Given's Rotation SVD example

Average Accuracy: 79.3% → 90.7%

Time: 11.9s

Precision: binary64

Cost: 20612

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

Math

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

↓

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

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)))) -1.0)
   (sqrt (* (/ p x) (/ p x)))
   (sqrt (+ 0.5 (/ (* x 0.5) (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)))) <= -1.0) {
		tmp = sqrt(((p / x) * (p / x)));
	} else {
		tmp = sqrt((0.5 + ((x * 0.5) / 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)))) <= -1.0) {
		tmp = Math.sqrt(((p / x) * (p / x)));
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.5) / Math.hypot(x, (p * 2.0)))));
	}
	return tmp;
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

↓

def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0:
		tmp = math.sqrt(((p / x) * (p / x)))
	else:
		tmp = math.sqrt((0.5 + ((x * 0.5) / 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)))) <= -1.0)
		tmp = sqrt(Float64(Float64(p / x) * Float64(p / x)));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.5) / hypot(x, Float64(p * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

↓

function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0)
		tmp = sqrt(((p / x) * (p / x)));
	else
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p * 2.0)))));
	end
	tmp_2 = 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], -1.0], N[Sqrt[N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(x * 0.5), $MachinePrecision] / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	79.3%
Target	79.3%
Herbie	90.7%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 16.7%

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

   2. Applied egg-rr 14.9%

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

      [Start] 16.7	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      expm1-log1p-u [=>] 16.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]
      expm1-udef [=>] 16.7	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Simplified 16.7%

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

      [Start] 14.9	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1 \]
      expm1-def [=>] 14.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]
      expm1-log1p [=>] 14.9	\[ \color{blue}{\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]
      associate-*r/ [=>] 16.7	\[ \sqrt{0.5 + \color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   4. Taylor expanded in x around -inf 51.9%

      \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

   5. Simplified 63.0%

      \[\leadsto \sqrt{\color{blue}{\frac{p}{x} \cdot \frac{p}{x}}} \]
      Proof

      [Start] 51.9	\[ \sqrt{\frac{{p}^{2}}{{x}^{2}}} \]
      unpow2 [=>] 51.9	\[ \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]
      unpow2 [=>] 51.9	\[ \sqrt{\frac{p \cdot p}{\color{blue}{x \cdot x}}} \]
      times-frac [=>] 63.0	\[ \sqrt{\color{blue}{\frac{p}{x} \cdot \frac{p}{x}}} \]

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.7%

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

   2. Applied egg-rr 98.7%

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

      [Start] 99.7	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      expm1-log1p-u [=>] 98.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]
      expm1-udef [=>] 98.7	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Simplified 99.7%

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

      [Start] 98.7	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1 \]
      expm1-def [=>] 98.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]
      expm1-log1p [=>] 99.7	\[ \color{blue}{\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]
      associate-*r/ [=>] 99.7	\[ \sqrt{0.5 + \color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

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

Alternatives

Alternative 1
Accuracy	68.0%
Cost	7904

\[\begin{array}{l} \mathbf{if}\;p \leq -16000000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -9.2 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{elif}\;p \leq -6.7 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.9 \cdot 10^{-152}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -1.35 \cdot 10^{-280}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.6 \cdot 10^{-264}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.45 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.25}{p}}\\ \end{array} \]

Alternative 2
Accuracy	68.0%
Cost	7520

\[\begin{array}{l} \mathbf{if}\;p \leq -15500000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2.25 \cdot 10^{-24}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -9.8 \cdot 10^{-152}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -3.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.25 \cdot 10^{-263}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.15 \cdot 10^{-226}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Accuracy	68.0%
Cost	7520

\[\begin{array}{l} \mathbf{if}\;p \leq -15500000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -8.5 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{elif}\;p \leq -4.2 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -3.4 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -5.5 \cdot 10^{-288}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 6.7 \cdot 10^{-264}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 6.5 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Accuracy	66.8%
Cost	7124

\[\begin{array}{l} \mathbf{if}\;p \leq -15500000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -3.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq -2.3 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.26 \cdot 10^{-304}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 2.3 \cdot 10^{-53}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Accuracy	26.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq 1.26 \cdot 10^{-304}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 6
Accuracy	17.0%
Cost	192

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

Reproduce

herbie shell --seed 2023138 
(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))))))))