Given's Rotation SVD example

Average Error: 13.7 → 5.8

Time: 7.1s

Precision: binary64

Cost: 20612

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

Math

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

↓

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

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

Target

Original	13.7
Target	13.7
Herbie	5.8

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.1

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

   2. Applied egg-rr 54.1

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

   3. Taylor expanded in x around -inf 28.6

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

   4. Simplified 28.6

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

   5. Applied egg-rr 22.5

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

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

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

4. Final simplification 5.8

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

Alternatives

Alternative 1
Error	20.7
Cost	7124

\[\begin{array}{l} \mathbf{if}\;p \leq -1.7224308668959296 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 9.596726413784006 \cdot 10^{-304}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 6.794751093088959 \cdot 10^{-165}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.971237565362955 \cdot 10^{-121}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 8.640475694562824 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 2
Error	21.5
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -1.7224308668959296 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.2009860048556524 \cdot 10^{-292}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 4.971237565362955 \cdot 10^{-121}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Error	46.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq 1.2009860048556524 \cdot 10^{-292}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 4
Error	53.1
Cost	192

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

Reproduce

herbie shell --seed 2022284 
(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))))))))