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

Percentage Accurate: 51.4% → 99.9%

Time: 16.1s

Alternatives: 7

Speedup: 1.7×

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: 51.4% 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: 99.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, 2 \cdot y\right)\\ {\left(\frac{x}{t_0}\right)}^{2} + \frac{\frac{y}{t_0} \cdot \frac{y}{-0.25}}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* 2.0 y))))
   (+ (pow (/ x t_0) 2.0) (/ (* (/ y t_0) (/ y -0.25)) t_0))))

C

double code(double x, double y) {
	double t_0 = hypot(x, (2.0 * y));
	return pow((x / t_0), 2.0) + (((y / t_0) * (y / -0.25)) / t_0);
}

Java

public static double code(double x, double y) {
	double t_0 = Math.hypot(x, (2.0 * y));
	return Math.pow((x / t_0), 2.0) + (((y / t_0) * (y / -0.25)) / t_0);
}

Python

def code(x, y):
	t_0 = math.hypot(x, (2.0 * y))
	return math.pow((x / t_0), 2.0) + (((y / t_0) * (y / -0.25)) / t_0)

Julia

function code(x, y)
	t_0 = hypot(x, Float64(2.0 * y))
	return Float64((Float64(x / t_0) ^ 2.0) + Float64(Float64(Float64(y / t_0) * Float64(y / -0.25)) / t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = hypot(x, (2.0 * y));
	tmp = ((x / t_0) ^ 2.0) + (((y / t_0) * (y / -0.25)) / t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(2.0 * y), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[Power[N[(x / t$95$0), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(y / t$95$0), $MachinePrecision] * N[(y / -0.25), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 48.8%

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

   1. div-inv 48.3%

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

   2. *-commutative 48.3%

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

   3. sub-neg 48.3%

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

   4. distribute-lft-in 48.3%

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

   5. *-commutative 48.3%

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

   6. div-inv 48.4%

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

   7. pow2 48.4%

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

   8. fma-def 48.4%

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

   9. *-commutative 48.4%

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

   10. associate-*l* 48.4%

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

   11. pow2 48.4%

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

3. Applied egg-rr 48.4%

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

5. Applied egg-rr 67.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2}\right)} - 1\right)} + \frac{1}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)} \cdot \left(y \cdot \left(y \cdot -4\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 67.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2}\right)\right)} + \frac{1}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)} \cdot \left(y \cdot \left(y \cdot -4\right)\right) \]

   2. expm1-log1p 67.5%

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

7. Simplified 67.5%

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

9. Applied egg-rr 71.7%

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

    1. unpow2 71.7%

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

    2. div-inv 71.7%

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

    3. times-frac 100.0%

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

    4. metadata-eval 100.0%

       \[\leadsto {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2} + \frac{\frac{y}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \cdot \frac{y}{\color{blue}{-0.25}}}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \]

11. Applied egg-rr 100.0%

    \[\leadsto {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2} + \frac{\color{blue}{\frac{y}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \cdot \frac{y}{-0.25}}}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \]

12. Final simplification 100.0%

    \[\leadsto {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2} + \frac{\frac{y}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \cdot \frac{y}{-0.25}}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \]

Alternative 2: 99.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, 2 \cdot y\right)\\ {\left(\frac{x}{t_0}\right)}^{2} + {\left(\frac{y}{t_0}\right)}^{2} \cdot -4 \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* 2.0 y))))
   (+ (pow (/ x t_0) 2.0) (* (pow (/ y t_0) 2.0) -4.0))))

C

double code(double x, double y) {
	double t_0 = hypot(x, (2.0 * y));
	return pow((x / t_0), 2.0) + (pow((y / t_0), 2.0) * -4.0);
}

Java

public static double code(double x, double y) {
	double t_0 = Math.hypot(x, (2.0 * y));
	return Math.pow((x / t_0), 2.0) + (Math.pow((y / t_0), 2.0) * -4.0);
}

Python

def code(x, y):
	t_0 = math.hypot(x, (2.0 * y))
	return math.pow((x / t_0), 2.0) + (math.pow((y / t_0), 2.0) * -4.0)

Julia

function code(x, y)
	t_0 = hypot(x, Float64(2.0 * y))
	return Float64((Float64(x / t_0) ^ 2.0) + Float64((Float64(y / t_0) ^ 2.0) * -4.0))
end

MATLAB

function tmp = code(x, y)
	t_0 = hypot(x, (2.0 * y));
	tmp = ((x / t_0) ^ 2.0) + (((y / t_0) ^ 2.0) * -4.0);
end

Wolfram

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

Derivation

1. Initial program 48.8%

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

   1. div-inv 48.3%

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

   2. *-commutative 48.3%

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

   3. sub-neg 48.3%

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

   4. distribute-lft-in 48.3%

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

   5. *-commutative 48.3%

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

   6. div-inv 48.4%

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

   7. pow2 48.4%

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

   8. fma-def 48.4%

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

   9. *-commutative 48.4%

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

   10. associate-*l* 48.4%

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

   11. pow2 48.4%

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

3. Applied egg-rr 48.4%

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

5. Applied egg-rr 67.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2}\right)} - 1\right)} + \frac{1}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)} \cdot \left(y \cdot \left(y \cdot -4\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 67.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2}\right)\right)} + \frac{1}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)} \cdot \left(y \cdot \left(y \cdot -4\right)\right) \]

   2. expm1-log1p 67.5%

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

7. Simplified 67.5%

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

9. Applied egg-rr 71.2%

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

    1. associate-/l* 71.2%

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

    2. associate-/r/ 71.2%

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

    3. *-un-lft-identity 71.2%

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

    4. metadata-eval 71.2%

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

    5. *-commutative 71.2%

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

    6. div-inv 71.2%

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

    7. unpow2 71.2%

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

    8. associate-*l* 99.8%

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

    9. div-inv 100.0%

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

    10. frac-times 100.0%

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

    11. metadata-eval 100.0%

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

    12. clear-num 100.0%

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

    13. pow2 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 3: 73.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, 2 \cdot y\right)\\ {\left(\frac{x}{t_0}\right)}^{2} + \frac{y \cdot -2}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* 2.0 y)))) (+ (pow (/ x t_0) 2.0) (/ (* y -2.0) t_0))))

C

double code(double x, double y) {
	double t_0 = hypot(x, (2.0 * y));
	return pow((x / t_0), 2.0) + ((y * -2.0) / t_0);
}

Java

public static double code(double x, double y) {
	double t_0 = Math.hypot(x, (2.0 * y));
	return Math.pow((x / t_0), 2.0) + ((y * -2.0) / t_0);
}

Python

def code(x, y):
	t_0 = math.hypot(x, (2.0 * y))
	return math.pow((x / t_0), 2.0) + ((y * -2.0) / t_0)

Julia

function code(x, y)
	t_0 = hypot(x, Float64(2.0 * y))
	return Float64((Float64(x / t_0) ^ 2.0) + Float64(Float64(y * -2.0) / t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = hypot(x, (2.0 * y));
	tmp = ((x / t_0) ^ 2.0) + ((y * -2.0) / t_0);
end

Wolfram

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

Derivation

1. Initial program 48.8%

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

   1. div-inv 48.3%

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

   2. *-commutative 48.3%

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

   3. sub-neg 48.3%

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

   4. distribute-lft-in 48.3%

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

   5. *-commutative 48.3%

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

   6. div-inv 48.4%

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

   7. pow2 48.4%

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

   8. fma-def 48.4%

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

   9. *-commutative 48.4%

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

   10. associate-*l* 48.4%

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

   11. pow2 48.4%

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

3. Applied egg-rr 48.4%

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

5. Applied egg-rr 67.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2}\right)} - 1\right)} + \frac{1}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)} \cdot \left(y \cdot \left(y \cdot -4\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 67.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2}\right)\right)} + \frac{1}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)} \cdot \left(y \cdot \left(y \cdot -4\right)\right) \]

   2. expm1-log1p 67.5%

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

7. Simplified 67.5%

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

9. Applied egg-rr 71.7%

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

10. Taylor expanded in y around inf 72.2%

    \[\leadsto {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2} + \frac{\color{blue}{-2 \cdot y}}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \]
11. Step-by-step derivation

    1. *-commutative 72.2%

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

12. Simplified 72.2%

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

13. Final simplification 72.2%

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

Alternative 4: 79.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-218}:\\ \;\;\;\;{\left(\frac{x \cdot 0.5}{y}\right)}^{2} + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+165}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+190}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 2e-218)
     (+ (pow (/ (* x 0.5) y) 2.0) -1.0)
     (if (<= (* x x) 1e+165)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (if (<= (* x x) 5e+190) -1.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 2e-218) {
		tmp = pow(((x * 0.5) / y), 2.0) + -1.0;
	} else if ((x * x) <= 1e+165) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 5e+190) {
		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) <= 2d-218) then
        tmp = (((x * 0.5d0) / y) ** 2.0d0) + (-1.0d0)
    else if ((x * x) <= 1d+165) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else if ((x * x) <= 5d+190) 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) <= 2e-218) {
		tmp = Math.pow(((x * 0.5) / y), 2.0) + -1.0;
	} else if ((x * x) <= 1e+165) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 5e+190) {
		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) <= 2e-218:
		tmp = math.pow(((x * 0.5) / y), 2.0) + -1.0
	elif (x * x) <= 1e+165:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	elif (x * x) <= 5e+190:
		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) <= 2e-218)
		tmp = Float64((Float64(Float64(x * 0.5) / y) ^ 2.0) + -1.0);
	elseif (Float64(x * x) <= 1e+165)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 5e+190)
		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) <= 2e-218)
		tmp = (((x * 0.5) / y) ^ 2.0) + -1.0;
	elseif ((x * x) <= 1e+165)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	elseif ((x * x) <= 5e+190)
		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], 2e-218], N[(N[Power[N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+165], 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], 5e+190], -1.0, 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 2.0000000000000001e-218

   1. Initial program 51.6%

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

      1. div-inv 50.5%

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

      2. *-commutative 50.5%

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

      3. sub-neg 50.5%

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

      4. distribute-lft-in 50.4%

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

      5. *-commutative 50.4%

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

      6. div-inv 50.5%

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

      7. pow2 50.5%

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

      8. fma-def 50.5%

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

      9. *-commutative 50.5%

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

      10. associate-*l* 50.5%

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

      11. pow2 50.5%

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

   3. Applied egg-rr 50.5%

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

   5. Applied egg-rr 50.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2}\right)} - 1\right)} + \frac{1}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)} \cdot \left(y \cdot \left(y \cdot -4\right)\right) \]
   6. Step-by-step derivation

      1. expm1-def 50.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot y\right)}\right)}^{2}\right)\right)} + \frac{1}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)} \cdot \left(y \cdot \left(y \cdot -4\right)\right) \]

      2. expm1-log1p 50.5%

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

   7. Simplified 50.5%

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

   8. Taylor expanded in x around 0 85.7%

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

   9. Taylor expanded in x around 0 86.2%

      \[\leadsto {\color{blue}{\left(0.5 \cdot \frac{x}{y}\right)}}^{2} + -1 \]
   10. Step-by-step derivation

       1. associate-*r/ 86.2%

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

   11. Simplified 86.2%

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

   if 2.0000000000000001e-218 < (*.f64 x x) < 9.99999999999999899e164

   1. Initial program 76.7%

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

   if 9.99999999999999899e164 < (*.f64 x x) < 5.00000000000000036e190

   1. Initial program 16.7%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000036e190 < (*.f64 x x)

   1. Initial program 23.8%

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

   2. Taylor expanded in x around inf 89.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-218}:\\ \;\;\;\;{\left(\frac{x \cdot 0.5}{y}\right)}^{2} + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+165}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+190}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 2.1 \cdot 10^{-217}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 9 \cdot 10^{+165}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+190}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 2.1e-217)
     -1.0
     (if (<= (* x x) 9e+165)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (if (<= (* x x) 5e+190) -1.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 2.1e-217) {
		tmp = -1.0;
	} else if ((x * x) <= 9e+165) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 5e+190) {
		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) <= 2.1d-217) then
        tmp = -1.0d0
    else if ((x * x) <= 9d+165) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else if ((x * x) <= 5d+190) 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) <= 2.1e-217) {
		tmp = -1.0;
	} else if ((x * x) <= 9e+165) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 5e+190) {
		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) <= 2.1e-217:
		tmp = -1.0
	elif (x * x) <= 9e+165:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	elif (x * x) <= 5e+190:
		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) <= 2.1e-217)
		tmp = -1.0;
	elseif (Float64(x * x) <= 9e+165)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 5e+190)
		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) <= 2.1e-217)
		tmp = -1.0;
	elseif ((x * x) <= 9e+165)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	elseif ((x * x) <= 5e+190)
		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], 2.1e-217], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 9e+165], 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], 5e+190], -1.0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.1e-217 or 8.9999999999999993e165 < (*.f64 x x) < 5.00000000000000036e190

   1. Initial program 49.5%

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

   2. Taylor expanded in x around 0 86.4%

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

   if 2.1e-217 < (*.f64 x x) < 8.9999999999999993e165

   1. Initial program 76.7%

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

   if 5.00000000000000036e190 < (*.f64 x x)

   1. Initial program 23.8%

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

   2. Taylor expanded in x around inf 89.5%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.1 \cdot 10^{-217}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 9 \cdot 10^{+165}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+190}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 62.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-39}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+19}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+95}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.1e-39)
   -1.0
   (if (<= x 6e-11)
     1.0
     (if (<= x 3.2e+19)
       -1.0
       (if (<= x 9.5e+82) 1.0 (if (<= x 2.2e+95) -1.0 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-39) {
		tmp = -1.0;
	} else if (x <= 6e-11) {
		tmp = 1.0;
	} else if (x <= 3.2e+19) {
		tmp = -1.0;
	} else if (x <= 9.5e+82) {
		tmp = 1.0;
	} else if (x <= 2.2e+95) {
		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 <= 1.1d-39) then
        tmp = -1.0d0
    else if (x <= 6d-11) then
        tmp = 1.0d0
    else if (x <= 3.2d+19) then
        tmp = -1.0d0
    else if (x <= 9.5d+82) then
        tmp = 1.0d0
    else if (x <= 2.2d+95) 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 <= 1.1e-39) {
		tmp = -1.0;
	} else if (x <= 6e-11) {
		tmp = 1.0;
	} else if (x <= 3.2e+19) {
		tmp = -1.0;
	} else if (x <= 9.5e+82) {
		tmp = 1.0;
	} else if (x <= 2.2e+95) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.1e-39:
		tmp = -1.0
	elif x <= 6e-11:
		tmp = 1.0
	elif x <= 3.2e+19:
		tmp = -1.0
	elif x <= 9.5e+82:
		tmp = 1.0
	elif x <= 2.2e+95:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-39)
		tmp = -1.0;
	elseif (x <= 6e-11)
		tmp = 1.0;
	elseif (x <= 3.2e+19)
		tmp = -1.0;
	elseif (x <= 9.5e+82)
		tmp = 1.0;
	elseif (x <= 2.2e+95)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.1e-39)
		tmp = -1.0;
	elseif (x <= 6e-11)
		tmp = 1.0;
	elseif (x <= 3.2e+19)
		tmp = -1.0;
	elseif (x <= 9.5e+82)
		tmp = 1.0;
	elseif (x <= 2.2e+95)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-39], -1.0, If[LessEqual[x, 6e-11], 1.0, If[LessEqual[x, 3.2e+19], -1.0, If[LessEqual[x, 9.5e+82], 1.0, If[LessEqual[x, 2.2e+95], -1.0, 1.0]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e-39 or 6e-11 < x < 3.2e19 or 9.50000000000000049e82 < x < 2.1999999999999999e95

   1. Initial program 53.3%

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

   2. Taylor expanded in x around 0 59.4%

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

   if 1.1e-39 < x < 6e-11 or 3.2e19 < x < 9.50000000000000049e82 or 2.1999999999999999e95 < x

   1. Initial program 33.3%

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

   2. Taylor expanded in x around inf 85.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-39}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+19}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+95}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 51.1% 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 48.8%

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

2. Taylor expanded in x around 0 49.3%

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

3. Final simplification 49.3%

   \[\leadsto -1 \]

Developer target: 51.8% 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 2023336 
(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))))