Given's Rotation SVD example

Average Error: 13.2 → 6.8

Time: 10.3s

Precision: binary64

Cost: 26884

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.95:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 0.5\right)}\\ \end{array} \]

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)))) -0.95)
   (/ (- p) x)
   (sqrt (fma x (/ 0.5 (hypot x (* p 2.0))) 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)))) <= -0.95) {
		tmp = -p / x;
	} else {
		tmp = sqrt(fma(x, (0.5 / hypot(x, (p * 2.0))), 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)))) <= -0.95)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(fma(x, Float64(0.5 / hypot(x, Float64(p * 2.0))), 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], -0.95], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

Target

Original	13.2
Target	13.2
Herbie	6.8

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.94999999999999996

   1. Initial program 52.7

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

   2. Simplified 52.6

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

      [Start] 52.7	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      distribute-lft-in [=>] 52.7	\[ \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      metadata-eval [=>] 52.7	\[ \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \]
      associate-*r/ [=>] 52.7	\[ \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      +-commutative [=>] 52.7	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}} \]
      fma-def [=>] 52.6	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}} \]
      associate-*l* [=>] 52.6	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(p \cdot p\right)}\right)}}} \]

   3. Taylor expanded in x around -inf 30.7

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

   4. Simplified 23.1

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

      [Start] 30.7	\[ \sqrt{\frac{{p}^{2}}{{x}^{2}}} \]
      unpow2 [=>] 30.7	\[ \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]
      unpow2 [=>] 30.7	\[ \sqrt{\frac{p \cdot p}{\color{blue}{x \cdot x}}} \]
      times-frac [=>] 23.1	\[ \sqrt{\color{blue}{\frac{p}{x} \cdot \frac{p}{x}}} \]

   5. Taylor expanded in p around -inf 27.3

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]

   6. Simplified 27.3

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

      [Start] 27.3	\[ -1 \cdot \frac{p}{x} \]
      mul-1-neg [=>] 27.3	\[ \color{blue}{-\frac{p}{x}} \]
      distribute-neg-frac [=>] 27.3	\[ \color{blue}{\frac{-p}{x}} \]

   if -0.94999999999999996 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 0.0

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

   2. Applied egg-rr 0.6

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

   3. Simplified 0.0

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

      [Start] 0.6	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1 \]
      expm1-def [=>] 0.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]
      expm1-log1p [=>] 0.0	\[ \color{blue}{\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]
      +-commutative [=>] 0.0	\[ \sqrt{\color{blue}{x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 0.5}} \]
      fma-def [=>] 0.0	\[ \sqrt{\color{blue}{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.95:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 0.5\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.8
Cost	20612

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.95:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]

Alternative 2
Error	20.5
Cost	7636

\[\begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq -9.6 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2.3 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4 \cdot 10^{-297}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 2.45 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.85 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{1 - \frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{elif}\;p \leq 2.5 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Error	20.5
Cost	7636

\[\begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq -2.2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.35 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4 \cdot 10^{-297}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.08 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 8 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{p \cdot p}{x \cdot x}, 1\right)\\ \mathbf{elif}\;p \leq 7.5 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	20.3
Cost	7256

\[\begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq -1.55 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -3.5 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4 \cdot 10^{-297}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.22 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 3.1 \cdot 10^{-56}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.7 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Error	34.6
Cost	652

\[\begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-137}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	34.8
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-137}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	40.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023060 
(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))))))))