Given's Rotation SVD example

Average Error: 13.6 → 5.7

Time: 10.8s

Precision: binary64

Cost: 33412

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}\\ \end{array} \]

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)))) -0.9998)
   (sqrt (* (/ p x) (/ 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)))) <= -0.9998) {
		tmp = sqrt(((p / x) * (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)))) <= -0.9998) {
		tmp = Math.sqrt(((p / x) * (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)))) <= -0.9998:
		tmp = math.sqrt(((p / x) * (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)))) <= -0.9998)
		tmp = sqrt(Float64(Float64(p / x) * 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)))) <= -0.9998)
		tmp = sqrt(((p / x) * (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], -0.9998], N[Sqrt[N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $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	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)))) < -0.99980000000000002

   1. Initial program 53.4

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

   2. Simplified 53.4

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
      Proof
      (sqrt.f64 (+.f64 1/2 (/.f64 (*.f64 1/2 x) (sqrt.f64 (fma.f64 x x (*.f64 4 (*.f64 p p))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (/.f64 (*.f64 1/2 x) (sqrt.f64 (fma.f64 x x (*.f64 4 (*.f64 p p))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (+.f64 (*.f64 1/2 1) (/.f64 (*.f64 1/2 x) (sqrt.f64 (fma.f64 x x (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 p) p))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (+.f64 (*.f64 1/2 1) (/.f64 (*.f64 1/2 x) (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) (*.f64 (*.f64 4 p) p))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (+.f64 (*.f64 1/2 1) (/.f64 (*.f64 1/2 x) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (+.f64 (*.f64 1/2 1) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 1/2 (+.f64 1 (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around -inf 29.6

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

   4. Simplified 22.2

      \[\leadsto \sqrt{\color{blue}{\frac{p}{x} \cdot \frac{p}{x}}} \]
      Proof
      (sqrt.f64 (*.f64 (/.f64 p x) (/.f64 p x))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 p p) (*.f64 x x)))): 1 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 p 2)) (*.f64 x x))): 4 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (pow.f64 p 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error

   if -0.99980000000000002 < (/.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 0.0

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

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.7
Cost	20612

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]

Alternative 2
Error	20.8
Cost	7700

\[\begin{array}{l} \mathbf{if}\;p \leq -3.65 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2.4 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -6 \cdot 10^{-196}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 3.5 \cdot 10^{-257}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7.2 \cdot 10^{-125}:\\ \;\;\;\;1.5 \cdot {\left(\frac{p}{x}\right)}^{3} - \frac{p}{x}\\ \mathbf{elif}\;p \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \end{array} \]

Alternative 3
Error	20.8
Cost	7640

\[\begin{array}{l} \mathbf{if}\;p \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2.8 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -2.7 \cdot 10^{-203}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 4.5 \cdot 10^{-260}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.1 \cdot 10^{-126}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \end{array} \]

Alternative 4
Error	20.7
Cost	7256

\[\begin{array}{l} \mathbf{if}\;p \leq -1.58 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.25 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -4.3 \cdot 10^{-197}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 4 \cdot 10^{-270}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.3 \cdot 10^{-126}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Error	35.5
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+69}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;x \leq -1.26 \cdot 10^{-111}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	36.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	40.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022343 
(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))))))))