Given's Rotation SVD example

Average Error: 13.0 → 6.7

Time: 10.6s

Precision: binary64

Cost: 33412

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

Math

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

↓

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

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

Target

Original	13.0
Target	13.0
Herbie	6.7

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

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 53.8

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

   2. Applied egg-rr 62.2

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

   3. Applied egg-rr 53.8

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

   4. Simplified 53.8

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

      [Start] 53.8	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1 \]
      expm1-def [=>] 53.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]
      expm1-log1p [=>] 53.8	\[ \color{blue}{\sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]
      *-commutative [=>] 53.8	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, \color{blue}{p \cdot 2}\right)}} \]

   5. Taylor expanded in x 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 -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\log \left(e^{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.7

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

Alternatives

Alternative 1
Error	6.7
Cost	27140

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 2
Error	6.7
Cost	20612

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

Alternative 3
Error	19.8
Cost	13508

\[\begin{array}{l} \mathbf{if}\;p \leq -2.6 \cdot 10^{-22}:\\ \;\;\;\;{\left({\left(0.5 + \frac{-0.25}{\frac{p}{x}}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	19.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;p \leq -5 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{0.5 + -0.25 \cdot \frac{x}{p}}\\ \mathbf{elif}\;p \leq 1.15 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Error	20.6
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -5.9 \cdot 10^{-294}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 8.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	19.7
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -6.2 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 4.6 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.05 \cdot 10^{-77}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7
Error	47.3
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq -5.9 \cdot 10^{-294}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 8
Error	53.5
Cost	192

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

Reproduce

herbie shell --seed 2023066 
(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))))))))