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

Percentage Accurate: 49.3% → 78.7%

Time: 6.8s

Alternatives: 6

Speedup: 19.0×

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

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: 49.3% 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

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: 78.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+165}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\frac{1}{x} \cdot \frac{y}{x}\right), -8, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (fma 0.5 (* (/ x y) (/ x y)) -1.0)))
   (if (<= (* x x) 5e-294)
     t_1
     (if (<= (* x x) 2e+88)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (if (<= (* x x) 2e+165)
         t_1
         (fma (* y (* (/ 1.0 x) (/ y x))) -8.0 1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = fma(0.5, ((x / y) * (x / y)), -1.0);
	double tmp;
	if ((x * x) <= 5e-294) {
		tmp = t_1;
	} else if ((x * x) <= 2e+88) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 2e+165) {
		tmp = t_1;
	} else {
		tmp = fma((y * ((1.0 / x) * (y / x))), -8.0, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0)
	tmp = 0.0
	if (Float64(x * x) <= 5e-294)
		tmp = t_1;
	elseif (Float64(x * x) <= 2e+88)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 2e+165)
		tmp = t_1;
	else
		tmp = fma(Float64(y * Float64(Float64(1.0 / x) * Float64(y / x))), -8.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-294], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 2e+88], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+165], t$95$1, N[(N[(y * N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -8.0 + 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.0000000000000003e-294 or 1.99999999999999992e88 < (*.f64 x x) < 1.9999999999999998e165

   1. Initial program 48.1%

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

   2. Taylor expanded in x around 0 79.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   3. Step-by-step derivation

      1. fma-neg 79.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 79.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]

      3. unpow2 79.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      4. times-frac 87.8%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]

      5. metadata-eval 87.8%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   4. Simplified 87.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]

   if 5.0000000000000003e-294 < (*.f64 x x) < 1.99999999999999992e88

   1. Initial program 82.5%

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

   if 1.9999999999999998e165 < (*.f64 x x)

   1. Initial program 25.3%

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

   2. Taylor expanded in x around inf 77.1%

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

      1. associate--l+ 77.1%

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

      2. distribute-rgt-out-- 77.1%

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

      3. metadata-eval 77.1%

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

      4. *-commutative 77.1%

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

      5. +-commutative 77.1%

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

      6. *-commutative 77.1%

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

      7. fma-def 77.1%

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

      8. unpow2 77.1%

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

      9. unpow2 77.1%

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

      10. times-frac 86.9%

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

   4. Simplified 86.9%

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

      1. add-log-exp 86.6%

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

      2. pow2 86.6%

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

   6. Applied egg-rr 86.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(e^{{\left(\frac{y}{x}\right)}^{2}}\right)}, -8, 1\right) \]
   7. Step-by-step derivation

      1. add-log-exp 86.9%

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

      2. unpow2 86.9%

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

      3. div-inv 86.9%

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

      4. associate-*l* 86.9%

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

   8. Applied egg-rr 86.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\frac{1}{x} \cdot \frac{y}{x}\right), -8, 1\right)\\ \end{array} \]

Alternative 2: 77.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{elif}\;x \cdot x \leq 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (fma 0.5 (* (/ x y) (/ x y)) -1.0)))
   (if (<= (* x x) 5e-294)
     t_1
     (if (<= (* x x) 2e+88)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (if (<= (* x x) 1e+181) t_1 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = fma(0.5, ((x / y) * (x / y)), -1.0);
	double tmp;
	if ((x * x) <= 5e-294) {
		tmp = t_1;
	} else if ((x * x) <= 2e+88) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 1e+181) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0)
	tmp = 0.0
	if (Float64(x * x) <= 5e-294)
		tmp = t_1;
	elseif (Float64(x * x) <= 2e+88)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 1e+181)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-294], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 2e+88], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+181], t$95$1, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.0000000000000003e-294 or 1.99999999999999992e88 < (*.f64 x x) < 9.9999999999999992e180

   1. Initial program 50.6%

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

   2. Taylor expanded in x around 0 77.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   3. Step-by-step derivation

      1. fma-neg 77.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 77.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]

      3. unpow2 77.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      4. times-frac 85.2%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]

      5. metadata-eval 85.2%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   4. Simplified 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]

   if 5.0000000000000003e-294 < (*.f64 x x) < 1.99999999999999992e88

   1. Initial program 82.5%

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

   if 9.9999999999999992e180 < (*.f64 x x)

   1. Initial program 21.5%

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

   2. Taylor expanded in x around inf 88.5%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 10^{+181}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 78.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+165}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (fma 0.5 (* (/ x y) (/ x y)) -1.0)))
   (if (<= (* x x) 5e-294)
     t_1
     (if (<= (* x x) 2e+88)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (if (<= (* x x) 2e+165) t_1 (fma (* (/ y x) (/ y x)) -8.0 1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = fma(0.5, ((x / y) * (x / y)), -1.0);
	double tmp;
	if ((x * x) <= 5e-294) {
		tmp = t_1;
	} else if ((x * x) <= 2e+88) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 2e+165) {
		tmp = t_1;
	} else {
		tmp = fma(((y / x) * (y / x)), -8.0, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0)
	tmp = 0.0
	if (Float64(x * x) <= 5e-294)
		tmp = t_1;
	elseif (Float64(x * x) <= 2e+88)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 2e+165)
		tmp = t_1;
	else
		tmp = fma(Float64(Float64(y / x) * Float64(y / x)), -8.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-294], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 2e+88], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+165], t$95$1, N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -8.0 + 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.0000000000000003e-294 or 1.99999999999999992e88 < (*.f64 x x) < 1.9999999999999998e165

   1. Initial program 48.1%

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

   2. Taylor expanded in x around 0 79.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   3. Step-by-step derivation

      1. fma-neg 79.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 79.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]

      3. unpow2 79.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      4. times-frac 87.8%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]

      5. metadata-eval 87.8%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   4. Simplified 87.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]

   if 5.0000000000000003e-294 < (*.f64 x x) < 1.99999999999999992e88

   1. Initial program 82.5%

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

   if 1.9999999999999998e165 < (*.f64 x x)

   1. Initial program 25.3%

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

   2. Taylor expanded in x around inf 77.1%

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

      1. associate--l+ 77.1%

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

      2. distribute-rgt-out-- 77.1%

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

      3. metadata-eval 77.1%

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

      4. *-commutative 77.1%

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

      5. +-commutative 77.1%

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

      6. *-commutative 77.1%

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

      7. fma-def 77.1%

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

      8. unpow2 77.1%

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

      9. unpow2 77.1%

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

      10. times-frac 86.9%

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

   4. Simplified 86.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \end{array} \]

Alternative 4: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 4.8 \cdot 10^{-294}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 6.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{elif}\;x \cdot x \leq 1.45 \cdot 10^{+165}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 4.8e-294)
     -1.0
     (if (<= (* x x) 6.2e+88)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (if (<= (* x x) 1.45e+165) -1.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 4.8e-294) {
		tmp = -1.0;
	} else if ((x * x) <= 6.2e+88) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 1.45e+165) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 4.8d-294) then
        tmp = -1.0d0
    else if ((x * x) <= 6.2d+88) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else if ((x * x) <= 1.45d+165) then
        tmp = -1.0d0
    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) <= 4.8e-294) {
		tmp = -1.0;
	} else if ((x * x) <= 6.2e+88) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 1.45e+165) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 4.8e-294:
		tmp = -1.0
	elif (x * x) <= 6.2e+88:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	elif (x * x) <= 1.45e+165:
		tmp = -1.0
	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) <= 4.8e-294)
		tmp = -1.0;
	elseif (Float64(x * x) <= 6.2e+88)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 1.45e+165)
		tmp = -1.0;
	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) <= 4.8e-294)
		tmp = -1.0;
	elseif ((x * x) <= 6.2e+88)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	elseif ((x * x) <= 1.45e+165)
		tmp = -1.0;
	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], 4.8e-294], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 6.2e+88], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1.45e+165], -1.0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.79999999999999994e-294 or 6.2000000000000003e88 < (*.f64 x x) < 1.45000000000000003e165

   1. Initial program 48.1%

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

   2. Taylor expanded in x around 0 87.2%

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

   if 4.79999999999999994e-294 < (*.f64 x x) < 6.2000000000000003e88

   1. Initial program 82.5%

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

   if 1.45000000000000003e165 < (*.f64 x x)

   1. Initial program 25.3%

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

   2. Taylor expanded in x around inf 86.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.8 \cdot 10^{-294}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 6.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 1.45 \cdot 10^{+165}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 60.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+58}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4e-50) 1.0 (if (<= y 5e+58) -1.0 (if (<= y 1.05e+76) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e-50) {
		tmp = 1.0;
	} else if (y <= 5e+58) {
		tmp = -1.0;
	} else if (y <= 1.05e+76) {
		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 (y <= 4d-50) then
        tmp = 1.0d0
    else if (y <= 5d+58) then
        tmp = -1.0d0
    else if (y <= 1.05d+76) 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 (y <= 4e-50) {
		tmp = 1.0;
	} else if (y <= 5e+58) {
		tmp = -1.0;
	} else if (y <= 1.05e+76) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e-50:
		tmp = 1.0
	elif y <= 5e+58:
		tmp = -1.0
	elif y <= 1.05e+76:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e-50)
		tmp = 1.0;
	elseif (y <= 5e+58)
		tmp = -1.0;
	elseif (y <= 1.05e+76)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-50)
		tmp = 1.0;
	elseif (y <= 5e+58)
		tmp = -1.0;
	elseif (y <= 1.05e+76)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e-50], 1.0, If[LessEqual[y, 5e+58], -1.0, If[LessEqual[y, 1.05e+76], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.00000000000000003e-50 or 4.99999999999999986e58 < y < 1.05000000000000003e76

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf 62.8%

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

   if 4.00000000000000003e-50 < y < 4.99999999999999986e58 or 1.05000000000000003e76 < y

   1. Initial program 37.1%

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

   2. Taylor expanded in x around 0 73.0%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+58}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 50.3% 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 50.0%

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

2. Taylor expanded in x around 0 46.5%

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

3. Final simplification 46.5%

   \[\leadsto -1 \]

Developer target: 49.8% 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 2023240 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

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