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

Average Error: 31.9 → 13.0

Time: 2.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ t_1 := \mathsf{fma}\left({\left(\frac{y}{x}\right)}^{2}, -8, 1\right)\\ t_2 := \mathsf{fma}\left(\frac{0.5}{y}, \frac{x \cdot x}{y}, -1\right)\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 x x (* -4.0 (* y y))) (fma x x (* y (* y 4.0)))))
        (t_1 (fma (pow (/ y x) 2.0) -8.0 1.0))
        (t_2 (fma (/ 0.5 y) (/ (* x x) y) -1.0)))
   (if (<= x -2.3e+86)
     t_1
     (if (<= x -5e-125)
       t_0
       (if (<= x 1.1e-110)
         t_2
         (if (<= x 2.9e-39)
           t_0
           (if (<= x 8e-14) t_2 (if (<= x 2.1e+69) t_0 t_1))))))))

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(x, x, (-4.0 * (y * y))) / fma(x, x, (y * (y * 4.0)));
	double t_1 = fma(pow((y / x), 2.0), -8.0, 1.0);
	double t_2 = fma((0.5 / y), ((x * x) / y), -1.0);
	double tmp;
	if (x <= -2.3e+86) {
		tmp = t_1;
	} else if (x <= -5e-125) {
		tmp = t_0;
	} else if (x <= 1.1e-110) {
		tmp = t_2;
	} else if (x <= 2.9e-39) {
		tmp = t_0;
	} else if (x <= 8e-14) {
		tmp = t_2;
	} else if (x <= 2.1e+69) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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(fma(x, x, Float64(-4.0 * Float64(y * y))) / fma(x, x, Float64(y * Float64(y * 4.0))))
	t_1 = fma((Float64(y / x) ^ 2.0), -8.0, 1.0)
	t_2 = fma(Float64(0.5 / y), Float64(Float64(x * x) / y), -1.0)
	tmp = 0.0
	if (x <= -2.3e+86)
		tmp = t_1;
	elseif (x <= -5e-125)
		tmp = t_0;
	elseif (x <= 1.1e-110)
		tmp = t_2;
	elseif (x <= 2.9e-39)
		tmp = t_0;
	elseif (x <= 8e-14)
		tmp = t_2;
	elseif (x <= 2.1e+69)
		tmp = t_0;
	else
		tmp = t_1;
	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[(x * x + N[(-4.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision] * -8.0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 / y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -2.3e+86], t$95$1, If[LessEqual[x, -5e-125], t$95$0, If[LessEqual[x, 1.1e-110], t$95$2, If[LessEqual[x, 2.9e-39], t$95$0, If[LessEqual[x, 8e-14], t$95$2, If[LessEqual[x, 2.1e+69], t$95$0, t$95$1]]]]]]]]]

Error

Bits error versus x

Bits error versus y

Target

Original	31.9
Target	31.6
Herbie	13.0

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

Derivation

1. Split input into 3 regimes

2. if x < -2.2999999999999999e86 or 2.10000000000000015e69 < x

   1. Initial program 48.7

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

   2. Simplified 48.7

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

   3. Taylor expanded in x around inf 18.1

      \[\leadsto \color{blue}{1 - 8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]

   4. Simplified 11.9

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

   5. Applied egg-rr 12.2

      \[\leadsto \color{blue}{\frac{{\left({\left(\frac{y}{x}\right)}^{2} \cdot -8\right)}^{3} + 1}{\left({\left(\frac{y}{x}\right)}^{2} \cdot -8\right) \cdot \left({\left(\frac{y}{x}\right)}^{2} \cdot -8\right) + \left(1 - \left({\left(\frac{y}{x}\right)}^{2} \cdot -8\right) \cdot 1\right)}} \]

   6. Applied egg-rr 12.4

      \[\leadsto \frac{{\left({\left(\frac{y}{x}\right)}^{2} \cdot -8\right)}^{3} + 1}{\color{blue}{\log \left({\left(e^{{\left(\frac{y}{x}\right)}^{4}}\right)}^{64}\right)} + \left(1 - \left({\left(\frac{y}{x}\right)}^{2} \cdot -8\right) \cdot 1\right)} \]

   7. Taylor expanded in y around 0 18.1

      \[\leadsto \color{blue}{1 - 8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]

   8. Simplified 11.9

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{y}{x}\right)}^{2}, -8, 1\right)} \]

   if -2.2999999999999999e86 < x < -4.99999999999999967e-125 or 1.1e-110 < x < 2.89999999999999988e-39 or 7.99999999999999999e-14 < x < 2.10000000000000015e69

   1. Initial program 16.3

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

   2. Simplified 16.3

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

   if -4.99999999999999967e-125 < x < 1.1e-110 or 2.89999999999999988e-39 < x < 7.99999999999999999e-14

   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. Simplified 27.1

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

   3. Taylor expanded in x around 0 16.2

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

   4. Simplified 11.2

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

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{y}{x}\right)}^{2}, -8, 1\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x \cdot x}{y}, -1\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x \cdot x}{y}, -1\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{y}{x}\right)}^{2}, -8, 1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))