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.84\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - y\right) \cdot x, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.84)))
   (- x (/ -1.0 y))
   (fma (* (- 1.0 y) x) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.84)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = fma(((1.0 - y) * x), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.84))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = fma(Float64(Float64(1.0 - y) * x), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.84]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.839999999999999969 < y

   1. Initial program 28.5%

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

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.3%

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

   if -1 < y < 0.839999999999999969

   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

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.84\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - y\right) \cdot x, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.82)))
   (- 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 <= 0.82)) {
		tmp = 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 <= 0.82))
		tmp = Float64(x - Float64(-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, 0.82]], $MachinePrecision]], N[(x - N[(-1.0 / 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 0.819999999999999951 < y

   1. Initial program 28.5%

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

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.3%

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

   if -1 < y < 0.819999999999999951

   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

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

   5. Applied rewrites 97.8%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -66000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 57:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* -1.0 (- x))))
   (if (<= y -66000000000.0)
     t_0
     (if (<= y -3.9e-36) (* y x) (if (<= y 57.0) 1.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 * -x;
	double tmp;
	if (y <= -66000000000.0) {
		tmp = t_0;
	} else if (y <= -3.9e-36) {
		tmp = y * x;
	} else if (y <= 57.0) {
		tmp = 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) * -x
    if (y <= (-66000000000.0d0)) then
        tmp = t_0
    else if (y <= (-3.9d-36)) then
        tmp = y * x
    else if (y <= 57.0d0) then
        tmp = 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 * -x;
	double tmp;
	if (y <= -66000000000.0) {
		tmp = t_0;
	} else if (y <= -3.9e-36) {
		tmp = y * x;
	} else if (y <= 57.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 * -x
	tmp = 0
	if y <= -66000000000.0:
		tmp = t_0
	elif y <= -3.9e-36:
		tmp = y * x
	elif y <= 57.0:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 * Float64(-x))
	tmp = 0.0
	if (y <= -66000000000.0)
		tmp = t_0;
	elseif (y <= -3.9e-36)
		tmp = Float64(y * x);
	elseif (y <= 57.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 * -x;
	tmp = 0.0;
	if (y <= -66000000000.0)
		tmp = t_0;
	elseif (y <= -3.9e-36)
		tmp = y * x;
	elseif (y <= 57.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 * (-x)), $MachinePrecision]}, If[LessEqual[y, -66000000000.0], t$95$0, If[LessEqual[y, -3.9e-36], N[(y * x), $MachinePrecision], If[LessEqual[y, 57.0], 1.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.6e10 or 57 < y

   1. Initial program 27.5%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 68.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 73.4%

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

   if -6.6e10 < y < -3.9000000000000001e-36

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 70.0%

       \[\leadsto y \cdot x \]

   if -3.9000000000000001e-36 < y < 57

   1. Initial program 99.9%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 66.4%

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

Alternative 13: 63.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(1 - x\right)\\ \mathbf{if}\;y \leq -66000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 57:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (- 1.0 x))))
   (if (<= y -66000000000.0)
     t_0
     (if (<= y -3.9e-36) (* y x) (if (<= y 57.0) 1.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (1.0 - x);
	double tmp;
	if (y <= -66000000000.0) {
		tmp = t_0;
	} else if (y <= -3.9e-36) {
		tmp = y * x;
	} else if (y <= 57.0) {
		tmp = 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 - (1.0d0 - x)
    if (y <= (-66000000000.0d0)) then
        tmp = t_0
    else if (y <= (-3.9d-36)) then
        tmp = y * x
    else if (y <= 57.0d0) then
        tmp = 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 - (1.0 - x);
	double tmp;
	if (y <= -66000000000.0) {
		tmp = t_0;
	} else if (y <= -3.9e-36) {
		tmp = y * x;
	} else if (y <= 57.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (1.0 - x)
	tmp = 0
	if y <= -66000000000.0:
		tmp = t_0
	elif y <= -3.9e-36:
		tmp = y * x
	elif y <= 57.0:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(1.0 - x))
	tmp = 0.0
	if (y <= -66000000000.0)
		tmp = t_0;
	elseif (y <= -3.9e-36)
		tmp = Float64(y * x);
	elseif (y <= 57.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (1.0 - x);
	tmp = 0.0;
	if (y <= -66000000000.0)
		tmp = t_0;
	elseif (y <= -3.9e-36)
		tmp = y * x;
	elseif (y <= 57.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -66000000000.0], t$95$0, If[LessEqual[y, -3.9e-36], N[(y * x), $MachinePrecision], If[LessEqual[y, 57.0], 1.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.6e10 or 57 < y

   1. Initial program 27.5%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 49.0

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

   5. Applied rewrites 49.0%

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

   if -6.6e10 < y < -3.9000000000000001e-36

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 70.0%

       \[\leadsto y \cdot x \]

   if -3.9000000000000001e-36 < y < 57

   1. Initial program 99.9%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 66.4%

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

Alternative 14: 86.5% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* -1.0 (- 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 * -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(-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 * (-x)), $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 28.5%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 72.0%

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

   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

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

   5. Applied rewrites 97.8%

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

4. Final simplification 82.6%

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

Alternative 15: 38.8% 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 58.1%

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

3. Taylor expanded in x around -inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

5. Applied rewrites 84.8%

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

7. Applied rewrites 69.0%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 28.6%

    \[\leadsto \color{blue}{1} \]
11. 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 2025017 
(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))))