Given's Rotation SVD example

Average Error: 13.6 → 5.9

Time: 8.4s

Precision: binary64

Cost: 33284

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

Math

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

↓

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

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 (* 0.5 (* 2.0 (* (/ p x) (/ p x)))))
   (sqrt (* 0.5 (exp (log1p (/ 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((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = sqrt((0.5 * exp(log1p((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((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = Math.sqrt((0.5 * Math.exp(Math.log1p((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((0.5 * (2.0 * ((p / x) * (p / x)))))
	else:
		tmp = math.sqrt((0.5 * math.exp(math.log1p((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 = sqrt(Float64(0.5 * Float64(2.0 * Float64(Float64(p / x) * Float64(p / x)))));
	else
		tmp = sqrt(Float64(0.5 * exp(log1p(Float64(x / hypot(x, Float64(p * 2.0)))))));
	end
	return 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[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[Exp[N[Log[1 + N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	13.6
Target	13.6
Herbie	5.9

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.2

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

   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 23.3

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}} \]
      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)))): 51 points increase in error, 29 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 -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

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.9

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

Alternatives

Alternative 1
Error	5.9
Cost	20612

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

Alternative 2
Error	20.7
Cost	7388

\[\begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq -9.5 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.65 \cdot 10^{-108}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq -7.8 \cdot 10^{-259}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -3.6 \cdot 10^{-305}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.5 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.75 \cdot 10^{-123}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.4 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Error	20.8
Cost	7388

\[\begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq -9 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -8.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq -2.15 \cdot 10^{-259}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p}{x} \cdot \frac{p}{x}, -0.5, 1\right)\\ \mathbf{elif}\;p \leq -3.6 \cdot 10^{-305}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.34 \cdot 10^{-126}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 6.4 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	35.7
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+123}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-106}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	35.5
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-107}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	60.3
Cost	64

\[0 \]

Alternative 7
Error	40.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022330 
(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))))))))