Given's Rotation SVD example

Average Error: 12.6 → 5.5

Time: 11.4s

Precision: binary64

Cost: 34308

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 + \frac{x}{\mathsf{hypot}\left(x, p + p\right)}\\ \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{{t_0}^{2} + -1}{t_0 + 1}}\\ \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

↓

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (/ x (hypot x (+ p p))))))
   (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.999)
     (sqrt (* 0.5 (* 2.0 (* (/ p x) (/ p x)))))
     (sqrt (* 0.5 (/ (+ (pow t_0 2.0) -1.0) (+ t_0 1.0)))))))

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 t_0 = 2.0 + (x / hypot(x, (p + p)));
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.999) {
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = sqrt((0.5 * ((pow(t_0, 2.0) + -1.0) / (t_0 + 1.0))));
	}
	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 t_0 = 2.0 + (x / Math.hypot(x, (p + p)));
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.999) {
		tmp = Math.sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = Math.sqrt((0.5 * ((Math.pow(t_0, 2.0) + -1.0) / (t_0 + 1.0))));
	}
	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):
	t_0 = 2.0 + (x / math.hypot(x, (p + p)))
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.999:
		tmp = math.sqrt((0.5 * (2.0 * ((p / x) * (p / x)))))
	else:
		tmp = math.sqrt((0.5 * ((math.pow(t_0, 2.0) + -1.0) / (t_0 + 1.0))))
	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)
	t_0 = Float64(2.0 + Float64(x / hypot(x, Float64(p + p))))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.999)
		tmp = sqrt(Float64(0.5 * Float64(2.0 * Float64(Float64(p / x) * Float64(p / x)))));
	else
		tmp = sqrt(Float64(0.5 * Float64(Float64((t_0 ^ 2.0) + -1.0) / Float64(t_0 + 1.0))));
	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)
	t_0 = 2.0 + (x / hypot(x, (p + p)));
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.999)
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	else
		tmp = sqrt((0.5 * (((t_0 ^ 2.0) + -1.0) / (t_0 + 1.0))));
	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_] := Block[{t$95$0 = N[(2.0 + N[(x / N[Sqrt[x ^ 2 + N[(p + p), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.999], N[Sqrt[N[(0.5 * N[(2.0 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	12.6
Target	12.6
Herbie	5.5

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.998999999999999999

   1. Initial program 53.3

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

   2. Applied egg-rr 53.3

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

   3. Taylor expanded in x around -inf 30.2

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

   4. Simplified 23.2

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}} \]
      Proof
      (*.f64 2 (*.f64 (/.f64 p x) (/.f64 p x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 p p) (*.f64 x x)))): 63 points increase in error, 23 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 p 2)) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (pow.f64 p 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error

   if -0.998999999999999999 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 0.0

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 0.0

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{{\left(2 + \frac{x}{\mathsf{hypot}\left(x, p + p\right)}\right)}^{2} + -1}{\left(2 + \frac{x}{\mathsf{hypot}\left(x, p + p\right)}\right) + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.5
Cost	27140

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \mathsf{log1p}\left(0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p + p\right)} + -1\right)\right)}\\ \end{array} \]

Alternative 2
Error	5.5
Cost	27012

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(x, p + p\right)}, x, 1\right)}\\ \end{array} \]

Alternative 3
Error	5.5
Cost	20740

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\left(2 + \frac{x}{\mathsf{hypot}\left(x, p + p\right)}\right) + -1\right)}\\ \end{array} \]

Alternative 4
Error	13.7
Cost	13704

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \mathbf{if}\;p \leq 3.887907752578724 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 9.380123697457626 \cdot 10^{-160}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.9
Cost	7256

\[\begin{array}{l} \mathbf{if}\;p \leq -0.0001666777039806335:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.149213346894678 \cdot 10^{-257}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -5.761679042347427 \cdot 10^{-307}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 3.887907752578724 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.380123697457626 \cdot 10^{-160}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 9.257484633322853 \cdot 10^{-83}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	20.0
Cost	7256

\[\begin{array}{l} \mathbf{if}\;p \leq -0.0001666777039806335:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + -0.5 \cdot \frac{x}{p}\right)}\\ \mathbf{elif}\;p \leq -1.149213346894678 \cdot 10^{-257}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -5.761679042347427 \cdot 10^{-307}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 3.887907752578724 \cdot 10^{-291}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.380123697457626 \cdot 10^{-160}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 9.257484633322853 \cdot 10^{-83}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7
Error	21.1
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -6.267657639217018 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 2.7929388547969235 \cdot 10^{-305}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 2.6675626654088793 \cdot 10^{-108}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8
Error	47.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq 2.7929388547969235 \cdot 10^{-305}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 9
Error	54.0
Cost	192

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

Reproduce

herbie shell --seed 2022311 
(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))))))))