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

Percentage Accurate: 50.0% → 99.9%

Time: 11.2s

Alternatives: 6

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: 50.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(y \cdot 2, x\right)\\ t_1 := \frac{y \cdot 2}{t\_0}\\ t_2 := {t\_1}^{2}\\ \left({\left(\frac{x}{t\_0}\right)}^{2} - t\_2\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{t\_1}\right)}^{2}, \sqrt[3]{{t\_1}^{4}}, t\_2\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot (* y 2.0) x)) (t_1 (/ (* y 2.0) t_0)) (t_2 (pow t_1 2.0)))
   (+
    (- (pow (/ x t_0) 2.0) t_2)
    (fma (- (pow (cbrt t_1) 2.0)) (cbrt (pow t_1 4.0)) t_2))))

C

double code(double x, double y) {
	double t_0 = hypot((y * 2.0), x);
	double t_1 = (y * 2.0) / t_0;
	double t_2 = pow(t_1, 2.0);
	return (pow((x / t_0), 2.0) - t_2) + fma(-pow(cbrt(t_1), 2.0), cbrt(pow(t_1, 4.0)), t_2);
}

Julia

function code(x, y)
	t_0 = hypot(Float64(y * 2.0), x)
	t_1 = Float64(Float64(y * 2.0) / t_0)
	t_2 = t_1 ^ 2.0
	return Float64(Float64((Float64(x / t_0) ^ 2.0) - t_2) + fma(Float64(-(cbrt(t_1) ^ 2.0)), cbrt((t_1 ^ 4.0)), t_2))
end

Wolfram

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

Derivation

1. Initial program 47.2%

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

   1. div-sub 47.2%

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

   2. associate-/l* 47.6%

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

   3. fmm-def 47.6%

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

   4. +-commutative 47.6%

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

   5. *-commutative 47.6%

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

   6. associate-*l* 47.6%

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

   7. fma-define 47.6%

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

   8. pow2 47.6%

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

   9. pow2 47.6%

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

   10. *-commutative 47.6%

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

   11. associate-*l* 47.4%

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

   12. pow2 47.4%

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

   13. +-commutative 47.4%

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

   14. *-commutative 47.4%

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

   15. associate-*l* 47.6%

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

   16. fma-define 47.6%

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

4. Applied egg-rr 47.6%

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

   1. fma-undefine 47.6%

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

   2. neg-sub0 47.6%

      \[\leadsto x \cdot \frac{x}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)} + \color{blue}{\left(0 - \frac{4 \cdot {y}^{2}}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}\right)} \]

   3. associate-+r- 47.6%

      \[\leadsto \color{blue}{\left(x \cdot \frac{x}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)} + 0\right) - \frac{4 \cdot {y}^{2}}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}} \]

6. Applied egg-rr 70.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{x}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}, 0\right) - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}} \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left({\left(\frac{x}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}}\right)}^{2}, \sqrt[3]{{\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{4}}, {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right)} \]
9. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(y \cdot 2, x\right)\\ \left(\frac{x}{t\_0} + \frac{y \cdot 2}{t\_0}\right) \cdot \frac{x - y \cdot 2}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.2%

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

   1. div-sub 47.2%

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

   2. associate-/l* 47.6%

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

   3. fmm-def 47.6%

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

   4. +-commutative 47.6%

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

   5. *-commutative 47.6%

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

   6. associate-*l* 47.6%

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

   7. fma-define 47.6%

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

   8. pow2 47.6%

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

   9. pow2 47.6%

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

   10. *-commutative 47.6%

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

   11. associate-*l* 47.4%

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

   12. pow2 47.4%

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

   13. +-commutative 47.4%

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

   14. *-commutative 47.4%

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

   15. associate-*l* 47.6%

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

   16. fma-define 47.6%

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

4. Applied egg-rr 47.6%

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

   1. fma-undefine 47.6%

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

   2. neg-sub0 47.6%

      \[\leadsto x \cdot \frac{x}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)} + \color{blue}{\left(0 - \frac{4 \cdot {y}^{2}}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}\right)} \]

   3. associate-+r- 47.6%

      \[\leadsto \color{blue}{\left(x \cdot \frac{x}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)} + 0\right) - \frac{4 \cdot {y}^{2}}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}} \]

6. Applied egg-rr 70.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{x}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}, 0\right) - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 70.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{x}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}, 0\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{x}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}, 0\right)}} - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} \]

   2. unpow2 70.1%

      \[\leadsto \sqrt{\mathsf{fma}\left(x, \frac{x}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}, 0\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{x}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}, 0\right)} - \color{blue}{\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)} \cdot \frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}} \]

   3. difference-of-squares 70.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(x, \frac{x}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}, 0\right)} + \frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right) \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{x}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}, 0\right)} - \frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)} \]

8. Applied egg-rr 100.0%

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

   1. sub-div 100.0%

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

10. Applied egg-rr 100.0%

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

Alternative 3: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 10^{-191}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{t\_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-191)
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
     (if (<= t_0 2e+143) (/ (- (* x x) t_0) (+ t_0 (* x x))) -1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-191) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 2e+143) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} 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 (t_0 <= 1d-191) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else if (t_0 <= 2d+143) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    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 (t_0 <= 1e-191) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 2e+143) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 1e-191:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	elif t_0 <= 2e+143:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-191)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (t_0 <= 2e+143)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	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 (t_0 <= 1e-191)
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	elseif (t_0 <= 2e+143)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	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[t$95$0, 1e-191], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+143], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1e-191

   1. Initial program 51.3%

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

   3. Taylor expanded in y around 0 71.9%

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

      1. unpow2 71.9%

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

      2. unpow2 71.9%

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

      3. times-frac 86.1%

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

   5. Applied egg-rr 86.1%

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

   if 1e-191 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e143

   1. Initial program 76.9%

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

   if 2e143 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 22.9%

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

   3. Taylor expanded in x around 0 82.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-191}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{+19}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.25e+19) -1.0 (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.25e+19) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.25d+19) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.25e+19) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.25e+19:
		tmp = -1.0
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.25e+19)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.25e+19)
		tmp = -1.0;
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.25e+19], -1.0, N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25e19

   1. Initial program 50.9%

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

   3. Taylor expanded in x around 0 62.4%

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

   if 2.25e19 < x

   1. Initial program 34.5%

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

   3. Taylor expanded in y around 0 88.1%

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

      1. unpow2 88.1%

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

      2. unpow2 88.1%

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

      3. times-frac 91.7%

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

   5. Applied egg-rr 91.7%

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

Alternative 5: 62.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+18}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.8e+18) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.8e+18) {
		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 <= 1.8d+18) 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 <= 1.8e+18) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.8e+18:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.8e+18)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.8e+18)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.8e+18], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.8e18

   1. Initial program 50.9%

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

   3. Taylor expanded in x around 0 62.4%

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

   if 1.8e18 < x

   1. Initial program 34.5%

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

   3. Taylor expanded in x around inf 91.5%

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

Alternative 6: 50.0% 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 47.2%

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

3. Taylor expanded in x around 0 50.5%

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

Developer Target 1: 50.5% 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 2024152 
(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))))