Given's Rotation SVD example

Average Error: 13.0 → 5.7

Time: 7.3s

Precision: binary64

Cost: 20612

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

Math

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

↓

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

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

Target

Original	13.0
Target	13.0
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)))) < -0.5

   1. Initial program 52.7

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

   2. Taylor expanded in x around -inf 28.6

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

   3. Simplified 28.9

      \[\leadsto \color{blue}{\frac{-\sqrt{2}}{\frac{x}{p \cdot \sqrt{0.5}}}} \]
      Proof

      [Start] 28.6	\[ -1 \cdot \frac{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot p\right)}{x} \]
      mul-1-neg [=>] 28.6	\[ \color{blue}{-\frac{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot p\right)}{x}} \]
      associate-/l* [=>] 28.9	\[ -\color{blue}{\frac{\sqrt{2}}{\frac{x}{\sqrt{0.5} \cdot p}}} \]
      distribute-neg-frac [=>] 28.9	\[ \color{blue}{\frac{-\sqrt{2}}{\frac{x}{\sqrt{0.5} \cdot p}}} \]
      *-commutative [=>] 28.9	\[ \frac{-\sqrt{2}}{\frac{x}{\color{blue}{p \cdot \sqrt{0.5}}}} \]

   4. Applied egg-rr 23.0

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

   if -0.5 < (/.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 0.7

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1} \]

   3. Simplified 0.0

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

      [Start] 0.7	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1 \]
      expm1-def [=>] 0.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]
      expm1-log1p [=>] 0.0	\[ \color{blue}{\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 5.7

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

Alternatives

Alternative 1
Error	14.0
Cost	13969

\[\begin{array}{l} t_0 := \sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \mathbf{if}\;p \leq -4.8 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -9.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 8.5 \cdot 10^{-269} \lor \neg \left(p \leq 8.8 \cdot 10^{-145}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{p}{x}\\ \end{array} \]

Alternative 2
Error	20.7
Cost	6992

\[\begin{array}{l} \mathbf{if}\;p \leq -4.2 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -7 \cdot 10^{-98}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{-47}:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3
Error	20.4
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -1.4 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 7.5 \cdot 10^{-48}:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	47.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;-\frac{p}{x}\\ \end{array} \]

Alternative 5
Error	53.4
Cost	192

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

Reproduce

herbie shell --seed 2023054 
(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))))))))