Given's Rotation SVD example

Average Error: 13.3 → 6.0

Time: 7.8s

Precision: binary64

Cost: 20612

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

Math

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

↓

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

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

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 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	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 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	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 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	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) / x) * Float64(-sqrt(Float64(0.5 * Float64(p * p)))));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	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 * (1.0 + (x / hypot((p * 2.0), x)))));
	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[(N[Sqrt[2.0], $MachinePrecision] / x), $MachinePrecision] * (-N[Sqrt[N[(0.5 * N[(p * p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	13.3
Target	13.3
Herbie	6.0

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

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 53.2

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

   2. Applied egg-rr 53.1

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

   3. Taylor expanded in x around -inf 27.8

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

   4. Simplified 27.8

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{x} \cdot \left(-p \cdot \sqrt{0.5}\right)} \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 2) x) (neg.f64 (*.f64 p (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) x) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 1/2) p)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (sqrt.f64 2) x) (*.f64 (sqrt.f64 1/2) p)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) p)) x))): 33 points increase in error, 44 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) p)) x))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 24.0

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

   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 0.0

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

4. Final simplification 6.0

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

Alternatives

Alternative 1
Error	20.2
Cost	13644

\[\begin{array}{l} \mathbf{if}\;p \leq -0.000467982557157476:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + -0.5 \cdot \frac{x}{p}\right)}\\ \mathbf{elif}\;p \leq -2.8312997360764375 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -2.4547422428817133 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(p \cdot \sqrt{0.5}\right)}{x}\\ \mathbf{elif}\;p \leq 3.927570449594663 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \left(1 + \frac{\frac{p}{x}}{\frac{x}{p}} \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2 + 0.25 \cdot \frac{x \cdot x}{p}}\right)}\\ \end{array} \]

Alternative 2
Error	20.3
Cost	13452

\[\begin{array}{l} \mathbf{if}\;p \leq -0.000467982557157476:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + -0.5 \cdot \frac{x}{p}\right)}\\ \mathbf{elif}\;p \leq -2.8312997360764375 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -2.4547422428817133 \cdot 10^{-201}:\\ \;\;\;\;{\left(\sqrt{\frac{p}{x}}\right)}^{2}\\ \mathbf{elif}\;p \leq 3.927570449594663 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \left(1 + \frac{\frac{p}{x}}{\frac{x}{p}} \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2 + 0.25 \cdot \frac{x \cdot x}{p}}\right)}\\ \end{array} \]

Alternative 3
Error	21.0
Cost	8016

\[\begin{array}{l} \mathbf{if}\;p \leq -0.000467982557157476:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + -0.5 \cdot \frac{x}{p}\right)}\\ \mathbf{elif}\;p \leq -2.8312997360764375 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -2.4547422428817133 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{elif}\;p \leq 3.927570449594663 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \left(1 + \frac{\frac{p}{x}}{\frac{x}{p}} \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2 + 0.25 \cdot \frac{x \cdot x}{p}}\right)}\\ \end{array} \]

Alternative 4
Error	21.2
Cost	7888

\[\begin{array}{l} \mathbf{if}\;p \leq -0.000467982557157476:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + -0.5 \cdot \frac{x}{p}\right)}\\ \mathbf{elif}\;p \leq -2.8312997360764375 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -2.4547422428817133 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{elif}\;p \leq 3.927570449594663 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \left(1 + \frac{\frac{p}{x}}{\frac{x}{p}} \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Error	21.1
Cost	7500

\[\begin{array}{l} \mathbf{if}\;p \leq -0.000467982557157476:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + -0.5 \cdot \frac{x}{p}\right)}\\ \mathbf{elif}\;p \leq -2.8312997360764375 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -2.4547422428817133 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{elif}\;p \leq 3.927570449594663 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	20.5
Cost	7108

\[\begin{array}{l} \mathbf{if}\;p \leq -0.000467982557157476:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + -0.5 \cdot \frac{x}{p}\right)}\\ \mathbf{elif}\;p \leq 3.927570449594663 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7
Error	20.4
Cost	6728

\[\begin{array}{l} \mathbf{if}\;p \leq -3.4403185646106954 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 3.927570449594663 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8
Error	28.7
Cost	6464

\[\sqrt{0.5} \]

Reproduce

herbie shell --seed 2022294 
(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))))))))