Given's Rotation SVD example

Average Error: 13.6 → 0.1

Time: 1.5min

Precision: binary64

Cost: 33412

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999998:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}\\ \end{array} \]

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)))) -0.9999998)
   (fabs (/ p x))
   (sqrt (* 0.5 (log (exp (+ 1.0 (/ 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)))) <= -0.9999998) {
		tmp = fabs((p / x));
	} else {
		tmp = sqrt((0.5 * log(exp((1.0 + (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)))) <= -0.9999998) {
		tmp = Math.abs((p / x));
	} else {
		tmp = Math.sqrt((0.5 * Math.log(Math.exp((1.0 + (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)))) <= -0.9999998:
		tmp = math.fabs((p / x))
	else:
		tmp = math.sqrt((0.5 * math.log(math.exp((1.0 + (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)))) <= -0.9999998)
		tmp = abs(Float64(p / x));
	else
		tmp = sqrt(Float64(0.5 * log(exp(Float64(1.0 + Float64(x / hypot(x, Float64(p * 2.0))))))));
	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)
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999998)
		tmp = abs((p / x));
	else
		tmp = sqrt((0.5 * log(exp((1.0 + (x / hypot(x, (p * 2.0))))))));
	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_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9999998], N[Abs[N[(p / x), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[Log[N[Exp[N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	13.6
Target	13.6
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.999999799999999994

   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 53.7

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

   3. Simplified 53.7

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{-2 \cdot \frac{p}{\frac{x}{p}} - x}}\right)} \]
      Proof
      (-.f64 (*.f64 -2 (/.f64 p (/.f64 x p))) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 p p) x))) x): 18 points increase in error, 2 points decrease in error
      (-.f64 (*.f64 -2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 p 2)) x)) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -2 (/.f64 (pow.f64 p 2) x)) (neg.f64 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -2 (/.f64 (pow.f64 p 2) x)) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in x around inf 30.5

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

   5. Simplified 24.4

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{2}{{\left(\sqrt[3]{\frac{x}{p}}\right)}^{6}}}} \]
      Proof
      (/.f64 2 (pow.f64 (cbrt.f64 (/.f64 x p)) 6)): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (pow.f64 (cbrt.f64 (/.f64 x p)) (Rewrite<= metadata-eval (*.f64 2 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (cbrt.f64 (/.f64 x p)) 3) (pow.f64 (cbrt.f64 (/.f64 x p)) 3)))): 24 points increase in error, 38 points decrease in error
      (/.f64 2 (*.f64 (Rewrite=> rem-cube-cbrt_binary64 (/.f64 x p)) (pow.f64 (cbrt.f64 (/.f64 x p)) 3))): 38 points increase in error, 62 points decrease in error
      (/.f64 2 (*.f64 (/.f64 x p) (Rewrite=> rem-cube-cbrt_binary64 (/.f64 x p)))): 37 points increase in error, 58 points decrease in error
      (/.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x x) (*.f64 p p)))): 58 points increase in error, 22 points decrease in error
      (/.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (*.f64 p p))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 x 2) (Rewrite<= unpow2_binary64 (pow.f64 p 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 (pow.f64 p 2)) (pow.f64 x 2))): 8 points increase in error, 13 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (pow.f64 p 2) (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (Rewrite=> unpow2_binary64 (*.f64 p p)) (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 p p) (Rewrite=> unpow2_binary64 (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 p x) (/.f64 p x)))): 28 points increase in error, 59 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 p)) x) (/.f64 p x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 x) p)) (/.f64 p x))): 14 points increase in error, 18 points decrease in error
      (*.f64 2 (*.f64 (*.f64 (/.f64 1 x) p) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 p)) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 (/.f64 1 x) p) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 x) p)))): 20 points increase in error, 18 points decrease in error
      (*.f64 2 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (/.f64 1 x) (/.f64 1 x)) (*.f64 p p)))): 60 points increase in error, 37 points decrease in error
      (*.f64 2 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 1 x) 2)) (*.f64 p p))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (pow.f64 (/.f64 1 x) 2) (Rewrite<= unpow2_binary64 (pow.f64 p 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 p 2) (pow.f64 (/.f64 1 x) 2)))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

   if -0.999999799999999994 < (/.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

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999998:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	20612

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999998:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 2
Error	20.3
Cost	8044

\[\begin{array}{l} t_0 := \left|\frac{p}{x}\right|\\ \mathbf{if}\;p \leq -6.6 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -3.05 \cdot 10^{-68}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -2.5 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -2.5 \cdot 10^{-201}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -5 \cdot 10^{-253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -3.7 \cdot 10^{-299}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.9 \cdot 10^{-292}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-261}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.8 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 8.4 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8.8 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Error	20.6
Cost	8044

\[\begin{array}{l} t_0 := \left|\frac{p}{x}\right|\\ \mathbf{if}\;p \leq -14000000:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{-2 \cdot \frac{p}{\frac{x}{p}} - x}\right)}\\ \mathbf{elif}\;p \leq -3.55 \cdot 10^{-68}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -2.8 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -3.1 \cdot 10^{-201}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -9.5 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -1.85 \cdot 10^{-299}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 6.6 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 6.5 \cdot 10^{-261}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4 \cdot 10^{-227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.8 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.15 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	20.1
Cost	6856

\[\begin{array}{l} \mathbf{if}\;p \leq -1.2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 2.2 \cdot 10^{-59}:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Error	24.5
Cost	6728

\[\begin{array}{l} \mathbf{if}\;p \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 6.2 \cdot 10^{-229}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	49.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 7
Error	62.9
Cost	64

\[-2 \]

Alternative 8
Error	55.1
Cost	64

\[4 \]

Reproduce

herbie shell --seed 2022325 
(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))))))))