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

Average Error: 31.9 → 12.3

Time: 8.9s

Precision: binary64

Cost: 8528

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := x \cdot x + y \cdot \left(y \cdot -4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-289}:\\ \;\;\;\;1 + \frac{\frac{y}{x}}{\frac{x}{y \cdot -8}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-89}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(x, x, t_0\right)}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;1 + \left(y \cdot -8\right) \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\frac{t_1}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

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))) (t_1 (+ (* x x) (* y (* y -4.0)))))
   (if (<= t_0 2e-289)
     (+ 1.0 (/ (/ y x) (/ x (* y -8.0))))
     (if (<= t_0 4e-89)
       (/ t_1 (fma x x t_0))
       (if (<= t_0 5e-36)
         (+ 1.0 (* (* y -8.0) (/ y (* x x))))
         (if (<= t_0 2e+228)
           (/ t_1 (+ t_0 (* x x)))
           (fma 0.5 (* (/ x y) (/ x y)) -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 t_1 = (x * x) + (y * (y * -4.0));
	double tmp;
	if (t_0 <= 2e-289) {
		tmp = 1.0 + ((y / x) / (x / (y * -8.0)));
	} else if (t_0 <= 4e-89) {
		tmp = t_1 / fma(x, x, t_0);
	} else if (t_0 <= 5e-36) {
		tmp = 1.0 + ((y * -8.0) * (y / (x * x)));
	} else if (t_0 <= 2e+228) {
		tmp = t_1 / (t_0 + (x * x));
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -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))
	t_1 = Float64(Float64(x * x) + Float64(y * Float64(y * -4.0)))
	tmp = 0.0
	if (t_0 <= 2e-289)
		tmp = Float64(1.0 + Float64(Float64(y / x) / Float64(x / Float64(y * -8.0))));
	elseif (t_0 <= 4e-89)
		tmp = Float64(t_1 / fma(x, x, t_0));
	elseif (t_0 <= 5e-36)
		tmp = Float64(1.0 + Float64(Float64(y * -8.0) * Float64(y / Float64(x * x))));
	elseif (t_0 <= 2e+228)
		tmp = Float64(t_1 / Float64(t_0 + Float64(x * x)));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -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]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-289], N[(1.0 + N[(N[(y / x), $MachinePrecision] / N[(x / N[(y * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-89], N[(t$95$1 / N[(x * x + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-36], N[(1.0 + N[(N[(y * -8.0), $MachinePrecision] * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+228], N[(t$95$1 / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]]]]

Target

Original	31.9
Target	31.6
Herbie	12.3

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

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e-289

   1. Initial program 30.1

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

   2. Taylor expanded in x around inf 16.2

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

   3. Simplified 16.2

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

      [Start] 16.2	\[ \left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}} \]
      cancel-sign-sub-inv [=>] 16.2	\[ \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) + \left(-4\right) \cdot \frac{{y}^{2}}{{x}^{2}}} \]
      metadata-eval [=>] 16.2	\[ \left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) + \color{blue}{-4} \cdot \frac{{y}^{2}}{{x}^{2}} \]
      associate-+l+ [=>] 16.2	\[ \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]
      associate-*r/ [=>] 16.2	\[ 1 + \left(\color{blue}{\frac{-4 \cdot {y}^{2}}{{x}^{2}}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      *-commutative [=>] 16.2	\[ 1 + \left(\frac{\color{blue}{{y}^{2} \cdot -4}}{{x}^{2}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      unpow2 [=>] 16.2	\[ 1 + \left(\frac{\color{blue}{\left(y \cdot y\right)} \cdot -4}{{x}^{2}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      associate-*r* [<=] 16.2	\[ 1 + \left(\frac{\color{blue}{y \cdot \left(y \cdot -4\right)}}{{x}^{2}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      unpow2 [=>] 16.2	\[ 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{\color{blue}{x \cdot x}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      associate-/l* [=>] 16.2	\[ 1 + \left(\color{blue}{\frac{y}{\frac{x \cdot x}{y \cdot -4}}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      associate-*r/ [=>] 16.2	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \color{blue}{\frac{-4 \cdot {y}^{2}}{{x}^{2}}}\right) \]
      *-commutative [=>] 16.2	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \frac{\color{blue}{{y}^{2} \cdot -4}}{{x}^{2}}\right) \]
      unpow2 [=>] 16.2	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \frac{\color{blue}{\left(y \cdot y\right)} \cdot -4}{{x}^{2}}\right) \]
      associate-*r* [<=] 16.2	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \frac{\color{blue}{y \cdot \left(y \cdot -4\right)}}{{x}^{2}}\right) \]
      unpow2 [=>] 16.2	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \frac{y \cdot \left(y \cdot -4\right)}{\color{blue}{x \cdot x}}\right) \]
      associate-/l* [=>] 16.2	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \color{blue}{\frac{y}{\frac{x \cdot x}{y \cdot -4}}}\right) \]

   4. Applied egg-rr 16.2

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

   5. Applied egg-rr 8.5

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

   if 2e-289 < (*.f64 (*.f64 y 4) y) < 4.00000000000000015e-89

   1. Initial program 15.0

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

   2. Simplified 15.0

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

      [Start] 15.0	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      *-commutative [=>] 15.0	\[ \frac{x \cdot x - \color{blue}{y \cdot \left(y \cdot 4\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-def [=>] 15.0	\[ \frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]
      *-commutative [=>] 15.0	\[ \frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   if 4.00000000000000015e-89 < (*.f64 (*.f64 y 4) y) < 5.00000000000000004e-36

   1. Initial program 16.6

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

   2. Taylor expanded in x around inf 27.0

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

   3. Simplified 27.0

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

      [Start] 27.0	\[ \left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}} \]
      cancel-sign-sub-inv [=>] 27.0	\[ \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) + \left(-4\right) \cdot \frac{{y}^{2}}{{x}^{2}}} \]
      metadata-eval [=>] 27.0	\[ \left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) + \color{blue}{-4} \cdot \frac{{y}^{2}}{{x}^{2}} \]
      associate-+l+ [=>] 27.0	\[ \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]
      associate-*r/ [=>] 27.0	\[ 1 + \left(\color{blue}{\frac{-4 \cdot {y}^{2}}{{x}^{2}}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      *-commutative [=>] 27.0	\[ 1 + \left(\frac{\color{blue}{{y}^{2} \cdot -4}}{{x}^{2}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      unpow2 [=>] 27.0	\[ 1 + \left(\frac{\color{blue}{\left(y \cdot y\right)} \cdot -4}{{x}^{2}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      associate-*r* [<=] 27.0	\[ 1 + \left(\frac{\color{blue}{y \cdot \left(y \cdot -4\right)}}{{x}^{2}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      unpow2 [=>] 27.0	\[ 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{\color{blue}{x \cdot x}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      associate-/l* [=>] 27.0	\[ 1 + \left(\color{blue}{\frac{y}{\frac{x \cdot x}{y \cdot -4}}} + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]
      associate-*r/ [=>] 27.0	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \color{blue}{\frac{-4 \cdot {y}^{2}}{{x}^{2}}}\right) \]
      *-commutative [=>] 27.0	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \frac{\color{blue}{{y}^{2} \cdot -4}}{{x}^{2}}\right) \]
      unpow2 [=>] 27.0	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \frac{\color{blue}{\left(y \cdot y\right)} \cdot -4}{{x}^{2}}\right) \]
      associate-*r* [<=] 27.0	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \frac{\color{blue}{y \cdot \left(y \cdot -4\right)}}{{x}^{2}}\right) \]
      unpow2 [=>] 27.0	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \frac{y \cdot \left(y \cdot -4\right)}{\color{blue}{x \cdot x}}\right) \]
      associate-/l* [=>] 27.0	\[ 1 + \left(\frac{y}{\frac{x \cdot x}{y \cdot -4}} + \color{blue}{\frac{y}{\frac{x \cdot x}{y \cdot -4}}}\right) \]

   4. Applied egg-rr 27.0

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

   if 5.00000000000000004e-36 < (*.f64 (*.f64 y 4) y) < 1.9999999999999998e228

   1. Initial program 16.0

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

   if 1.9999999999999998e228 < (*.f64 (*.f64 y 4) y)

   1. Initial program 54.2

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

   2. Simplified 54.2

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

      [Start] 54.2	\[ \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      *-commutative [=>] 54.2	\[ \frac{x \cdot x - \color{blue}{y \cdot \left(y \cdot 4\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
      fma-def [=>] 54.2	\[ \frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]
      *-commutative [=>] 54.2	\[ \frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Taylor expanded in x around 0 16.6

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

   4. Simplified 9.7

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

      [Start] 16.6	\[ 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1 \]
      fma-neg [=>] 16.6	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]
      unpow2 [=>] 16.6	\[ \mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]
      unpow2 [=>] 16.6	\[ \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]
      times-frac [=>] 9.7	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]
      metadata-eval [=>] 9.7	\[ \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 12.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-289}:\\ \;\;\;\;1 + \frac{\frac{y}{x}}{\frac{x}{y \cdot -8}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-89}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-36}:\\ \;\;\;\;1 + \left(y \cdot -8\right) \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.3
Cost	8528

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{t_0 + x \cdot x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-289}:\\ \;\;\;\;1 + \frac{\frac{y}{x}}{\frac{x}{y \cdot -8}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;1 + \left(y \cdot -8\right) \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 2
Error	12.4
Cost	2768

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{t_0 + x \cdot x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-289}:\\ \;\;\;\;1 + \frac{\frac{y}{x}}{\frac{x}{y \cdot -8}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;1 + \left(y \cdot -8\right) \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y \cdot 4}\\ \end{array} \]

Alternative 3
Error	16.2
Cost	1868

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-139}:\\ \;\;\;\;1 + \frac{\frac{y}{x}}{\frac{x}{y \cdot -8}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-91}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t_0 \leq 10^{-11}:\\ \;\;\;\;1 + \left(y \cdot -8\right) \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y \cdot 4}\\ \end{array} \]

Alternative 4
Error	17.5
Cost	1496

\[\begin{array}{l} t_0 := 1 + \left(y \cdot -8\right) \cdot \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -240000000000:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-78}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+123}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{+158}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Error	17.3
Cost	1496

\[\begin{array}{l} t_0 := 1 + \left(y \cdot -8\right) \cdot \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -2700:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -3.45 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-81}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-18}:\\ \;\;\;\;1 + \frac{\frac{y}{x}}{\frac{x}{y \cdot -8}}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+123}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{+158}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6
Error	16.7
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -460000:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-46}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-78}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7
Error	31.2
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022354 
(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))))