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

Percentage Accurate: 50.9% → 81.0%

Time: 12.4s

Precision: binary64

Cost: 1992

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-148}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{y}{x}} \cdot 0.5 + -1\\ \end{array} \]

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))))
   (if (<= t_0 5e-148)
     (+ 1.0 (* -4.0 (* (/ y x) (/ y x))))
     (if (<= t_0 5e+184)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (+ (* (/ x (* y (/ y x))) 0.5) -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 tmp;
	if (t_0 <= 5e-148) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 5e+184) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = ((x / (y * (y / x))) * 0.5) + -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) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 5d-148) then
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) * (y / x)))
    else if (t_0 <= 5d+184) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    else
        tmp = ((x / (y * (y / x))) * 0.5d0) + (-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 tmp;
	if (t_0 <= 5e-148) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 5e+184) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = ((x / (y * (y / x))) * 0.5) + -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)
	tmp = 0
	if t_0 <= 5e-148:
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)))
	elif t_0 <= 5e+184:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = ((x / (y * (y / x))) * 0.5) + -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))
	tmp = 0.0
	if (t_0 <= 5e-148)
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (t_0 <= 5e+184)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y * Float64(y / x))) * 0.5) + -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);
	tmp = 0.0;
	if (t_0 <= 5e-148)
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	elseif (t_0 <= 5e+184)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	else
		tmp = ((x / (y * (y / x))) * 0.5) + -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]}, If[LessEqual[t$95$0, 5e-148], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+184], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Target

Original	50.9%
Target	51.3%
Herbie	81.0%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 4.9999999999999999e-148

   1. Initial program 61.7%

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

   2. Taylor expanded in x around inf 58.4%

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

   3. Simplified 58.4%

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

      [Start] 58.4%	\[ \frac{{x}^{2}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      unpow2 [=>] 58.4%	\[ \frac{\color{blue}{x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Taylor expanded in x around inf 88.0%

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

   5. Simplified 93.4%

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

      [Start] 88.0%	\[ 1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}} \]
      unpow2 [=>] 88.0%	\[ 1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]
      unpow2 [=>] 88.0%	\[ 1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]
      times-frac [=>] 93.4%	\[ 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   if 4.9999999999999999e-148 < (*.f64 (*.f64 y 4) y) < 4.9999999999999999e184

   1. Initial program 77.8%

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

   if 4.9999999999999999e184 < (*.f64 (*.f64 y 4) y)

   1. Initial program 19.4%

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

   2. Taylor expanded in x around 0 85.5%

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

   3. Simplified 92.6%

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

      [Start] 85.5%	\[ 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1 \]
      fma-neg [=>] 85.5%	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]
      unpow2 [=>] 85.5%	\[ \mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]
      unpow2 [=>] 85.5%	\[ \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]
      times-frac [=>] 92.6%	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]
      metadata-eval [=>] 92.6%	\[ \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   4. Applied egg-rr 92.6%

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

      [Start] 92.6%	\[ \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right) \]
      fma-udef [=>] 92.6%	\[ \color{blue}{0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1} \]
      *-commutative [=>] 92.6%	\[ \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5} + -1 \]
      pow2 [=>] 92.6%	\[ \color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot 0.5 + -1 \]

   5. Applied egg-rr 92.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{x} \cdot y}} \cdot 0.5 + -1 \]
      Step-by-step derivation

      [Start] 92.6%	\[ {\left(\frac{x}{y}\right)}^{2} \cdot 0.5 + -1 \]
      unpow2 [=>] 92.6%	\[ \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.5 + -1 \]
      clear-num [=>] 92.6%	\[ \left(\color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{x}{y}\right) \cdot 0.5 + -1 \]
      frac-times [=>] 92.6%	\[ \color{blue}{\frac{1 \cdot x}{\frac{y}{x} \cdot y}} \cdot 0.5 + -1 \]
      *-un-lft-identity [<=] 92.6%	\[ \frac{\color{blue}{x}}{\frac{y}{x} \cdot y} \cdot 0.5 + -1 \]

3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-148}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{y}{x}} \cdot 0.5 + -1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.0%
Cost	1992

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-148}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{y}{x}} \cdot 0.5 + -1\\ \end{array} \]

Alternative 2
Accuracy	74.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-64} \lor \neg \left(y \leq 8.6 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{x}{y \cdot \frac{y}{x}} \cdot 0.5 + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 3
Accuracy	74.1%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-65}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+65}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Accuracy	73.9%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-64}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Accuracy	50.2%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023263 
(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))))