Given's Rotation SVD example

Average Error: 13.3 → 5.7

Time: 11.2s

Precision: binary64

Cost: 52996

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\\ \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\sqrt[3]{{t_0}^{2}}, \sqrt[3]{t_0}, 1\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ x (hypot x (* p 2.0)))))
   (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
     (sqrt (* 0.5 (* 2.0 (* (/ p x) (/ p x)))))
     (sqrt (* 0.5 (fma (cbrt (pow t_0 2.0)) (cbrt 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 = x / hypot(x, (p * 2.0));
	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 * fma(cbrt(pow(t_0, 2.0)), cbrt(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(x / hypot(x, Float64(p * 2.0)))
	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 * fma(cbrt((t_0 ^ 2.0)), cbrt(t_0), 1.0)));
	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_] := Block[{t$95$0 = N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, 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[(N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$0, 1/3], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	13.3
Target	13.3
Herbie	5.7

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

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 54.0

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

   2. Applied egg-rr 54.0

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

   3. Taylor expanded in x around -inf 30.3

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

   4. Simplified 22.8

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}} \]
      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)))): 61 points increase in error, 31 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, 1 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

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

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}}, \sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}, 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.7
Cost	27140

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, p + p\right)}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 2
Error	5.7
Cost	20612

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

Alternative 3
Error	20.7
Cost	14104

\[\begin{array}{l} t_0 := \frac{p \cdot {1}^{0.5}}{x}\\ \mathbf{if}\;p \leq -1.828289886302339 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -8.195036608424947 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -6.701495220375145 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.840345046035984 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.9352514848513003 \cdot 10^{-267}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8.116769719726647 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{2}}{x} \cdot \left(\sqrt{0.5} \cdot \left(-p\right)\right)\\ \mathbf{elif}\;p \leq 1.558362308605261 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\frac{-2}{x} \cdot \left(p \cdot p\right) - x}\right)}\\ \end{array} \]

Alternative 4
Error	20.7
Cost	14104

\[\begin{array}{l} t_0 := \frac{p \cdot {1}^{0.5}}{x}\\ \mathbf{if}\;p \leq -1.828289886302339 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -8.195036608424947 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -6.701495220375145 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.840345046035984 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.9352514848513003 \cdot 10^{-267}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8.116769719726647 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \frac{-p}{x}\right)\\ \mathbf{elif}\;p \leq 1.558362308605261 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\frac{-2}{x} \cdot \left(p \cdot p\right) - x}\right)}\\ \end{array} \]

Alternative 5
Error	20.8
Cost	14104

\[\begin{array}{l} t_0 := \frac{p \cdot {1}^{0.5}}{x}\\ \mathbf{if}\;p \leq -1.828289886302339 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -8.195036608424947 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -6.701495220375145 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.840345046035984 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.9352514848513003 \cdot 10^{-267}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8.116769719726647 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{-\sqrt{2}}{\frac{x}{p}}\\ \mathbf{elif}\;p \leq 1.558362308605261 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\frac{-2}{x} \cdot \left(p \cdot p\right) - x}\right)}\\ \end{array} \]

Alternative 6
Error	20.5
Cost	8020

\[\begin{array}{l} t_0 := \frac{p \cdot {1}^{0.5}}{x}\\ \mathbf{if}\;p \leq -1.828289886302339 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -8.195036608424947 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -6.701495220375145 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.840345046035984 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.558362308605261 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\frac{-2}{x} \cdot \left(p \cdot p\right) - x}\right)}\\ \end{array} \]

Alternative 7
Error	20.2
Cost	7312

\[\begin{array}{l} t_0 := \frac{p \cdot {1}^{0.5}}{x}\\ \mathbf{if}\;p \leq -1.828289886302339 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -8.195036608424947 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -6.701495220375145 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.840345046035984 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.558362308605261 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8
Error	20.2
Cost	6728

\[\begin{array}{l} \mathbf{if}\;p \leq -1.1008293975446658 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.558362308605261 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 9
Error	28.3
Cost	6464

\[\sqrt{0.5} \]

Reproduce

herbie shell --seed 2022308 
(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))))))))