Given's Rotation SVD example

Average Accuracy: 79.7% → 90.9%

Time: 11.3s

Precision: binary64

Cost: 33540.00

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99999:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left({\left(0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{3}\right)}^{0.3333333333333333}}\\ \end{array} \]

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)))) -0.99999)
   (sqrt (* (/ p x) (/ p x)))
   (sqrt
    (pow
     (pow (+ 0.5 (* x (/ 0.5 (hypot x (* p 2.0))))) 3.0)
     0.3333333333333333))))

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)))) <= -0.99999) {
		tmp = sqrt(((p / x) * (p / x)));
	} else {
		tmp = sqrt(pow(pow((0.5 + (x * (0.5 / hypot(x, (p * 2.0))))), 3.0), 0.3333333333333333));
	}
	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)))) <= -0.99999) {
		tmp = Math.sqrt(((p / x) * (p / x)));
	} else {
		tmp = Math.sqrt(Math.pow(Math.pow((0.5 + (x * (0.5 / Math.hypot(x, (p * 2.0))))), 3.0), 0.3333333333333333));
	}
	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)))) <= -0.99999:
		tmp = math.sqrt(((p / x) * (p / x)))
	else:
		tmp = math.sqrt(math.pow(math.pow((0.5 + (x * (0.5 / math.hypot(x, (p * 2.0))))), 3.0), 0.3333333333333333))
	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)))) <= -0.99999)
		tmp = sqrt(Float64(Float64(p / x) * Float64(p / x)));
	else
		tmp = sqrt(((Float64(0.5 + Float64(x * Float64(0.5 / hypot(x, Float64(p * 2.0))))) ^ 3.0) ^ 0.3333333333333333));
	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)))) <= -0.99999)
		tmp = sqrt(((p / x) * (p / x)));
	else
		tmp = sqrt((((0.5 + (x * (0.5 / hypot(x, (p * 2.0))))) ^ 3.0) ^ 0.3333333333333333));
	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], -0.99999], N[Sqrt[N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Power[N[Power[N[(0.5 + N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision]]

Target

Original	79.7%
Target	79.7%
Herbie	90.9%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.999990000000000046

   1. Initial program 16.8%

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

   2. Simplified 16.8%

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

      [Start] 16.8	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      distribute-lft-in [=>] 16.8	\[ \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      metadata-eval [=>] 16.8	\[ \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \]
      associate-*r/ [=>] 16.8	\[ \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      +-commutative [=>] 16.8	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}} \]
      fma-def [=>] 16.8	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}} \]
      associate-*l* [=>] 16.8	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(p \cdot p\right)}\right)}}} \]

   3. Taylor expanded in x around -inf 51.6%

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

   4. Simplified 62.8%

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

      [Start] 51.6	\[ \sqrt{\frac{{p}^{2}}{{x}^{2}}} \]
      unpow2 [=>] 51.6	\[ \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]
      unpow2 [=>] 51.6	\[ \sqrt{\frac{p \cdot p}{\color{blue}{x \cdot x}}} \]
      times-frac [=>] 62.8	\[ \sqrt{\color{blue}{\frac{p}{x} \cdot \frac{p}{x}}} \]

   if -0.999990000000000046 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      distribute-lft-in [=>] 99.9	\[ \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      metadata-eval [=>] 99.9	\[ \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \]
      associate-*r/ [=>] 99.9	\[ \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      +-commutative [=>] 99.9	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}} \]
      fma-def [=>] 99.9	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}} \]
      associate-*l* [=>] 99.9	\[ \sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(p \cdot p\right)}\right)}}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{3}\right)}^{0.3333333333333333}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99999:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left({\left(0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{3}\right)}^{0.3333333333333333}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.9%
Cost	20612.00

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99999:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]

Alternative 2
Accuracy	68.2%
Cost	7504.00

\[\begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq -8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -5.2 \cdot 10^{-288}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.05 \cdot 10^{-203}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.7 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{1 - \frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{elif}\;p \leq 4.8 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 6.5 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Accuracy	68.1%
Cost	7256.00

\[\begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq -2.65 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -6.2 \cdot 10^{-288}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-203}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.4 \cdot 10^{-149}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.2 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 4.1 \cdot 10^{-64}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Accuracy	67.1%
Cost	6860.00

\[\begin{array}{l} \mathbf{if}\;p \leq -3.1 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2.1 \cdot 10^{-294}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Accuracy	26.1%
Cost	388.00

\[\begin{array}{l} \mathbf{if}\;p \leq -2.1 \cdot 10^{-294}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 6
Accuracy	16.4%
Cost	192.00

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

Reproduce

herbie shell --seed 2023096 
(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))))))))