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

Percentage Accurate: 51.5% → 81.4%

Time: 7.5s

Alternatives: 5

Speedup: 3.2×

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

real(8) function code(x, y)
    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: 51.5% 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

real(8) function code(x, y)
    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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-295}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{\frac{y}{x}}}{y} + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-295)
     (+ (* 0.5 (/ (/ x (/ y x)) y)) -1.0)
     (if (<= (* x x) 2e+281) (/ (- (* x x) t_0) (+ (* x x) t_0)) 1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-295) {
		tmp = (0.5 * ((x / (y / x)) / y)) + -1.0;
	} else if ((x * x) <= 2e+281) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 1d-295) then
        tmp = (0.5d0 * ((x / (y / x)) / y)) + (-1.0d0)
    else if ((x * x) <= 2d+281) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-295) {
		tmp = (0.5 * ((x / (y / x)) / y)) + -1.0;
	} else if ((x * x) <= 2e+281) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 1e-295:
		tmp = (0.5 * ((x / (y / x)) / y)) + -1.0
	elif (x * x) <= 2e+281:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-295)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / Float64(y / x)) / y)) + -1.0);
	elseif (Float64(x * x) <= 2e+281)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 1e-295)
		tmp = (0.5 * ((x / (y / x)) / y)) + -1.0;
	elseif ((x * x) <= 2e+281)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-295], N[(N[(0.5 * N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+281], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.00000000000000006e-295

   1. Initial program 63.1%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

         \[\leadsto {x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{{y}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{y}^{2}} \cdot {x}^{2}\right)}\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{{x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}}\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{x}^{2}}{y \cdot \color{blue}{y}}\right)\right)\right) \]

      16. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\frac{{x}^{2}}{y}}{\color{blue}{y}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{{x}^{2}}{y}\right), \color{blue}{y}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2}\right), y\right), y\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), y\right), y\right)\right)\right) \]

      20. *-lowering-*.f64 90.4%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right), y\right)\right)\right) \]

   5. Simplified 90.4%

      \[\leadsto \color{blue}{-1 + 0.5 \cdot \frac{\frac{x \cdot x}{y}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{\frac{x \cdot x}{y}}{y}\right), \color{blue}{-1}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\frac{x \cdot x}{y}}{y}\right)\right), -1\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{x \cdot x}{y}\right), y\right)\right), -1\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(x \cdot \frac{x}{y}\right), y\right)\right), -1\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y}{x}}\right), y\right)\right), -1\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{x}{\frac{y}{x}}\right), y\right)\right), -1\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{x}\right)\right), y\right)\right), -1\right) \]

      9. /-lowering-/.f64 91.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), y\right)\right), -1\right) \]

   7. Applied egg-rr 91.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{x}{\frac{y}{x}}}{y} + -1} \]

   if 1.00000000000000006e-295 < (*.f64 x x) < 2.0000000000000001e281

   1. Initial program 84.4%

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

   if 2.0000000000000001e281 < (*.f64 x x)

   1. Initial program 2.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 80.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-295}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{\frac{y}{x}}}{y} + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.001:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{\frac{y}{x}}}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 0.001) (+ (* 0.5 (/ (/ x (/ y x)) y)) -1.0) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (0.5 * ((x / (y / x)) / y)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 0.001d0) then
        tmp = (0.5d0 * ((x / (y / x)) / y)) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 0.001) {
		tmp = (0.5 * ((x / (y / x)) / y)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 0.001:
		tmp = (0.5 * ((x / (y / x)) / y)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 0.001)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / Float64(y / x)) / y)) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 0.001)
		tmp = (0.5 * ((x / (y / x)) / y)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[(N[(0.5 * N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-3

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

         \[\leadsto {x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{{y}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{y}^{2}} \cdot {x}^{2}\right)}\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{{x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}}\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{x}^{2}}{y \cdot \color{blue}{y}}\right)\right)\right) \]

      16. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\frac{{x}^{2}}{y}}{\color{blue}{y}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{{x}^{2}}{y}\right), \color{blue}{y}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2}\right), y\right), y\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), y\right), y\right)\right)\right) \]

      20. *-lowering-*.f64 79.2%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right), y\right)\right)\right) \]

   5. Simplified 79.2%

      \[\leadsto \color{blue}{-1 + 0.5 \cdot \frac{\frac{x \cdot x}{y}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{\frac{x \cdot x}{y}}{y}\right), \color{blue}{-1}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\frac{x \cdot x}{y}}{y}\right)\right), -1\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{x \cdot x}{y}\right), y\right)\right), -1\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(x \cdot \frac{x}{y}\right), y\right)\right), -1\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y}{x}}\right), y\right)\right), -1\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{x}{\frac{y}{x}}\right), y\right)\right), -1\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{x}\right)\right), y\right)\right), -1\right) \]

      9. /-lowering-/.f64 79.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), y\right)\right), -1\right) \]

   7. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{x}{\frac{y}{x}}}{y} + -1} \]

   if 1e-3 < (*.f64 x x)

   1. Initial program 37.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 74.3%

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

Alternative 3: 75.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10.6:\\ \;\;\;\;-1 + 0.5 \cdot \frac{\frac{x \cdot x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 10.6) (+ -1.0 (* 0.5 (/ (/ (* x x) y) y))) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 10.6) {
		tmp = -1.0 + (0.5 * (((x * x) / y) / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 10.6d0) then
        tmp = (-1.0d0) + (0.5d0 * (((x * x) / y) / y))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 10.6) {
		tmp = -1.0 + (0.5 * (((x * x) / y) / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 10.6:
		tmp = -1.0 + (0.5 * (((x * x) / y) / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 10.6)
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(Float64(x * x) / y) / y)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 10.6)
		tmp = -1.0 + (0.5 * (((x * x) / y) / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 10.6], N[(-1.0 + N[(0.5 * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 10.5999999999999996

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

         \[\leadsto {x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{{y}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{y}^{2}} \cdot {x}^{2}\right)}\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{{x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}}\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{x}^{2}}{y \cdot \color{blue}{y}}\right)\right)\right) \]

      16. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\frac{{x}^{2}}{y}}{\color{blue}{y}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{{x}^{2}}{y}\right), \color{blue}{y}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2}\right), y\right), y\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), y\right), y\right)\right)\right) \]

      20. *-lowering-*.f64 79.2%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right), y\right)\right)\right) \]

   5. Simplified 79.2%

      \[\leadsto \color{blue}{-1 + 0.5 \cdot \frac{\frac{x \cdot x}{y}}{y}} \]

   if 10.5999999999999996 < (*.f64 x x)

   1. Initial program 37.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 74.3%

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

Alternative 4: 62.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 0.9) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.9) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.9d0) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.9) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.9:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.9)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.9)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.9], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 54.9%

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

   if 0.900000000000000022 < x

   1. Initial program 41.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 67.7%

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

Alternative 5: 51.2% accurate, 19.0× 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

real(8) function code(x, y)
    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 52.7%

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

3. Taylor expanded in x around 0

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

5. Simplified 48.9%

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

Developer Target 1: 51.9% accurate, 0.1× 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

real(8) function code(x, y)
    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 2024161 
(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))))