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

Average Error: 31.9 → 12.5

Time: 2.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{t_0 + x \cdot x}\\ \mathbf{elif}\;t_0 \leq 10^{-27}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(x, x, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, {\left(\frac{x}{y}\right)}^{2}, -1\right)\\ \end{array} \]

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 0.0)
     1.0
     (if (<= t_0 2e-123)
       (/ (+ (* x x) (* y (* y -4.0))) (+ t_0 (* x x)))
       (if (<= t_0 1e-27)
         1.0
         (if (<= t_0 2e+206)
           (/ (fma x x (* -4.0 (* y y))) (fma x x t_0))
           (fma 0.5 (pow (/ x y) 2.0) -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 tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0;
	} else if (t_0 <= 2e-123) {
		tmp = ((x * x) + (y * (y * -4.0))) / (t_0 + (x * x));
	} else if (t_0 <= 1e-27) {
		tmp = 1.0;
	} else if (t_0 <= 2e+206) {
		tmp = fma(x, x, (-4.0 * (y * y))) / fma(x, x, t_0);
	} else {
		tmp = fma(0.5, pow((x / y), 2.0), -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))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = 1.0;
	elseif (t_0 <= 2e-123)
		tmp = Float64(Float64(Float64(x * x) + Float64(y * Float64(y * -4.0))) / Float64(t_0 + Float64(x * x)));
	elseif (t_0 <= 1e-27)
		tmp = 1.0;
	elseif (t_0 <= 2e+206)
		tmp = Float64(fma(x, x, Float64(-4.0 * Float64(y * y))) / fma(x, x, t_0));
	else
		tmp = fma(0.5, (Float64(x / y) ^ 2.0), -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[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], 1.0, If[LessEqual[t$95$0, 2e-123], N[(N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-27], 1.0, If[LessEqual[t$95$0, 2e+206], N[(N[(x * x + N[(-4.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]]]]]]

Target

Original	31.9
Target	31.5
Herbie	12.5

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y 4) y) < 0.0 or 2.0000000000000001e-123 < (*.f64 (*.f64 y 4) y) < 1e-27

   1. Initial program 27.0

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

   2. Taylor expanded in x around inf 11.8

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

   if 0.0 < (*.f64 (*.f64 y 4) y) < 2.0000000000000001e-123

   1. Initial program 16.4

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

   if 1e-27 < (*.f64 (*.f64 y 4) y) < 2.0000000000000001e206

   1. Initial program 16.0

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

   2. Simplified 16.0

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

   if 2.0000000000000001e206 < (*.f64 (*.f64 y 4) y)

   1. Initial program 52.4

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

   2. Taylor expanded in x around 0 15.7

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

   3. Simplified 9.8

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

   4. Applied egg-rr 9.4

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}, -1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{-27}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, {\left(\frac{x}{y}\right)}^{2}, -1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(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))))