Given's Rotation SVD example

Average Accuracy: 78.2% → 90.6%

Time: 8.4s

Precision: binary64

Cost: 20612

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

Math

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

↓

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

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)))) -0.5)
   (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)))) <= -0.5) {
		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)))) <= -0.5) {
		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)))) <= -0.5:
		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)))) <= -0.5)
		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)))) <= -0.5)
		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], -0.5], 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	78.2%
Target	78.2%
Herbie	90.6%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.5

   1. Initial program 15.9%

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

   2. Simplified 15.9%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
      Proof

      [Start] 15.9	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      distribute-lft-in [=>] 15.9	\[ \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      metadata-eval [=>] 15.9	\[ \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \]
      associate-*r/ [=>] 15.9	\[ \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      +-commutative [=>] 15.9	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}} \]
      fma-def [=>] 15.9	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}} \]
      associate-*l* [=>] 15.9	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(p \cdot p\right)}\right)}}} \]

   3. Taylor expanded in x around -inf 51.5%

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

   4. Simplified 63.8%

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

      [Start] 51.5	\[ \sqrt{\frac{{p}^{2}}{{x}^{2}}} \]
      unpow2 [=>] 51.5	\[ \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]
      unpow2 [=>] 51.5	\[ \sqrt{\frac{p \cdot p}{\color{blue}{x \cdot x}}} \]
      times-frac [=>] 63.8	\[ \sqrt{\color{blue}{\frac{p}{x} \cdot \frac{p}{x}}} \]

   if -0.5 < (/.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. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
      Proof

      [Start] 100.0	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      distribute-lft-in [=>] 100.0	\[ \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      metadata-eval [=>] 100.0	\[ \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \]
      associate-*r/ [=>] 100.0	\[ \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      +-commutative [=>] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}} \]
      fma-def [=>] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}} \]
      associate-*l* [=>] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(p \cdot p\right)}\right)}}} \]

   3. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}} \]
      fma-udef [=>] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{x \cdot x + 4 \cdot \left(p \cdot p\right)}}}} \]
      add-sqr-sqrt [=>] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{x \cdot x + \color{blue}{\sqrt{4 \cdot \left(p \cdot p\right)} \cdot \sqrt{4 \cdot \left(p \cdot p\right)}}}}} \]
      hypot-def [=>] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{4 \cdot \left(p \cdot p\right)}\right)}}} \]
      *-commutative [=>] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{\left(p \cdot p\right) \cdot 4}}\right)}} \]
      sqrt-prod [=>] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{p \cdot p} \cdot \sqrt{4}}\right)}} \]
      sqrt-prod [=>] 49.2	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)} \cdot \sqrt{4}\right)}} \]
      add-sqr-sqrt [<=] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, \color{blue}{p} \cdot \sqrt{4}\right)}} \]
      metadata-eval [=>] 100.0	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, p \cdot \color{blue}{2}\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

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

Alternatives

Alternative 1
Accuracy	67.5%
Cost	6992

\[\begin{array}{l} \mathbf{if}\;p \leq -1.4 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2.85 \cdot 10^{-180}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -1.75 \cdot 10^{-227}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 8.2 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 2
Accuracy	67.0%
Cost	6992

\[\begin{array}{l} \mathbf{if}\;p \leq -6 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{0.5 + -0.25 \cdot \frac{x}{p}}\\ \mathbf{elif}\;p \leq -1.8 \cdot 10^{-177}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -7.2 \cdot 10^{-228}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 4.7 \cdot 10^{-117}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Accuracy	68.0%
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -7.6 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -4.6 \cdot 10^{-305}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 2.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Accuracy	27.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq -1.02 \cdot 10^{-299}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 5
Accuracy	16.9%
Cost	192

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

Reproduce

herbie shell --seed 2023137 
(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))))))))