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

Average Error: 31.1 → 13.8

Time: 12.4s

Precision: binary64

Cost: 14800

Math

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

↓

\[\begin{array}{l} t_0 := -1 + \mathsf{log1p}\left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\\ t_1 := \mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 10^{-66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{t_1}\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x}\right)}^{2} \cdot -8 + 1\\ \end{array} \]

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 (+ -1.0 (log1p (* 0.5 (pow (/ x y) 2.0)))))
        (t_1 (fma x x (* y (* y 4.0)))))
   (if (<= (* x x) 5e-157)
     t_0
     (if (<= (* x x) 1e-66)
       (/ (fma x x (* (* y y) -4.0)) t_1)
       (if (<= (* x x) 4e+91)
         t_0
         (if (<= (* x x) 1e+268)
           (/ (fma y (* y -4.0) (* x x)) t_1)
           (+ (* (pow (/ y x) 2.0) -8.0) 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 = -1.0 + log1p((0.5 * pow((x / y), 2.0)));
	double t_1 = fma(x, x, (y * (y * 4.0)));
	double tmp;
	if ((x * x) <= 5e-157) {
		tmp = t_0;
	} else if ((x * x) <= 1e-66) {
		tmp = fma(x, x, ((y * y) * -4.0)) / t_1;
	} else if ((x * x) <= 4e+91) {
		tmp = t_0;
	} else if ((x * x) <= 1e+268) {
		tmp = fma(y, (y * -4.0), (x * x)) / t_1;
	} else {
		tmp = (pow((y / x), 2.0) * -8.0) + 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(-1.0 + log1p(Float64(0.5 * (Float64(x / y) ^ 2.0))))
	t_1 = fma(x, x, Float64(y * Float64(y * 4.0)))
	tmp = 0.0
	if (Float64(x * x) <= 5e-157)
		tmp = t_0;
	elseif (Float64(x * x) <= 1e-66)
		tmp = Float64(fma(x, x, Float64(Float64(y * y) * -4.0)) / t_1);
	elseif (Float64(x * x) <= 4e+91)
		tmp = t_0;
	elseif (Float64(x * x) <= 1e+268)
		tmp = Float64(fma(y, Float64(y * -4.0), Float64(x * x)) / t_1);
	else
		tmp = Float64(Float64((Float64(y / x) ^ 2.0) * -8.0) + 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[(-1.0 + N[Log[1 + N[(0.5 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * x + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-157], t$95$0, If[LessEqual[N[(x * x), $MachinePrecision], 1e-66], N[(N[(x * x + N[(N[(y * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 4e+91], t$95$0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+268], N[(N[(y * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision] * -8.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]

Target

Original	31.1
Target	30.8
Herbie	13.8

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 5.0000000000000002e-157 or 9.9999999999999998e-67 < (*.f64 x x) < 4.00000000000000032e91

   1. Initial program 23.1

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

   2. Simplified 23.1

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

      [Start] 23.1	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      cancel-sign-sub-inv [=>] 23.1	\[ \frac{\color{blue}{x \cdot x + \left(-y \cdot 4\right) \cdot y}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      +-commutative [=>] 23.1	\[ \frac{\color{blue}{\left(-y \cdot 4\right) \cdot y + x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      *-commutative [=>] 23.1	\[ \frac{\color{blue}{y \cdot \left(-y \cdot 4\right)} + x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-def [=>] 23.1	\[ \frac{\color{blue}{\mathsf{fma}\left(y, -y \cdot 4, x \cdot x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      distribute-rgt-neg-in [=>] 23.1	\[ \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(-4\right)}, x \cdot x\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      metadata-eval [=>] 23.1	\[ \frac{\mathsf{fma}\left(y, y \cdot \color{blue}{-4}, x \cdot x\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-def [=>] 23.1	\[ \frac{\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]
      *-commutative [=>] 23.1	\[ \frac{\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Taylor expanded in y around inf 21.2

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{{y}^{2}} - \left(-0.25 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right)} \]

   4. Simplified 17.1

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

      [Start] 21.2	\[ 0.25 \cdot \frac{{x}^{2}}{{y}^{2}} - \left(-0.25 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]
      associate--r+ [=>] 21.2	\[ \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{{y}^{2}} - -0.25 \cdot \frac{{x}^{2}}{{y}^{2}}\right) - 1} \]
      distribute-rgt-out-- [=>] 21.2	\[ \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot \left(0.25 - -0.25\right)} - 1 \]
      metadata-eval [=>] 21.2	\[ \frac{{x}^{2}}{{y}^{2}} \cdot \color{blue}{0.5} - 1 \]
      fma-neg [=>] 21.2	\[ \color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{{y}^{2}}, 0.5, -1\right)} \]
      unpow2 [=>] 21.2	\[ \mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{y}^{2}}, 0.5, -1\right) \]
      unpow2 [=>] 21.2	\[ \mathsf{fma}\left(\frac{x \cdot x}{\color{blue}{y \cdot y}}, 0.5, -1\right) \]
      times-frac [=>] 17.1	\[ \mathsf{fma}\left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, 0.5, -1\right) \]
      metadata-eval [=>] 17.1	\[ \mathsf{fma}\left(\frac{x}{y} \cdot \frac{x}{y}, 0.5, \color{blue}{-1}\right) \]

   5. Applied egg-rr 17.8

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

   6. Taylor expanded in x around 0 20.7

      \[\leadsto \log \color{blue}{\left(0.5 \cdot \frac{e^{-1} \cdot {x}^{2}}{{y}^{2}} + e^{-1}\right)} \]

   7. Simplified 16.0

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

      [Start] 20.7	\[ \log \left(0.5 \cdot \frac{e^{-1} \cdot {x}^{2}}{{y}^{2}} + e^{-1}\right) \]
      fma-def [=>] 20.7	\[ \log \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{e^{-1} \cdot {x}^{2}}{{y}^{2}}, e^{-1}\right)\right)} \]
      *-commutative [=>] 20.7	\[ \log \left(\mathsf{fma}\left(0.5, \frac{\color{blue}{{x}^{2} \cdot e^{-1}}}{{y}^{2}}, e^{-1}\right)\right) \]
      associate-/l* [=>] 20.7	\[ \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{{x}^{2}}{\frac{{y}^{2}}{e^{-1}}}}, e^{-1}\right)\right) \]
      associate-/r/ [=>] 20.7	\[ \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot e^{-1}}, e^{-1}\right)\right) \]
      unpow2 [=>] 20.7	\[ \log \left(\mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{{y}^{2}} \cdot e^{-1}, e^{-1}\right)\right) \]
      unpow2 [=>] 20.7	\[ \log \left(\mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot e^{-1}, e^{-1}\right)\right) \]
      times-frac [=>] 16.0	\[ \log \left(\mathsf{fma}\left(0.5, \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot e^{-1}, e^{-1}\right)\right) \]
      unpow2 [<=] 16.0	\[ \log \left(\mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot e^{-1}, e^{-1}\right)\right) \]

   8. Applied egg-rr 16.0

      \[\leadsto \color{blue}{\log \left(e^{-1} \cdot \left(1 + 0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\right) + 0} \]

   9. Simplified 16.0

      \[\leadsto \color{blue}{-1 + \mathsf{log1p}\left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)} \]
      Proof

      [Start] 16.0	\[ \log \left(e^{-1} \cdot \left(1 + 0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\right) + 0 \]
      +-rgt-identity [=>] 16.0	\[ \color{blue}{\log \left(e^{-1} \cdot \left(1 + 0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\right)} \]
      +-commutative [=>] 16.0	\[ \log \left(e^{-1} \cdot \color{blue}{\left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2} + 1\right)}\right) \]
      log-prod [=>] 16.0	\[ \color{blue}{\log \left(e^{-1}\right) + \log \left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2} + 1\right)} \]
      rem-log-exp [=>] 16.0	\[ \color{blue}{-1} + \log \left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2} + 1\right) \]
      +-commutative [<=] 16.0	\[ -1 + \log \color{blue}{\left(1 + 0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)} \]
      log1p-def [=>] 16.0	\[ -1 + \color{blue}{\mathsf{log1p}\left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)} \]

   if 5.0000000000000002e-157 < (*.f64 x x) < 9.9999999999999998e-67

   1. Initial program 15.6

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

   2. Simplified 15.6

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

      [Start] 15.6	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-neg [=>] 15.6	\[ \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      *-commutative [=>] 15.6	\[ \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      associate-*r* [=>] 15.6	\[ \frac{\mathsf{fma}\left(x, x, -\color{blue}{\left(y \cdot y\right) \cdot 4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      distribute-rgt-neg-in [=>] 15.6	\[ \frac{\mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot y\right) \cdot \left(-4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      metadata-eval [=>] 15.6	\[ \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot \color{blue}{-4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-def [=>] 15.6	\[ \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]
      *-commutative [=>] 15.6	\[ \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   if 4.00000000000000032e91 < (*.f64 x x) < 9.9999999999999997e267

   1. Initial program 15.3

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

   2. Simplified 15.3

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

      [Start] 15.3	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      cancel-sign-sub-inv [=>] 15.3	\[ \frac{\color{blue}{x \cdot x + \left(-y \cdot 4\right) \cdot y}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      +-commutative [=>] 15.3	\[ \frac{\color{blue}{\left(-y \cdot 4\right) \cdot y + x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      *-commutative [=>] 15.3	\[ \frac{\color{blue}{y \cdot \left(-y \cdot 4\right)} + x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-def [=>] 15.3	\[ \frac{\color{blue}{\mathsf{fma}\left(y, -y \cdot 4, x \cdot x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      distribute-rgt-neg-in [=>] 15.3	\[ \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(-4\right)}, x \cdot x\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      metadata-eval [=>] 15.3	\[ \frac{\mathsf{fma}\left(y, y \cdot \color{blue}{-4}, x \cdot x\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-def [=>] 15.3	\[ \frac{\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]
      *-commutative [=>] 15.3	\[ \frac{\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   if 9.9999999999999997e267 < (*.f64 x x)

   1. Initial program 58.0

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

   2. Simplified 58.0

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

      [Start] 58.0	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-neg [=>] 58.0	\[ \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      *-commutative [=>] 58.0	\[ \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      associate-*r* [=>] 58.0	\[ \frac{\mathsf{fma}\left(x, x, -\color{blue}{\left(y \cdot y\right) \cdot 4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      distribute-rgt-neg-in [=>] 58.0	\[ \frac{\mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot y\right) \cdot \left(-4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      metadata-eval [=>] 58.0	\[ \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot \color{blue}{-4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-def [=>] 58.0	\[ \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]
      *-commutative [=>] 58.0	\[ \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Taylor expanded in x around inf 15.9

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

   4. Simplified 8.4

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

      [Start] 15.9	\[ \left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}} \]
      associate--l+ [=>] 15.9	\[ \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]
      +-commutative [=>] 15.9	\[ \color{blue}{\left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) + 1} \]
      distribute-rgt-out-- [=>] 15.9	\[ \color{blue}{\frac{{y}^{2}}{{x}^{2}} \cdot \left(-4 - 4\right)} + 1 \]
      metadata-eval [=>] 15.9	\[ \frac{{y}^{2}}{{x}^{2}} \cdot \color{blue}{-8} + 1 \]
      fma-def [=>] 15.9	\[ \color{blue}{\mathsf{fma}\left(\frac{{y}^{2}}{{x}^{2}}, -8, 1\right)} \]
      unpow2 [=>] 15.9	\[ \mathsf{fma}\left(\frac{\color{blue}{y \cdot y}}{{x}^{2}}, -8, 1\right) \]
      unpow2 [=>] 15.9	\[ \mathsf{fma}\left(\frac{y \cdot y}{\color{blue}{x \cdot x}}, -8, 1\right) \]
      times-frac [=>] 8.4	\[ \mathsf{fma}\left(\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, -8, 1\right) \]

   5. Applied egg-rr 8.4

      \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{2} \cdot -8 + 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-157}:\\ \;\;\;\;-1 + \mathsf{log1p}\left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{-66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+91}:\\ \;\;\;\;-1 + \mathsf{log1p}\left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x}\right)}^{2} \cdot -8 + 1\\ \end{array} \]

Alternatives

Alternative 1
Error	13.8
Cost	14280

\[\begin{array}{l} t_0 := -1 + \mathsf{log1p}\left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\\ t_1 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 10^{-66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(x, x, t_1\right)}\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 10^{+268}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x}\right)}^{2} \cdot -8 + 1\\ \end{array} \]

Alternative 2
Error	13.8
Cost	14092

\[\begin{array}{l} t_0 := \frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ t_1 := -1 + \mathsf{log1p}\left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 10^{+268}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x}\right)}^{2} \cdot -8 + 1\\ \end{array} \]

Alternative 3
Error	14.6
Cost	7952

\[\begin{array}{l} t_0 := \frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-157}:\\ \;\;\;\;-1 + \frac{0.5 \cdot \frac{x}{\frac{y}{x}}}{y}\\ \mathbf{elif}\;x \cdot x \leq 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+100}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+268}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x}\right)}^{2} \cdot -8 + 1\\ \end{array} \]

Alternative 4
Error	14.6
Cost	2256

\[\begin{array}{l} t_0 := \frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-157}:\\ \;\;\;\;-1 + \frac{0.5 \cdot \frac{x}{\frac{y}{x}}}{y}\\ \mathbf{elif}\;x \cdot x \leq 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+100}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+268}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y \cdot 4}{x}\\ \end{array} \]

Alternative 5
Error	16.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+58} \lor \neg \left(x \leq 2.2 \cdot 10^{+62}\right):\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y \cdot 4}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6
Error	16.5
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+58} \lor \neg \left(x \leq 1.55 \cdot 10^{+62}\right):\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y \cdot 4}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\left(x \cdot \frac{x}{y}\right) \cdot 0.25}{y}\\ \end{array} \]

Alternative 7
Error	16.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{+57} \lor \neg \left(x \leq 3.6 \cdot 10^{+61}\right):\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{y \cdot 4}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{0.5 \cdot \frac{x}{\frac{y}{x}}}{y}\\ \end{array} \]

Alternative 8
Error	18.5
Cost	856

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+147}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{+123}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-65}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-102}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9
Error	32.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022364 
(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))))