Given's Rotation SVD example

Average Error: 13.1 → 6.7

Time: 7.3s

Precision: binary64

Cost: 33668

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999999:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \log \left(e^{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p + p\right)}\right)}\right)\right)}\\ \end{array} \]

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)))) -0.9999999)
   (/ (- p) x)
   (sqrt (* 0.5 (* 2.0 (log (exp (* 0.5 (+ 1.0 (/ x (hypot x (+ p p))))))))))))

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)))) <= -0.9999999) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * (2.0 * log(exp((0.5 * (1.0 + (x / hypot(x, (p + p))))))))));
	}
	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)))) <= -0.9999999) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt((0.5 * (2.0 * Math.log(Math.exp((0.5 * (1.0 + (x / Math.hypot(x, (p + p))))))))));
	}
	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)))) <= -0.9999999:
		tmp = -p / x
	else:
		tmp = math.sqrt((0.5 * (2.0 * math.log(math.exp((0.5 * (1.0 + (x / math.hypot(x, (p + p))))))))))
	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)))) <= -0.9999999)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(2.0 * log(exp(Float64(0.5 * Float64(1.0 + Float64(x / hypot(x, Float64(p + p))))))))));
	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)))) <= -0.9999999)
		tmp = -p / x;
	else
		tmp = sqrt((0.5 * (2.0 * log(exp((0.5 * (1.0 + (x / hypot(x, (p + p))))))))));
	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], -0.9999999], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(2.0 * N[Log[N[Exp[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p + p), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	13.1
Target	13.1
Herbie	6.7

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.999999900000000053

   1. Initial program 52.5

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

   2. Taylor expanded in x around -inf 30.6

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

   3. Simplified 30.6

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{p \cdot p}{x \cdot x}\right)}} \]
      Proof
      (*.f64 2 (/.f64 (*.f64 p p) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 p 2)) (*.f64 x x))): 1 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (pow.f64 p 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))): 1 points increase in error, 0 points decrease in error

   4. Taylor expanded in p around -inf 26.7

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]

   5. Simplified 26.7

      \[\leadsto \color{blue}{\frac{-p}{x}} \]
      Proof
      (/.f64 (neg.f64 p) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 p)) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 p x))): 0 points increase in error, 0 points decrease in error

   if -0.999999900000000053 < (/.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

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

   3. Applied egg-rr 0.4

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{{\left(\sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 1}\right)}^{3}}} \]

   4. Applied egg-rr 0.0

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

   5. Simplified 0.0

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{1 + \frac{x}{\mathsf{hypot}\left(x, p + p\right)}}}\right)\right)}} \]
      Proof
      (*.f64 2 (log.f64 (sqrt.f64 (exp.f64 (+.f64 1 (/.f64 x (hypot.f64 x (+.f64 p p)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (log.f64 (sqrt.f64 (exp.f64 (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 x (hypot.f64 x (+.f64 p p))) 1)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= count-2_binary64 (+.f64 (log.f64 (sqrt.f64 (exp.f64 (+.f64 (/.f64 x (hypot.f64 x (+.f64 p p))) 1)))) (log.f64 (sqrt.f64 (exp.f64 (+.f64 (/.f64 x (hypot.f64 x (+.f64 p p))) 1)))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.0

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999999:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \log \left(e^{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p + p\right)}\right)}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.7
Cost	33284

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999999:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}}\\ \end{array} \]

Alternative 2
Error	6.7
Cost	20612

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999999:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 3
Error	20.3
Cost	7888

\[\begin{array}{l} \mathbf{if}\;p \leq -8.2 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 4.8 \cdot 10^{-293}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\frac{p \cdot p}{\frac{x}{-2}} - x}\right)}\\ \end{array} \]

Alternative 4
Error	20.2
Cost	6992

\[\begin{array}{l} \mathbf{if}\;p \leq -7.6 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-293}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.6 \cdot 10^{-199}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Error	20.7
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -4.2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 2.7 \cdot 10^{-294}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 3.25 \cdot 10^{-52}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	46.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq 2.7 \cdot 10^{-294}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 7
Error	53.3
Cost	192

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

Reproduce

herbie shell --seed 2022332 
(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))))))))