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

Percentage Accurate: 50.5% → 81.4%

Time: 2.9s

Alternatives: 7

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: 50.5% 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: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4 \cdot y\_m, y\_m, x \cdot x\right)\\ \mathbf{if}\;y\_m \leq 3 \cdot 10^{-158}:\\ \;\;\;\;\mathsf{fma}\left(-8, \frac{y\_m}{x} \cdot \frac{y\_m}{x}, 1\right)\\ \mathbf{elif}\;y\_m \leq 1.4 \cdot 10^{+98}:\\ \;\;\;\;\frac{x \cdot x}{t\_0} - \frac{\left(4 \cdot y\_m\right) \cdot y\_m}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (fma (* 4.0 y_m) y_m (* x x))))
   (if (<= y_m 3e-158)
     (fma -8.0 (* (/ y_m x) (/ y_m x)) 1.0)
     (if (<= y_m 1.4e+98)
       (- (/ (* x x) t_0) (/ (* (* 4.0 y_m) y_m) t_0))
       -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = fma((4.0 * y_m), y_m, (x * x));
	double tmp;
	if (y_m <= 3e-158) {
		tmp = fma(-8.0, ((y_m / x) * (y_m / x)), 1.0);
	} else if (y_m <= 1.4e+98) {
		tmp = ((x * x) / t_0) - (((4.0 * y_m) * y_m) / t_0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = fma(Float64(4.0 * y_m), y_m, Float64(x * x))
	tmp = 0.0
	if (y_m <= 3e-158)
		tmp = fma(-8.0, Float64(Float64(y_m / x) * Float64(y_m / x)), 1.0);
	elseif (y_m <= 1.4e+98)
		tmp = Float64(Float64(Float64(x * x) / t_0) - Float64(Float64(Float64(4.0 * y_m) * y_m) / t_0));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$95$m, 3e-158], 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, 1.4e+98], N[(N[(N[(x * x), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 3e-158

   1. Initial program 51.2%

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

   3. 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 52.8

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

   5. Applied rewrites 52.8%

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

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

      3. lift-/.f64 N/A

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

      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 57.8

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

   7. Applied rewrites 57.8%

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

   if 3e-158 < y < 1.4e98

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-sub N/A

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

      10. lower--.f64 N/A

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

   4. Applied rewrites 82.2%

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

   if 1.4e98 < y

   1. Initial program 15.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.6%

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

Alternative 2: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{-158}:\\ \;\;\;\;\mathsf{fma}\left(-8, \frac{y\_m}{x} \cdot \frac{y\_m}{x}, 1\right)\\ \mathbf{elif}\;y\_m \leq 1.4 \cdot 10^{+98}:\\ \;\;\;\;\frac{x \cdot x - \left(y\_m \cdot 4\right) \cdot y\_m}{\mathsf{fma}\left(x, x, \left(4 \cdot y\_m\right) \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3e-158)
   (fma -8.0 (* (/ y_m x) (/ y_m x)) 1.0)
   (if (<= y_m 1.4e+98)
     (/ (- (* x x) (* (* y_m 4.0) y_m)) (fma x x (* (* 4.0 y_m) y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3e-158) {
		tmp = fma(-8.0, ((y_m / x) * (y_m / x)), 1.0);
	} else if (y_m <= 1.4e+98) {
		tmp = ((x * x) - ((y_m * 4.0) * y_m)) / fma(x, x, ((4.0 * y_m) * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 3e-158)
		tmp = fma(-8.0, Float64(Float64(y_m / x) * Float64(y_m / x)), 1.0);
	elseif (y_m <= 1.4e+98)
		tmp = Float64(Float64(Float64(x * x) - Float64(Float64(y_m * 4.0) * y_m)) / fma(x, x, Float64(Float64(4.0 * y_m) * y_m)));
	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, 3e-158], 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, 1.4e+98], N[(N[(N[(x * x), $MachinePrecision] - N[(N[(y$95$m * 4.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 3e-158

   1. Initial program 51.2%

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

   3. 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 52.8

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

   5. Applied rewrites 52.8%

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

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

      3. lift-/.f64 N/A

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

      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 57.8

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

   7. Applied rewrites 57.8%

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

   if 3e-158 < y < 1.4e98

   1. Initial program 82.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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. lower-fma.f64 82.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

   if 1.4e98 < y

   1. Initial program 15.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.6%

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

Alternative 3: 81.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{-158}:\\ \;\;\;\;\mathsf{fma}\left(-8, \frac{y\_m}{x} \cdot \frac{y\_m}{x}, 1\right)\\ \mathbf{elif}\;y\_m \leq 1.4 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m \cdot y\_m, -4, x \cdot x\right)}{\mathsf{fma}\left(x, x, \left(4 \cdot y\_m\right) \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3e-158)
   (fma -8.0 (* (/ y_m x) (/ y_m x)) 1.0)
   (if (<= y_m 1.4e+98)
     (/ (fma (* y_m y_m) -4.0 (* x x)) (fma x x (* (* 4.0 y_m) y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3e-158) {
		tmp = fma(-8.0, ((y_m / x) * (y_m / x)), 1.0);
	} else if (y_m <= 1.4e+98) {
		tmp = fma((y_m * y_m), -4.0, (x * x)) / fma(x, x, ((4.0 * y_m) * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 3e-158)
		tmp = fma(-8.0, Float64(Float64(y_m / x) * Float64(y_m / x)), 1.0);
	elseif (y_m <= 1.4e+98)
		tmp = Float64(fma(Float64(y_m * y_m), -4.0, Float64(x * x)) / fma(x, x, Float64(Float64(4.0 * y_m) * y_m)));
	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, 3e-158], 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, 1.4e+98], N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 3e-158

   1. Initial program 51.2%

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

   3. 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 52.8

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

   5. Applied rewrites 52.8%

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

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

      3. lift-/.f64 N/A

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

      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 57.8

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

   7. Applied rewrites 57.8%

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

   if 3e-158 < y < 1.4e98

   1. Initial program 82.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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. lower-fma.f64 82.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 82.2

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

   6. Applied rewrites 82.2%

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

   if 1.4e98 < y

   1. Initial program 15.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.6%

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

Alternative 4: 75.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-8, \frac{y\_m}{x} \cdot \frac{y\_m}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.1e-7) (fma -8.0 (* (/ y_m x) (/ y_m x)) 1.0) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.1e-7) {
		tmp = fma(-8.0, ((y_m / x) * (y_m / x)), 1.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.1e-7)
		tmp = fma(-8.0, Float64(Float64(y_m / x) * Float64(y_m / x)), 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, 1.1e-7], N[(-8.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.1000000000000001e-7

   1. Initial program 56.5%

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

   3. Taylor expanded in 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 55.8

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

   5. Applied rewrites 55.8%

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

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

      3. lift-/.f64 N/A

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

      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 60.0

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

   7. Applied rewrites 60.0%

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

   if 1.1000000000000001e-7 < y

   1. Initial program 28.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.1%

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

Alternative 5: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y\_m \cdot y\_m}{x}, -8, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.1e-7) (/ (fma (/ (* y_m y_m) x) -8.0 x) x) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.1e-7) {
		tmp = fma(((y_m * y_m) / x), -8.0, x) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.1e-7)
		tmp = Float64(fma(Float64(Float64(y_m * y_m) / x), -8.0, x) / x);
	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, 1.1e-7], N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] / x), $MachinePrecision] * -8.0 + x), $MachinePrecision] / x), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.1000000000000001e-7

   1. Initial program 56.5%

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

   3. Taylor expanded in 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 55.8

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

   5. Applied rewrites 55.8%

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

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{-8 \cdot {y}^{2}}{{x}^{2}} + {x}^{\color{blue}{0}} \]

      9. metadata-eval N/A

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

      10. pow-div N/A

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

      11. div-add N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 33.5

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

   7. Applied rewrites 33.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\left(y \cdot y\right) \cdot -8 + x \cdot x}{\color{blue}{x} \cdot x} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(y \cdot y\right) \cdot -8 + x \cdot x}{x}}{\color{blue}{x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(y \cdot y\right) \cdot -8 + x \cdot x}{x}}{\color{blue}{x}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(y \cdot y\right) \cdot -8 + x \cdot x}{x}}{x} \]

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 34.2

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

   9. Applied rewrites 34.2%

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

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 59.5

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

   12. Applied rewrites 59.5%

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

   if 1.1000000000000001e-7 < y

   1. Initial program 28.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.1%

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

Alternative 6: 74.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 1.2e-17) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.2e-17) {
		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 (y_m <= 1.2d-17) 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 (y_m <= 1.2e-17) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.2e-17:
		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 <= 1.2e-17)
		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 (y_m <= 1.2e-17)
		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[y$95$m, 1.2e-17], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.19999999999999993e-17

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 57.9%

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

   if 1.19999999999999993e-17 < y

   1. Initial program 29.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.8%

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

Alternative 7: 51.3% 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 49.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.2%

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

Developer Target 1: 50.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t\_0\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t\_3}{x \cdot x + t\_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t\_1} - t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t\_1}}\right)}^{2} - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2025064 
(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))))