Given's Rotation SVD example

Average Accuracy: 79.5% → 89.4%

Time: 8.1s

Precision: binary64

Cost: 20612

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

Math

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

↓

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

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)))) -1.0)
   (/ (- 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)))) <= -1.0) {
		tmp = -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)))) <= -1.0) {
		tmp = -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)))) <= -1.0:
		tmp = -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)))) <= -1.0)
		tmp = Float64(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)))) <= -1.0)
		tmp = -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], -1.0], N[((-p) / x), $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	79.5%
Target	79.5%
Herbie	89.4%

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

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 17.0%

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

   2. Applied egg-rr 15.2%

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

      [Start] 17.0	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      expm1-log1p-u [=>] 17.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]
      expm1-udef [=>] 17.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Simplified 17.0%

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

      [Start] 15.2	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1 \]
      expm1-def [=>] 15.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]
      expm1-log1p [=>] 15.2	\[ \color{blue}{\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]
      associate-*r/ [=>] 17.0	\[ \sqrt{0.5 + \color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   4. Taylor expanded in x around -inf 57.6%

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

   5. Simplified 57.6%

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

      [Start] 57.6	\[ -1 \cdot \frac{p}{x} \]
      mul-1-neg [=>] 57.6	\[ \color{blue}{-\frac{p}{x}} \]
      distribute-neg-frac [=>] 57.6	\[ \color{blue}{\frac{-p}{x}} \]

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.6%

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

   2. Applied egg-rr 98.6%

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

      [Start] 99.6	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      expm1-log1p-u [=>] 98.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]
      expm1-udef [=>] 98.6	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Simplified 99.6%

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

      [Start] 98.6	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1 \]
      expm1-def [=>] 98.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]
      expm1-log1p [=>] 99.6	\[ \color{blue}{\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]
      associate-*r/ [=>] 99.6	\[ \sqrt{0.5 + \color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

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

Alternatives

Alternative 1
Accuracy	67.4%
Cost	7376

\[\begin{array}{l} \mathbf{if}\;p \leq -4.2 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 10^{-220}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.4 \cdot 10^{-136}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 9.2 \cdot 10^{+26}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \end{array} \]

Alternative 2
Accuracy	67.6%
Cost	6992

\[\begin{array}{l} \mathbf{if}\;p \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 4 \cdot 10^{-219}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.65 \cdot 10^{-136}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 6.5 \cdot 10^{+26}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Accuracy	66.4%
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -1.15 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 8.2 \cdot 10^{-280}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.45 \cdot 10^{-130}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Accuracy	26.1%
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq 8.2 \cdot 10^{-280}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 5
Accuracy	16.8%
Cost	192

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

Reproduce

herbie shell --seed 2023130 
(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))))))))