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

Average Error: 31.8 → 13.0

Time: 2.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{0.5}{y}, \frac{x \cdot x}{y}, -1\right)\\ t_1 := {\left(\frac{y}{x}\right)}^{2} \cdot -8 + 1\\ t_2 := \frac{\mathsf{fma}\left(y \cdot -4, y, x \cdot x\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.48 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+139}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ 0.5 y) (/ (* x x) y) -1.0))
        (t_1 (+ (* (pow (/ y x) 2.0) -8.0) 1.0))
        (t_2 (/ (fma (* y -4.0) y (* x x)) (+ (* x x) (* y (* y 4.0))))))
   (if (<= x -1.6e+90)
     t_1
     (if (<= x -3.9e+77)
       t_0
       (if (<= x -4.3e+18)
         1.0
         (if (<= x -1.48e-100)
           t_2
           (if (<= x 1.45e-48) t_0 (if (<= x 1.7e+139) t_2 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((0.5 / y), ((x * x) / y), -1.0);
	double t_1 = (pow((y / x), 2.0) * -8.0) + 1.0;
	double t_2 = fma((y * -4.0), y, (x * x)) / ((x * x) + (y * (y * 4.0)));
	double tmp;
	if (x <= -1.6e+90) {
		tmp = t_1;
	} else if (x <= -3.9e+77) {
		tmp = t_0;
	} else if (x <= -4.3e+18) {
		tmp = 1.0;
	} else if (x <= -1.48e-100) {
		tmp = t_2;
	} else if (x <= 1.45e-48) {
		tmp = t_0;
	} else if (x <= 1.7e+139) {
		tmp = t_2;
	} 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 = fma(Float64(0.5 / y), Float64(Float64(x * x) / y), -1.0)
	t_1 = Float64(Float64((Float64(y / x) ^ 2.0) * -8.0) + 1.0)
	t_2 = Float64(fma(Float64(y * -4.0), y, Float64(x * x)) / Float64(Float64(x * x) + Float64(y * Float64(y * 4.0))))
	tmp = 0.0
	if (x <= -1.6e+90)
		tmp = t_1;
	elseif (x <= -3.9e+77)
		tmp = t_0;
	elseif (x <= -4.3e+18)
		tmp = 1.0;
	elseif (x <= -1.48e-100)
		tmp = t_2;
	elseif (x <= 1.45e-48)
		tmp = t_0;
	elseif (x <= 1.7e+139)
		tmp = t_2;
	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[(0.5 / y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision] * -8.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * -4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e+90], t$95$1, If[LessEqual[x, -3.9e+77], t$95$0, If[LessEqual[x, -4.3e+18], 1.0, If[LessEqual[x, -1.48e-100], t$95$2, If[LessEqual[x, 1.45e-48], t$95$0, If[LessEqual[x, 1.7e+139], t$95$2, t$95$1]]]]]]]]]

Error

Bits error versus x

Bits error versus y

Target

Original	31.8
Target	31.5
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 4 regimes

2. if x < -1.59999999999999999e90 or 1.7000000000000001e139 < x

   1. Initial program 54.7

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

   2. Taylor expanded in x around inf 15.8

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

   3. Simplified 9.4

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

   4. Applied egg-rr 9.4

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

   if -1.59999999999999999e90 < x < -3.8999999999999998e77 or -1.48000000000000002e-100 < x < 1.4500000000000001e-48

   1. Initial program 25.4

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

   2. Taylor expanded in x around 0 17.4

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

   3. Simplified 12.6

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

   if -3.8999999999999998e77 < x < -4.3e18

   1. Initial program 14.6

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

   2. Taylor expanded in x around inf 24.6

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

   if -4.3e18 < x < -1.48000000000000002e-100 or 1.4500000000000001e-48 < x < 1.7000000000000001e139

   1. Initial program 16.2

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

   2. Applied egg-rr 16.2

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

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+90}:\\ \;\;\;\;{\left(\frac{y}{x}\right)}^{2} \cdot -8 + 1\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x \cdot x}{y}, -1\right)\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.48 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, x \cdot x\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x \cdot x}{y}, -1\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, x \cdot x\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x}\right)}^{2} \cdot -8 + 1\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))