Given's Rotation SVD example

Average Accuracy: 78.8% → 99.4%

Time: 8.9s

Precision: binary64

Cost: 27012

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999999995:\\ \;\;\;\;{\left({\left(\sqrt[3]{\frac{p}{x}}\right)}^{2}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(x, p \cdot 2\right)}, x, 1\right)}\\ \end{array} \]

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)))) -0.999999995)
   (pow (pow (cbrt (/ p x)) 2.0) 1.5)
   (sqrt (* 0.5 (fma (/ 1.0 (hypot x (* p 2.0))) x 1.0)))))

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)))) <= -0.999999995) {
		tmp = pow(pow(cbrt((p / x)), 2.0), 1.5);
	} else {
		tmp = sqrt((0.5 * fma((1.0 / hypot(x, (p * 2.0))), x, 1.0)));
	}
	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)))) <= -0.999999995)
		tmp = (cbrt(Float64(p / x)) ^ 2.0) ^ 1.5;
	else
		tmp = sqrt(Float64(0.5 * fma(Float64(1.0 / hypot(x, Float64(p * 2.0))), x, 1.0)));
	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], -0.999999995], 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[(N[(1.0 / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	78.8%
Target	78.8%
Herbie	99.4%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.99999999500000003

   1. Initial program 23.1%

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

   2. Taylor expanded in x around -inf 46.7%

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

   3. Simplified 56.0%

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

   4. Applied egg-rr 97.1%

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

   if -0.99999999500000003 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.8%

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

   2. Applied egg-rr 99.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, x, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999999995:\\ \;\;\;\;{\left({\left(\sqrt[3]{\frac{p}{x}}\right)}^{2}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(x, p \cdot 2\right)}, x, 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999999995:\\ \;\;\;\;{\left({\left(\sqrt[3]{\frac{p}{x}}\right)}^{2}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 2
Accuracy	89.3%
Cost	20612

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

Alternative 3
Accuracy	68.1%
Cost	7892

\[\begin{array}{l} \mathbf{if}\;p \leq -1.85 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -3.8 \cdot 10^{-285}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 2.35 \cdot 10^{-240}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.45 \cdot 10^{-205}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.5 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 + -2 \cdot \frac{p \cdot p}{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Accuracy	68.3%
Cost	7124

\[\begin{array}{l} \mathbf{if}\;p \leq -1.7 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2.1 \cdot 10^{-282}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.35 \cdot 10^{-241}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.9 \cdot 10^{-204}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 4 \cdot 10^{-57}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Accuracy	67.2%
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -1.85 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -6.8 \cdot 10^{-286}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 5.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Accuracy	26.7%
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq -6.8 \cdot 10^{-286}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 7
Accuracy	16.9%
Cost	192

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

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(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))))))))