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

Average Error: 32.1 → 15.8

Time: 1.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{0.5}{y}, x \cdot \frac{x}{y}, -1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+115}:\\ \;\;\;\;{\left(\sqrt[3]{t_0}\right)}^{3}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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 (fma (/ 0.5 y) (* x (/ x y)) -1.0)))
   (if (<= x -5e+162)
     1.0
     (if (<= x -2.4e+115)
       (pow (cbrt t_0) 3.0)
       (if (<= x -3e-83)
         (/ (fma x x (* -4.0 (* y y))) (fma x x (* y (* y 4.0))))
         (if (<= x 3.1e-94) t_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 = fma((0.5 / y), (x * (x / y)), -1.0);
	double tmp;
	if (x <= -5e+162) {
		tmp = 1.0;
	} else if (x <= -2.4e+115) {
		tmp = pow(cbrt(t_0), 3.0);
	} else if (x <= -3e-83) {
		tmp = fma(x, x, (-4.0 * (y * y))) / fma(x, x, (y * (y * 4.0)));
	} else if (x <= 3.1e-94) {
		tmp = t_0;
	} else {
		tmp = 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 = fma(Float64(0.5 / y), Float64(x * Float64(x / y)), -1.0)
	tmp = 0.0
	if (x <= -5e+162)
		tmp = 1.0;
	elseif (x <= -2.4e+115)
		tmp = cbrt(t_0) ^ 3.0;
	elseif (x <= -3e-83)
		tmp = Float64(fma(x, x, Float64(-4.0 * Float64(y * y))) / fma(x, x, Float64(y * Float64(y * 4.0))));
	elseif (x <= 3.1e-94)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return 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[(0.5 / y), $MachinePrecision] * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -5e+162], 1.0, If[LessEqual[x, -2.4e+115], N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[x, -3e-83], N[(N[(x * x + N[(-4.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e-94], t$95$0, 1.0]]]]]

Error

Bits error versus x

Bits error versus y

Target

Original	32.1
Target	31.8
Herbie	15.8

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

Derivation

1. Split input into 4 regimes

2. if x < -4.9999999999999997e162 or 3.0999999999999998e-94 < x

   1. Initial program 42.6

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

   2. Taylor expanded in x around inf 17.6

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

   if -4.9999999999999997e162 < x < -2.4e115

   1. Initial program 26.7

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

   2. Taylor expanded in x around 0 49.2

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

   3. Simplified 49.2

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

   4. Applied egg-rr 46.0

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\frac{0.5}{y}, \frac{x}{y} \cdot x, -1\right)}\right)}^{3}} \]

   if -2.4e115 < x < -3.0000000000000001e-83

   1. Initial program 15.4

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

   2. Simplified 15.4

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

   if -3.0000000000000001e-83 < x < 3.0999999999999998e-94

   1. Initial program 26.9

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

   2. Taylor expanded in x around 0 16.5

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

   3. Simplified 11.0

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

   4. Taylor expanded in x around 0 11.0

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

   5. Simplified 10.6

      \[\leadsto \mathsf{fma}\left(\frac{0.5}{y}, \color{blue}{x \cdot \frac{x}{y}}, -1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 15.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+115}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left(\frac{0.5}{y}, x \cdot \frac{x}{y}, -1\right)}\right)}^{3}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, x \cdot \frac{x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))