Given's Rotation SVD example

Percentage Accurate: 79.5% → 91.0%

Time: 17.3s

Precision: binary64

Cost: 26884

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

Math

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

↓

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

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 (fma (/ 0.5 (hypot x (* p 2.0))) x 0.5))))

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(fma((0.5 / hypot(x, (p * 2.0))), x, 0.5));
	}
	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(fma(Float64(0.5 / hypot(x, Float64(p * 2.0))), x, 0.5));
	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], -1.0], N[((-N[Sqrt[N[(p * p), $MachinePrecision]], $MachinePrecision]) / x), $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * x + 0.5), $MachinePrecision]], $MachinePrecision]]

Target

Original	79.5%
Target	79.5%
Herbie	91.0%

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

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 15.8%

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

   2. Taylor expanded in x around -inf 60.4%

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

   3. Applied egg-rr 64.9%

      \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{p \cdot p}}}{x} \]
      Step-by-step derivation

      [Start] 60.4%	\[ -1 \cdot \frac{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot p\right)}{x} \]
      associate-*r* [=>] 60.1%	\[ -1 \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot p}}{x} \]
      sqrt-unprod [=>] 61.1%	\[ -1 \cdot \frac{\color{blue}{\sqrt{2 \cdot 0.5}} \cdot p}{x} \]
      metadata-eval [=>] 61.1%	\[ -1 \cdot \frac{\sqrt{\color{blue}{1}} \cdot p}{x} \]
      metadata-eval [=>] 61.1%	\[ -1 \cdot \frac{\color{blue}{1} \cdot p}{x} \]
      *-un-lft-identity [<=] 61.1%	\[ -1 \cdot \frac{\color{blue}{p}}{x} \]
      add-sqr-sqrt [=>] 54.3%	\[ -1 \cdot \frac{\color{blue}{\sqrt{p} \cdot \sqrt{p}}}{x} \]
      sqrt-unprod [=>] 64.9%	\[ -1 \cdot \frac{\color{blue}{\sqrt{p \cdot p}}}{x} \]

   if -1 < (/.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%

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

      [Start] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      clear-num [=>] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \color{blue}{\frac{1}{\frac{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}{x}}}\right)} \]
      associate-/r/ [=>] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \color{blue}{\frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot x}\right)} \]
      +-commutative [=>] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}} \cdot x\right)} \]
      add-sqr-sqrt [=>] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}} \cdot x\right)} \]
      hypot-def [=>] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \frac{1}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}} \cdot x\right)} \]
      associate-*l* [=>] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)} \cdot x\right)} \]
      sqrt-prod [=>] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)} \cdot x\right)} \]
      metadata-eval [=>] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)} \cdot x\right)} \]
      sqrt-unprod [<=] 50.5%	\[ \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)} \cdot x\right)} \]
      add-sqr-sqrt [<=] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)} \cdot x\right)} \]

   3. Applied egg-rr 98.9%

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

      [Start] 100.0%	\[ \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x\right)} \]
      expm1-log1p-u [=>] 99.0%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x\right)}\right)\right)} \]
      expm1-udef [=>] 98.9%	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x\right)}\right)} - 1} \]
      distribute-lft-in [=>] 98.9%	\[ e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x\right)}}\right)} - 1 \]
      metadata-eval [=>] 98.9%	\[ e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.5} + 0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x\right)}\right)} - 1 \]
      associate-*l/ [=>] 98.9%	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \color{blue}{\frac{1 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}}\right)} - 1 \]
      *-un-lft-identity [<=] 98.9%	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{\color{blue}{x}}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1 \]
      associate-*r/ [=>] 98.9%	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + \color{blue}{\frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}}\right)} - 1 \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}}}} \]
      Step-by-step derivation

      [Start] 98.9%	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1 \]
      expm1-def [=>] 99.0%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]
      expm1-log1p [=>] 100.0%	\[ \color{blue}{\sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]
      associate-/l* [=>] 100.0%	\[ \sqrt{0.5 + \color{blue}{\frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]
      *-commutative [=>] 100.0%	\[ \sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, \color{blue}{p \cdot 2}\right)}{x}}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}, x, 0.5\right)}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ \sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}}} \]
      +-commutative [=>] 100.0%	\[ \sqrt{\color{blue}{\frac{0.5}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}} + 0.5}} \]
      associate-/r/ [=>] 100.0%	\[ \sqrt{\color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)} \cdot x} + 0.5} \]
      fma-def [=>] 100.0%	\[ \sqrt{\color{blue}{\mathsf{fma}\left(\frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}, x, 0.5\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

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

Alternatives

Alternative 1
Accuracy	91.0%
Cost	26884

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

Alternative 2
Accuracy	91.0%
Cost	20612

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

Alternative 3
Accuracy	80.3%
Cost	13836

\[\begin{array}{l} t_0 := \frac{-\sqrt{p \cdot p}}{x}\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 4
Accuracy	68.3%
Cost	7388

\[\begin{array}{l} \mathbf{if}\;p \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq -2.5 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -3.4 \cdot 10^{-233}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.45 \cdot 10^{-120}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 4.6 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Accuracy	64.3%
Cost	7180

\[\begin{array}{l} t_0 := \frac{-\sqrt{p \cdot p}}{x}\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 40000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	44.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-75}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	44.5%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	36.2%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023263 
(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))))))))