Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Percentage Accurate: 50.3% → 78.8%

Time: 2.5s

Alternatives: 4

Speedup: 48.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_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
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 50.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_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
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.42 \cdot 10^{-138}:\\ \;\;\;\;\mathsf{fma}\left(-8, \frac{y\_m}{x} \cdot \frac{y\_m}{x}, 1\right)\\ \mathbf{elif}\;y\_m \leq 8.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m \cdot y\_m, -4, x \cdot x\right)}{\mathsf{fma}\left(4 \cdot y\_m, y\_m, x \cdot x\right)}\\ \mathbf{elif}\;y\_m \leq 1.6 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2.42e-138)
   (fma -8.0 (* (/ y_m x) (/ y_m x)) 1.0)
   (if (<= y_m 8.8e+20)
     (/ (fma (* y_m y_m) -4.0 (* x x)) (fma (* 4.0 y_m) y_m (* x x)))
     (if (<= y_m 1.6e+40) 1.0 -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 2.42e-138) {
		tmp = fma(-8.0, ((y_m / x) * (y_m / x)), 1.0);
	} else if (y_m <= 8.8e+20) {
		tmp = fma((y_m * y_m), -4.0, (x * x)) / fma((4.0 * y_m), y_m, (x * x));
	} else if (y_m <= 1.6e+40) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 2.42e-138)
		tmp = fma(-8.0, Float64(Float64(y_m / x) * Float64(y_m / x)), 1.0);
	elseif (y_m <= 8.8e+20)
		tmp = Float64(fma(Float64(y_m * y_m), -4.0, Float64(x * x)) / fma(Float64(4.0 * y_m), y_m, Float64(x * x)));
	elseif (y_m <= 1.6e+40)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 2.42e-138], N[(-8.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 8.8e+20], N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.6e+40], 1.0, -1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.41999999999999993e-138

   1. Initial program 44.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto -8 \cdot \frac{{y}^{2}}{{x}^{2}} + \color{blue}{1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \color{blue}{\frac{{y}^{2}}{{x}^{2}}}, 1\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{{y}^{2}}{\color{blue}{{x}^{2}}}, 1\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{{\color{blue}{x}}^{2}}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{{\color{blue}{x}}^{2}}, 1\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{x \cdot \color{blue}{x}}, 1\right) \]

      7. lift-*.f64 51.6

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{x \cdot \color{blue}{x}}, 1\right) \]

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-8, \frac{y \cdot y}{x \cdot x}, 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{\color{blue}{x} \cdot x}, 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{\color{blue}{x \cdot x}}, 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{x \cdot \color{blue}{x}}, 1\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y}{x} \cdot \color{blue}{\frac{y}{x}}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y}{x} \cdot \color{blue}{\frac{y}{x}}, 1\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y}{x} \cdot \frac{\color{blue}{y}}{x}, 1\right) \]

      7. lower-/.f64 61.1

         \[\leadsto \mathsf{fma}\left(-8, \frac{y}{x} \cdot \frac{y}{\color{blue}{x}}, 1\right) \]

   7. Applied rewrites 61.1%

      \[\leadsto \mathsf{fma}\left(-8, \frac{y}{x} \cdot \color{blue}{\frac{y}{x}}, 1\right) \]

   if 2.41999999999999993e-138 < y < 8.8e20

   1. Initial program 94.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{x \cdot x} + \left(y \cdot 4\right) \cdot y} \]

      3. pow2 N/A

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{x}^{2}} + \left(y \cdot 4\right) \cdot y} \]

      4. +-commutative N/A

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y + {x}^{2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right)} \cdot y + {x}^{2}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y} + {x}^{2}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, {x}^{2}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(\color{blue}{4 \cdot y}, y, {x}^{2}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(\color{blue}{4 \cdot y}, y, {x}^{2}\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(4 \cdot y, y, \color{blue}{x \cdot x}\right)} \]

      11. lift-*.f64 94.2

          \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(4 \cdot y, y, \color{blue}{x \cdot x}\right)} \]

   4. Applied rewrites 94.2%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot y}}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x} - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x - \color{blue}{\left(y \cdot 4\right)} \cdot y}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot y}}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot x - \color{blue}{\left(4 \cdot y\right)} \cdot y}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\color{blue}{{x}^{2}} - \left(4 \cdot y\right) \cdot y}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{{x}^{2} - \color{blue}{4 \cdot \left(y \cdot y\right)}}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{{x}^{2} - 4 \cdot \color{blue}{{y}^{2}}}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

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

         \[\leadsto \frac{\color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot {y}^{2}}}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{{x}^{2} + \color{blue}{-4} \cdot {y}^{2}}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\color{blue}{-4 \cdot {y}^{2} + {x}^{2}}}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{y}^{2} \cdot -4} + {x}^{2}}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({y}^{2}, -4, {x}^{2}\right)}}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      14. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot y}, -4, {x}^{2}\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot y}, -4, {x}^{2}\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      16. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -4, \color{blue}{x \cdot x}\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

      17. lift-*.f64 94.2

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -4, \color{blue}{x \cdot x}\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

   6. Applied rewrites 94.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot y, -4, x \cdot x\right)}}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \]

   if 8.8e20 < y < 1.5999999999999999e40

   1. Initial program 20.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 100.0%

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

   if 1.5999999999999999e40 < y

   1. Initial program 28.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.8%

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

Alternative 2: 63.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-40}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-8, \frac{y\_m}{x} \cdot \frac{y\_m}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= x 3.8e-40) -1.0 (fma -8.0 (* (/ y_m x) (/ y_m x)) 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (x <= 3.8e-40) {
		tmp = -1.0;
	} else {
		tmp = fma(-8.0, ((y_m / x) * (y_m / x)), 1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (x <= 3.8e-40)
		tmp = -1.0;
	else
		tmp = fma(-8.0, Float64(Float64(y_m / x) * Float64(y_m / x)), 1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[x, 3.8e-40], -1.0, N[(-8.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7999999999999999e-40

   1. Initial program 47.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.7%

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

   if 3.7999999999999999e-40 < x

   1. Initial program 46.6%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto -8 \cdot \frac{{y}^{2}}{{x}^{2}} + \color{blue}{1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \color{blue}{\frac{{y}^{2}}{{x}^{2}}}, 1\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{{y}^{2}}{\color{blue}{{x}^{2}}}, 1\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{{\color{blue}{x}}^{2}}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{{\color{blue}{x}}^{2}}, 1\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{x \cdot \color{blue}{x}}, 1\right) \]

      7. lift-*.f64 68.0

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{x \cdot \color{blue}{x}}, 1\right) \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-8, \frac{y \cdot y}{x \cdot x}, 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{\color{blue}{x} \cdot x}, 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{\color{blue}{x \cdot x}}, 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y \cdot y}{x \cdot \color{blue}{x}}, 1\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y}{x} \cdot \color{blue}{\frac{y}{x}}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y}{x} \cdot \color{blue}{\frac{y}{x}}, 1\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-8, \frac{y}{x} \cdot \frac{\color{blue}{y}}{x}, 1\right) \]

      7. lower-/.f64 73.5

         \[\leadsto \mathsf{fma}\left(-8, \frac{y}{x} \cdot \frac{y}{\color{blue}{x}}, 1\right) \]

   7. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(-8, \frac{y}{x} \cdot \color{blue}{\frac{y}{x}}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 62.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-40}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= x 5e-40) -1.0 1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (x <= 5e-40) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (x <= 5d-40) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (x <= 5e-40) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if x <= 5e-40:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (x <= 5e-40)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (x <= 5e-40)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[x, 5e-40], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999965e-40

   1. Initial program 47.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.7%

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

   if 4.99999999999999965e-40 < x

   1. Initial program 46.6%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 72.8%

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

Alternative 4: 50.9% accurate, 48.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ -1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

y_m = fabs(y);
double code(double x, double y_m) {
	return -1.0;
}

Fortran

y_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = -1.0d0
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return -1.0;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

y_m = abs(y)
function code(x, y_m)
	return -1.0
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

1. Initial program 47.3%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 50.4%

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

Reproduce

herbie shell --seed 2025084 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 9743233849626781/10000000000000000) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4))))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))