Given's Rotation SVD example

Average Accuracy: 78.6% → 99.3%

Time: 14.3s

Precision: binary64

Cost: 33284

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;{\left({\left(\sqrt[3]{\frac{p}{x}}\right)}^{2}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}}\\ \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

↓

(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.9998)
   (pow (pow (cbrt (/ p x)) 2.0) 1.5)
   (sqrt (* 0.5 (exp (log1p (/ x (hypot x (* p 2.0)))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

↓

double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9998) {
		tmp = pow(pow(cbrt((p / x)), 2.0), 1.5);
	} else {
		tmp = sqrt((0.5 * exp(log1p((x / hypot(x, (p * 2.0)))))));
	}
	return tmp;
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

↓

public static double code(double p, double x) {
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9998) {
		tmp = Math.pow(Math.pow(Math.cbrt((p / x)), 2.0), 1.5);
	} else {
		tmp = Math.sqrt((0.5 * Math.exp(Math.log1p((x / Math.hypot(x, (p * 2.0)))))));
	}
	return tmp;
}

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

↓

function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.9998)
		tmp = (cbrt(Float64(p / x)) ^ 2.0) ^ 1.5;
	else
		tmp = sqrt(Float64(0.5 * exp(log1p(Float64(x / hypot(x, Float64(p * 2.0)))))));
	end
	return tmp
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9998], N[Power[N[Power[N[Power[N[(p / x), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], 1.5], $MachinePrecision], N[Sqrt[N[(0.5 * N[Exp[N[Log[1 + N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	78.6%
Target	78.6%
Herbie	99.3%

\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.99980000000000002

   1. Initial program 16.2%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]

   2. Taylor expanded in x around -inf 51.3%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]

   3. Simplified 63.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}} \]
      Proof

      [Start] 51.3	\[ \sqrt{0.5 \cdot \left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)} \]
      unpow2 [=>] 51.3	\[ \sqrt{0.5 \cdot \left(2 \cdot \frac{\color{blue}{p \cdot p}}{{x}^{2}}\right)} \]
      unpow2 [=>] 51.3	\[ \sqrt{0.5 \cdot \left(2 \cdot \frac{p \cdot p}{\color{blue}{x \cdot x}}\right)} \]
      times-frac [=>] 63.8	\[ \sqrt{0.5 \cdot \left(2 \cdot \color{blue}{\left(\frac{p}{x} \cdot \frac{p}{x}\right)}\right)} \]

   4. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{{\left({\left(\sqrt[3]{\frac{p}{x}}\right)}^{2}\right)}^{1.5}} \]
      Proof

      [Start] 63.8	\[ \sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)} \]
      pow1/2 [=>] 63.8	\[ \color{blue}{{\left(0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)\right)}^{0.5}} \]
      add-cube-cbrt [=>] 63.1	\[ {\color{blue}{\left(\left(\sqrt[3]{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)} \cdot \sqrt[3]{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\right) \cdot \sqrt[3]{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\right)}}^{0.5} \]
      pow3 [=>] 63.1	\[ {\color{blue}{\left({\left(\sqrt[3]{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\right)}^{3}\right)}}^{0.5} \]
      metadata-eval [<=] 63.1	\[ {\left({\left(\sqrt[3]{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\right)}^{\color{blue}{\left(1 + 2\right)}}\right)}^{0.5} \]
      pow-pow [=>] 63.1	\[ \color{blue}{{\left(\sqrt[3]{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)}} \]
      associate-*r* [=>] 63.1	\[ {\left(\sqrt[3]{\color{blue}{\left(0.5 \cdot 2\right) \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)}}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      metadata-eval [=>] 63.1	\[ {\left(\sqrt[3]{\color{blue}{1} \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      *-un-lft-identity [<=] 63.1	\[ {\left(\sqrt[3]{\color{blue}{\frac{p}{x} \cdot \frac{p}{x}}}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      cbrt-prod [=>] 97.3	\[ {\color{blue}{\left(\sqrt[3]{\frac{p}{x}} \cdot \sqrt[3]{\frac{p}{x}}\right)}}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      pow2 [=>] 97.3	\[ {\color{blue}{\left({\left(\sqrt[3]{\frac{p}{x}}\right)}^{2}\right)}}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      metadata-eval [=>] 97.3	\[ {\left({\left(\sqrt[3]{\frac{p}{x}}\right)}^{2}\right)}^{\left(\color{blue}{3} \cdot 0.5\right)} \]
      metadata-eval [=>] 97.3	\[ {\left({\left(\sqrt[3]{\frac{p}{x}}\right)}^{2}\right)}^{\color{blue}{1.5}} \]

   if -0.99980000000000002 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]

   2. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
      Proof

      [Start] 100.0	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      add-exp-log [=>] 100.0	\[ \sqrt{0.5 \cdot \color{blue}{e^{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]
      log1p-udef [<=] 100.0	\[ \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]
      +-commutative [=>] 100.0	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]
      add-sqr-sqrt [=>] 100.0	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]
      hypot-def [=>] 100.0	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]
      associate-*l* [=>] 100.0	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]
      sqrt-prod [=>] 100.0	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]
      metadata-eval [=>] 100.0	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]
      sqrt-unprod [<=] 49.4	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]
      add-sqr-sqrt [<=] 100.0	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;{\left({\left(\sqrt[3]{\frac{p}{x}}\right)}^{2}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;{\left({\left(\sqrt[3]{\frac{p}{x}}\right)}^{2}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 2
Accuracy	90.8%
Cost	20612

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 3
Accuracy	58.8%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\frac{p \cdot p}{\frac{x}{-2}} - x}\right)}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	57.7%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Accuracy	67.2%
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -1.85 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -4.8 \cdot 10^{-290}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 5000000000:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Accuracy	67.9%
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -5.8 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 3.2 \cdot 10^{-39}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5200000000:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7
Accuracy	27.0%
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq -4.8 \cdot 10^{-290}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 8
Accuracy	16.8%
Cost	192

\[\frac{p}{x} \]

Reproduce

herbie shell --seed 2023142 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))