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

Average Accuracy: 50.1% → 79.6%

Time: 14.7s

Precision: binary64

Cost: 7044

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+109}:\\ \;\;\;\;{\left(\frac{x}{y}\right)}^{2} \cdot 0.5 + -1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y \cdot 4}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ (* x x) (* y (* y -4.0))) (+ (* x x) (* y (* y 4.0))))))
   (if (<= y -3e+109)
     (+ (* (pow (/ x y) 2.0) 0.5) -1.0)
     (if (<= y -2e-73)
       t_0
       (if (<= y 7.8e-47)
         1.0
         (if (<= y 2.2e+71) t_0 (+ -1.0 (* (/ x y) (/ x (* y 4.0))))))))))

C

double code(double x, double y) {
	return ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y));
}

↓

double code(double x, double y) {
	double t_0 = ((x * x) + (y * (y * -4.0))) / ((x * x) + (y * (y * 4.0)));
	double tmp;
	if (y <= -3e+109) {
		tmp = (pow((x / y), 2.0) * 0.5) + -1.0;
	} else if (y <= -2e-73) {
		tmp = t_0;
	} else if (y <= 7.8e-47) {
		tmp = 1.0;
	} else if (y <= 2.2e+71) {
		tmp = t_0;
	} else {
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) - ((y * 4.0d0) * y)) / ((x * x) + ((y * 4.0d0) * y))
end function

↓

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 = ((x * x) + (y * (y * (-4.0d0)))) / ((x * x) + (y * (y * 4.0d0)))
    if (y <= (-3d+109)) then
        tmp = (((x / y) ** 2.0d0) * 0.5d0) + (-1.0d0)
    else if (y <= (-2d-73)) then
        tmp = t_0
    else if (y <= 7.8d-47) then
        tmp = 1.0d0
    else if (y <= 2.2d+71) then
        tmp = t_0
    else
        tmp = (-1.0d0) + ((x / y) * (x / (y * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y));
}

↓

public static double code(double x, double y) {
	double t_0 = ((x * x) + (y * (y * -4.0))) / ((x * x) + (y * (y * 4.0)));
	double tmp;
	if (y <= -3e+109) {
		tmp = (Math.pow((x / y), 2.0) * 0.5) + -1.0;
	} else if (y <= -2e-73) {
		tmp = t_0;
	} else if (y <= 7.8e-47) {
		tmp = 1.0;
	} else if (y <= 2.2e+71) {
		tmp = t_0;
	} else {
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)));
	}
	return tmp;
}

Python

def code(x, y):
	return ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y))

↓

def code(x, y):
	t_0 = ((x * x) + (y * (y * -4.0))) / ((x * x) + (y * (y * 4.0)))
	tmp = 0
	if y <= -3e+109:
		tmp = (math.pow((x / y), 2.0) * 0.5) + -1.0
	elif y <= -2e-73:
		tmp = t_0
	elif y <= 7.8e-47:
		tmp = 1.0
	elif y <= 2.2e+71:
		tmp = t_0
	else:
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)))
	return tmp

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) - Float64(Float64(y * 4.0) * y)) / Float64(Float64(x * x) + Float64(Float64(y * 4.0) * y)))
end

↓

function code(x, y)
	t_0 = Float64(Float64(Float64(x * x) + Float64(y * Float64(y * -4.0))) / Float64(Float64(x * x) + Float64(y * Float64(y * 4.0))))
	tmp = 0.0
	if (y <= -3e+109)
		tmp = Float64(Float64((Float64(x / y) ^ 2.0) * 0.5) + -1.0);
	elseif (y <= -2e-73)
		tmp = t_0;
	elseif (y <= 7.8e-47)
		tmp = 1.0;
	elseif (y <= 2.2e+71)
		tmp = t_0;
	else
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / Float64(y * 4.0))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = ((x * x) + (y * (y * -4.0))) / ((x * x) + (y * (y * 4.0)));
	tmp = 0.0;
	if (y <= -3e+109)
		tmp = (((x / y) ^ 2.0) * 0.5) + -1.0;
	elseif (y <= -2e-73)
		tmp = t_0;
	elseif (y <= 7.8e-47)
		tmp = 1.0;
	elseif (y <= 2.2e+71)
		tmp = t_0;
	else
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e+109], N[(N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y, -2e-73], t$95$0, If[LessEqual[y, 7.8e-47], 1.0, If[LessEqual[y, 2.2e+71], t$95$0, N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	50.1%
Target	50.5%
Herbie	79.6%

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if y < -3.00000000000000015e109

   1. Initial program 16.1%

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

   2. Simplified 16.1%

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

      [Start] 16.1	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-neg [=>] 16.1	\[ \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      *-commutative [=>] 16.1	\[ \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      associate-*r* [=>] 16.1	\[ \frac{\mathsf{fma}\left(x, x, -\color{blue}{\left(y \cdot y\right) \cdot 4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      distribute-rgt-neg-in [=>] 16.1	\[ \frac{\mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot y\right) \cdot \left(-4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      metadata-eval [=>] 16.1	\[ \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot \color{blue}{-4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-def [=>] 16.1	\[ \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]
      *-commutative [=>] 16.1	\[ \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Taylor expanded in x around 0 74.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]

   4. Simplified 84.1%

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

      [Start] 74.0	\[ 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1 \]
      fma-neg [=>] 74.0	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]
      unpow2 [=>] 74.0	\[ \mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]
      unpow2 [=>] 74.0	\[ \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]
      times-frac [=>] 84.1	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]
      metadata-eval [=>] 84.1	\[ \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   5. Applied egg-rr 84.1%

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

      [Start] 84.1	\[ \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right) \]
      fma-udef [=>] 84.1	\[ \color{blue}{0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1} \]
      *-commutative [=>] 84.1	\[ \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5} + -1 \]
      pow2 [=>] 84.1	\[ \color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot 0.5 + -1 \]

   if -3.00000000000000015e109 < y < -1.99999999999999999e-73 or 7.79999999999999956e-47 < y < 2.19999999999999995e71

   1. Initial program 74.8%

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

   if -1.99999999999999999e-73 < y < 7.79999999999999956e-47

   1. Initial program 61.2%

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

   2. Taylor expanded in x around inf 80.0%

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

   if 2.19999999999999995e71 < y

   1. Initial program 24.9%

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

   2. Taylor expanded in x around 0 22.9%

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

   3. Simplified 22.9%

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

      [Start] 22.9	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{4 \cdot {y}^{2}} \]
      *-commutative [=>] 22.9	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]
      unpow2 [=>] 22.9	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot y\right)} \cdot 4} \]
      associate-*r* [<=] 22.9	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]

   4. Applied egg-rr 81.0%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y \cdot 4} - 1} \]
      Proof

      [Start] 22.9	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]
      *-commutative [<=] 22.9	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y}} \]
      div-sub [=>] 22.9	\[ \color{blue}{\frac{x \cdot x}{\left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y}} \]
      *-commutative [=>] 22.9	\[ \frac{x \cdot x}{\color{blue}{y \cdot \left(y \cdot 4\right)}} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y} \]
      times-frac [=>] 23.2	\[ \color{blue}{\frac{x}{y} \cdot \frac{x}{y \cdot 4}} - \frac{\left(y \cdot 4\right) \cdot y}{\left(y \cdot 4\right) \cdot y} \]
      *-inverses [=>] 81.0	\[ \frac{x}{y} \cdot \frac{x}{y \cdot 4} - \color{blue}{1} \]

3. Recombined 4 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+109}:\\ \;\;\;\;{\left(\frac{x}{y}\right)}^{2} \cdot 0.5 + -1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y \cdot 4}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.8%
Cost	2645

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := -1 + \frac{x}{y} \cdot \frac{x}{y \cdot 4}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{-21}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 7 \cdot 10^{+130} \lor \neg \left(t_0 \leq 7 \cdot 10^{+156}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-4 \cdot \left(y \cdot y\right)}{x \cdot x}\\ \end{array} \]

Alternative 2
Accuracy	79.6%
Cost	1744

\[\begin{array}{l} t_0 := \frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+109}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{\frac{x}{\frac{y}{x}}}{y}\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y \cdot 4}\\ \end{array} \]

Alternative 3
Accuracy	74.0%
Cost	1233

\[\begin{array}{l} t_0 := -1 + 0.5 \cdot \frac{\frac{x}{\frac{y}{x}}}{y}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-31}:\\ \;\;\;\;1 + \frac{-4 \cdot \left(y \cdot y\right)}{x \cdot x}\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{-71} \lor \neg \left(y \leq 1.15 \cdot 10^{-45}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	73.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+76}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-32}:\\ \;\;\;\;1 + \frac{-4 \cdot \left(y \cdot y\right)}{x \cdot x}\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-72}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-45}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Accuracy	73.5%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+76}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-73}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6
Accuracy	50.4%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023133 
(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))))