Given's Rotation SVD example

Average Error: 13.0 → 6.5

Time: 7.6s

Precision: binary64

Cost: 20612

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99999999999995:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

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)))) -0.99999999999995)
   (/ (- p) x)
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

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)))) <= -0.99999999999995) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	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)))) <= -0.99999999999995) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	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)))) <= -0.99999999999995:
		tmp = -p / x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	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)))) <= -0.99999999999995)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	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)))) <= -0.99999999999995)
		tmp = -p / x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	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], -0.99999999999995], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	13.0
Target	13.0
Herbie	6.5

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.99999999999995004

   1. Initial program 53.7

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

   2. Taylor expanded in x around -inf 30.9

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

   3. Simplified 30.9

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

      [Start] 30.9	\[ \sqrt{0.5 \cdot \left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)} \]
      associate-*r/ [=>] 30.9	\[ \sqrt{0.5 \cdot \color{blue}{\frac{2 \cdot {p}^{2}}{{x}^{2}}}} \]
      unpow2 [=>] 30.9	\[ \sqrt{0.5 \cdot \frac{2 \cdot \color{blue}{\left(p \cdot p\right)}}{{x}^{2}}} \]
      unpow2 [=>] 30.9	\[ \sqrt{0.5 \cdot \frac{2 \cdot \left(p \cdot p\right)}{\color{blue}{x \cdot x}}} \]

   4. Taylor expanded in p around -inf 26.4

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

   5. Simplified 26.4

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

      [Start] 26.4	\[ -1 \cdot \frac{p}{x} \]
      associate-*r/ [=>] 26.4	\[ \color{blue}{\frac{-1 \cdot p}{x}} \]
      neg-mul-1 [<=] 26.4	\[ \frac{\color{blue}{-p}}{x} \]

   if -0.99999999999995004 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99999999999995:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.5
Cost	8160

\[\begin{array}{l} \mathbf{if}\;p \leq -7.1 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.9 \cdot 10^{-119}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -3.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq -1.85 \cdot 10^{-183}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -5.2 \cdot 10^{-300}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 6.8 \cdot 10^{-94}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 2.1 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 2
Error	20.6
Cost	8096

\[\begin{array}{l} \mathbf{if}\;p \leq -7.1 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -9 \cdot 10^{-120}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -2.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq -3.4 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -5.2 \cdot 10^{-300}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.75 \cdot 10^{-96}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 2.1 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 0.00015:\\ \;\;\;\;{\left(\frac{1}{\frac{x \cdot x}{p \cdot p}}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Error	20.6
Cost	7521

\[\begin{array}{l} \mathbf{if}\;p \leq -8 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -9.5 \cdot 10^{-119}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -1.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq -1.55 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -5.2 \cdot 10^{-300}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 7.4 \cdot 10^{-96} \lor \neg \left(p \leq 1.62 \cdot 10^{-24}\right) \land p \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	20.7
Cost	7125

\[\begin{array}{l} \mathbf{if}\;p \leq -8 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -5.2 \cdot 10^{-300}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 6 \cdot 10^{-94} \lor \neg \left(p \leq 2.1 \cdot 10^{-24}\right) \land p \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Error	47.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq -5.2 \cdot 10^{-300}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 6
Error	53.7
Cost	192

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

Reproduce

herbie shell --seed 2023059 
(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))))))))