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

Average Accuracy: 50.8% → 79.9%

Time: 8.1s

Precision: binary64

Cost: 2256

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ t_2 := \frac{\frac{x}{y}}{\frac{y}{x}} \cdot 0.5 + -1\\ \mathbf{if}\;x \cdot x \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{x}{y}} \cdot -8 + 1\\ \end{array} \]

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 (* y (* y 4.0)))
        (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0)))
        (t_2 (+ (* (/ (/ x y) (/ y x)) 0.5) -1.0)))
   (if (<= (* x x) 0.0)
     t_2
     (if (<= (* x x) 2e-217)
       t_1
       (if (<= (* x x) 2e-140)
         t_2
         (if (<= (* x x) 5e+82) t_1 (+ (* (/ (/ y x) (/ x y)) -8.0) 1.0)))))))

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 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double t_2 = (((x / y) / (y / x)) * 0.5) + -1.0;
	double tmp;
	if ((x * x) <= 0.0) {
		tmp = t_2;
	} else if ((x * x) <= 2e-217) {
		tmp = t_1;
	} else if ((x * x) <= 2e-140) {
		tmp = t_2;
	} else if ((x * x) <= 5e+82) {
		tmp = t_1;
	} else {
		tmp = (((y / x) / (x / y)) * -8.0) + 1.0;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    t_2 = (((x / y) / (y / x)) * 0.5d0) + (-1.0d0)
    if ((x * x) <= 0.0d0) then
        tmp = t_2
    else if ((x * x) <= 2d-217) then
        tmp = t_1
    else if ((x * x) <= 2d-140) then
        tmp = t_2
    else if ((x * x) <= 5d+82) then
        tmp = t_1
    else
        tmp = (((y / x) / (x / y)) * (-8.0d0)) + 1.0d0
    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 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double t_2 = (((x / y) / (y / x)) * 0.5) + -1.0;
	double tmp;
	if ((x * x) <= 0.0) {
		tmp = t_2;
	} else if ((x * x) <= 2e-217) {
		tmp = t_1;
	} else if ((x * x) <= 2e-140) {
		tmp = t_2;
	} else if ((x * x) <= 5e+82) {
		tmp = t_1;
	} else {
		tmp = (((y / x) / (x / y)) * -8.0) + 1.0;
	}
	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 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	t_2 = (((x / y) / (y / x)) * 0.5) + -1.0
	tmp = 0
	if (x * x) <= 0.0:
		tmp = t_2
	elif (x * x) <= 2e-217:
		tmp = t_1
	elif (x * x) <= 2e-140:
		tmp = t_2
	elif (x * x) <= 5e+82:
		tmp = t_1
	else:
		tmp = (((y / x) / (x / y)) * -8.0) + 1.0
	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(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	t_2 = Float64(Float64(Float64(Float64(x / y) / Float64(y / x)) * 0.5) + -1.0)
	tmp = 0.0
	if (Float64(x * x) <= 0.0)
		tmp = t_2;
	elseif (Float64(x * x) <= 2e-217)
		tmp = t_1;
	elseif (Float64(x * x) <= 2e-140)
		tmp = t_2;
	elseif (Float64(x * x) <= 5e+82)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(y / x) / Float64(x / y)) * -8.0) + 1.0);
	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 = y * (y * 4.0);
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	t_2 = (((x / y) / (y / x)) * 0.5) + -1.0;
	tmp = 0.0;
	if ((x * x) <= 0.0)
		tmp = t_2;
	elseif ((x * x) <= 2e-217)
		tmp = t_1;
	elseif ((x * x) <= 2e-140)
		tmp = t_2;
	elseif ((x * x) <= 5e+82)
		tmp = t_1;
	else
		tmp = (((y / x) / (x / y)) * -8.0) + 1.0;
	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[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 0.0], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 2e-217], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 2e-140], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 5e+82], t$95$1, N[(N[(N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]]

Target

Original	50.8%
Target	51.3%
Herbie	79.9%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 0.0 or 2.00000000000000016e-217 < (*.f64 x x) < 2e-140

   1. Initial program 59.5%

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

   2. Taylor expanded in x around 0 82.4%

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

   3. Simplified 93.2%

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

      [Start] 82.4	\[ 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1 \]
      fma-neg [=>] 82.4	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]
      unpow2 [=>] 82.4	\[ \mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]
      unpow2 [=>] 82.4	\[ \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]
      times-frac [=>] 93.2	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]
      metadata-eval [=>] 93.2	\[ \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   4. Applied egg-rr 93.2%

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

      [Start] 93.2	\[ \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right) \]
      fma-udef [=>] 93.2	\[ \color{blue}{0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1} \]
      *-commutative [=>] 93.2	\[ \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5} + -1 \]
      pow2 [=>] 93.2	\[ \color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot 0.5 + -1 \]

   5. Applied egg-rr 93.2%

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

      [Start] 93.2	\[ {\left(\frac{x}{y}\right)}^{2} \cdot 0.5 + -1 \]
      unpow2 [=>] 93.2	\[ \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.5 + -1 \]
      clear-num [=>] 93.2	\[ \left(\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right) \cdot 0.5 + -1 \]
      un-div-inv [=>] 93.2	\[ \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \cdot 0.5 + -1 \]

   if 0.0 < (*.f64 x x) < 2.00000000000000016e-217 or 2e-140 < (*.f64 x x) < 5.00000000000000015e82

   1. Initial program 93.7%

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

   if 5.00000000000000015e82 < (*.f64 x x)

   1. Initial program 27.5%

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

   2. Taylor expanded in x around inf 76.4%

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

   3. Simplified 83.4%

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

      [Start] 76.4	\[ \left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}} \]
      associate--l+ [=>] 76.4	\[ \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]
      distribute-rgt-out-- [=>] 76.4	\[ 1 + \color{blue}{\frac{{y}^{2}}{{x}^{2}} \cdot \left(-4 - 4\right)} \]
      metadata-eval [=>] 76.4	\[ 1 + \frac{{y}^{2}}{{x}^{2}} \cdot \color{blue}{-8} \]
      *-commutative [<=] 76.4	\[ 1 + \color{blue}{-8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
      +-commutative [<=] 76.4	\[ \color{blue}{-8 \cdot \frac{{y}^{2}}{{x}^{2}} + 1} \]
      *-commutative [=>] 76.4	\[ \color{blue}{\frac{{y}^{2}}{{x}^{2}} \cdot -8} + 1 \]
      fma-def [=>] 76.4	\[ \color{blue}{\mathsf{fma}\left(\frac{{y}^{2}}{{x}^{2}}, -8, 1\right)} \]
      unpow2 [=>] 76.4	\[ \mathsf{fma}\left(\frac{\color{blue}{y \cdot y}}{{x}^{2}}, -8, 1\right) \]
      unpow2 [=>] 76.4	\[ \mathsf{fma}\left(\frac{y \cdot y}{\color{blue}{x \cdot x}}, -8, 1\right) \]
      times-frac [=>] 83.4	\[ \mathsf{fma}\left(\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, -8, 1\right) \]

   4. Applied egg-rr 83.4%

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

      [Start] 83.4	\[ \mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right) \]
      fma-udef [=>] 83.4	\[ \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8 + 1} \]
      pow2 [=>] 83.4	\[ \color{blue}{{\left(\frac{y}{x}\right)}^{2}} \cdot -8 + 1 \]

   5. Applied egg-rr 83.4%

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

      [Start] 83.4	\[ {\left(\frac{y}{x}\right)}^{2} \cdot -8 + 1 \]
      unpow2 [=>] 83.4	\[ \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8 + 1 \]
      clear-num [=>] 83.4	\[ \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \cdot -8 + 1 \]
      un-div-inv [=>] 83.4	\[ \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \cdot -8 + 1 \]

3. Recombined 3 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} \cdot 0.5 + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-217}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} \cdot 0.5 + -1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+82}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{x}{y}} \cdot -8 + 1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.5%
Cost	1497

\[\begin{array}{l} t_0 := \frac{\frac{x}{y}}{\frac{y}{x}} \cdot 0.5 + -1\\ t_1 := \frac{\frac{y}{x}}{\frac{x}{y}} \cdot -8 + 1\\ \mathbf{if}\;x \leq -1.86 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+33} \lor \neg \left(x \leq 5.5 \cdot 10^{+61}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Accuracy	73.0%
Cost	1232

\[\begin{array}{l} t_0 := \frac{\frac{x}{y}}{\frac{y}{x}} \cdot 0.5 + -1\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-23}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{+22}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+62}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Accuracy	72.4%
Cost	856

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-148}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-74}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 10^{+30}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+61}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	49.4%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023160 
(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))))