Given's Rotation SVD example

Average Accuracy: 78.8% → 91.1%

Time: 7.3s

Precision: binary64

Cost: 20612

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

Math

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

↓

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

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)))) -1.0)
   (/ (- (sqrt (* p p))) x)
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

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 * p)) / x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	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 * p)) / x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	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 * p)) / x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	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 = Float64(Float64(-sqrt(Float64(p * p))) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	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 * p)) / x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	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[((-N[Sqrt[N[(p * p), $MachinePrecision]], $MachinePrecision]) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	78.8%
Target	78.8%
Herbie	91.1%

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

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 15.0%

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

   2. Taylor expanded in x around -inf 55.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot p\right)}{x}} \]

   3. Simplified 55.2%

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

      [Start] 55.2	\[ -1 \cdot \frac{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot p\right)}{x} \]
      mul-1-neg [=>] 55.2	\[ \color{blue}{-\frac{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot p\right)}{x}} \]
      associate-/l* [=>] 55.0	\[ -\color{blue}{\frac{\sqrt{2}}{\frac{x}{\sqrt{0.5} \cdot p}}} \]
      associate-/r/ [=>] 55.2	\[ -\color{blue}{\frac{\sqrt{2}}{x} \cdot \left(\sqrt{0.5} \cdot p\right)} \]
      distribute-lft-neg-in [=>] 55.2	\[ \color{blue}{\left(-\frac{\sqrt{2}}{x}\right) \cdot \left(\sqrt{0.5} \cdot p\right)} \]
      *-commutative [=>] 55.2	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot \color{blue}{\left(p \cdot \sqrt{0.5}\right)} \]

   4. Applied egg-rr 64.4%

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

      [Start] 55.2	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot \left(p \cdot \sqrt{0.5}\right) \]
      add-sqr-sqrt [=>] 48.2	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot \color{blue}{\left(\sqrt{p \cdot \sqrt{0.5}} \cdot \sqrt{p \cdot \sqrt{0.5}}\right)} \]
      sqrt-unprod [=>] 64.1	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot \color{blue}{\sqrt{\left(p \cdot \sqrt{0.5}\right) \cdot \left(p \cdot \sqrt{0.5}\right)}} \]
      pow1/2 [=>] 64.1	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot \color{blue}{{\left(\left(p \cdot \sqrt{0.5}\right) \cdot \left(p \cdot \sqrt{0.5}\right)\right)}^{0.5}} \]
      *-commutative [=>] 64.1	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot {\left(\color{blue}{\left(\sqrt{0.5} \cdot p\right)} \cdot \left(p \cdot \sqrt{0.5}\right)\right)}^{0.5} \]
      *-commutative [=>] 64.1	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot {\left(\left(\sqrt{0.5} \cdot p\right) \cdot \color{blue}{\left(\sqrt{0.5} \cdot p\right)}\right)}^{0.5} \]
      swap-sqr [=>] 64.0	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot {\color{blue}{\left(\left(\sqrt{0.5} \cdot \sqrt{0.5}\right) \cdot \left(p \cdot p\right)\right)}}^{0.5} \]
      add-sqr-sqrt [<=] 64.4	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot {\left(\color{blue}{0.5} \cdot \left(p \cdot p\right)\right)}^{0.5} \]

   5. Simplified 64.4%

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

      [Start] 64.4	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot {\left(0.5 \cdot \left(p \cdot p\right)\right)}^{0.5} \]
      unpow1/2 [=>] 64.4	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot \color{blue}{\sqrt{0.5 \cdot \left(p \cdot p\right)}} \]
      associate-*r* [=>] 64.4	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot \sqrt{\color{blue}{\left(0.5 \cdot p\right) \cdot p}} \]
      *-commutative [=>] 64.4	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot \sqrt{\color{blue}{p \cdot \left(0.5 \cdot p\right)}} \]

   6. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(p \cdot \left(p \cdot 0.5\right)\right)\right)}^{0.5}}{x}} \]
      Proof

      [Start] 64.4	\[ \left(-\frac{\sqrt{2}}{x}\right) \cdot \sqrt{p \cdot \left(0.5 \cdot p\right)} \]
      distribute-lft-neg-out [=>] 64.4	\[ \color{blue}{-\frac{\sqrt{2}}{x} \cdot \sqrt{p \cdot \left(0.5 \cdot p\right)}} \]
      neg-sub0 [=>] 64.4	\[ \color{blue}{0 - \frac{\sqrt{2}}{x} \cdot \sqrt{p \cdot \left(0.5 \cdot p\right)}} \]
      associate-*l/ [=>] 64.5	\[ 0 - \color{blue}{\frac{\sqrt{2} \cdot \sqrt{p \cdot \left(0.5 \cdot p\right)}}{x}} \]
      pow1/2 [=>] 64.5	\[ 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{p \cdot \left(0.5 \cdot p\right)}}{x} \]
      pow1/2 [=>] 64.5	\[ 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(p \cdot \left(0.5 \cdot p\right)\right)}^{0.5}}}{x} \]
      pow-prod-down [=>] 64.8	\[ 0 - \frac{\color{blue}{{\left(2 \cdot \left(p \cdot \left(0.5 \cdot p\right)\right)\right)}^{0.5}}}{x} \]
      *-commutative [=>] 64.8	\[ 0 - \frac{{\left(2 \cdot \left(p \cdot \color{blue}{\left(p \cdot 0.5\right)}\right)\right)}^{0.5}}{x} \]

   7. Simplified 64.8%

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

      [Start] 64.8	\[ 0 - \frac{{\left(2 \cdot \left(p \cdot \left(p \cdot 0.5\right)\right)\right)}^{0.5}}{x} \]
      neg-sub0 [<=] 64.8	\[ \color{blue}{-\frac{{\left(2 \cdot \left(p \cdot \left(p \cdot 0.5\right)\right)\right)}^{0.5}}{x}} \]
      distribute-neg-frac [=>] 64.8	\[ \color{blue}{\frac{-{\left(2 \cdot \left(p \cdot \left(p \cdot 0.5\right)\right)\right)}^{0.5}}{x}} \]
      unpow1/2 [=>] 64.8	\[ \frac{-\color{blue}{\sqrt{2 \cdot \left(p \cdot \left(p \cdot 0.5\right)\right)}}}{x} \]
      *-commutative [=>] 64.8	\[ \frac{-\sqrt{\color{blue}{\left(p \cdot \left(p \cdot 0.5\right)\right) \cdot 2}}}{x} \]
      associate-*r* [=>] 64.8	\[ \frac{-\sqrt{\color{blue}{\left(\left(p \cdot p\right) \cdot 0.5\right)} \cdot 2}}{x} \]
      associate-*l* [=>] 64.8	\[ \frac{-\sqrt{\color{blue}{\left(p \cdot p\right) \cdot \left(0.5 \cdot 2\right)}}}{x} \]
      metadata-eval [=>] 64.8	\[ \frac{-\sqrt{\left(p \cdot p\right) \cdot \color{blue}{1}}}{x} \]
      *-rgt-identity [=>] 64.8	\[ \frac{-\sqrt{\color{blue}{p \cdot p}}}{x} \]

   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 99.7%

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

      [Start] 99.7	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      add-sqr-sqrt [=>] 99.7	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}} + x \cdot x}}\right)} \]
      hypot-def [=>] 99.7	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(\sqrt{\left(4 \cdot p\right) \cdot p}, x\right)}}\right)} \]
      associate-*l* [=>] 99.7	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(\sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}, x\right)}\right)} \]
      sqrt-prod [=>] 99.7	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(\color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}, x\right)}\right)} \]
      metadata-eval [=>] 99.7	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(\color{blue}{2} \cdot \sqrt{p \cdot p}, x\right)}\right)} \]
      sqrt-unprod [<=] 49.6	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}, x\right)}\right)} \]
      add-sqr-sqrt [<=] 99.7	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot \color{blue}{p}, x\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

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

Alternatives

Alternative 1
Accuracy	68.4%
Cost	7124

\[\begin{array}{l} \mathbf{if}\;p \leq -1.45 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -3.7 \cdot 10^{-104}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -5.3 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.65 \cdot 10^{-192}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.8 \cdot 10^{-81}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 2
Accuracy	67.9%
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -5.5 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 10^{-74}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Accuracy	27.0%
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq 9.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 4
Accuracy	16.9%
Cost	192

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

Reproduce

herbie shell --seed 2023151 
(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))))))))