Given's Rotation SVD example

Average Error: 13.0 → 5.8

Time: 26.8s

Precision: binary64

Cost: 27204

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(x, 2 \cdot p\right)\\ \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \leq -0.9999995:\\ \;\;\;\;-\frac{\sqrt{p \cdot p}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{t_0 + x}{t_0}}\\ \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

↓

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (hypot x (* 2.0 p))))
   (if (<= (/ x (sqrt (+ (* (* 4.0 p) p) (* x x)))) -0.9999995)
     (- (/ (sqrt (* p p)) x))
     (sqrt (* 0.5 (/ (+ t_0 x) t_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 t_0 = hypot(x, (2.0 * p));
	double tmp;
	if ((x / sqrt((((4.0 * p) * p) + (x * x)))) <= -0.9999995) {
		tmp = -(sqrt((p * p)) / x);
	} else {
		tmp = sqrt((0.5 * ((t_0 + x) / t_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 t_0 = Math.hypot(x, (2.0 * p));
	double tmp;
	if ((x / Math.sqrt((((4.0 * p) * p) + (x * x)))) <= -0.9999995) {
		tmp = -(Math.sqrt((p * p)) / x);
	} else {
		tmp = Math.sqrt((0.5 * ((t_0 + x) / t_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):
	t_0 = math.hypot(x, (2.0 * p))
	tmp = 0
	if (x / math.sqrt((((4.0 * p) * p) + (x * x)))) <= -0.9999995:
		tmp = -(math.sqrt((p * p)) / x)
	else:
		tmp = math.sqrt((0.5 * ((t_0 + x) / t_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)
	t_0 = hypot(x, Float64(2.0 * p))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))) <= -0.9999995)
		tmp = Float64(-Float64(sqrt(Float64(p * p)) / x));
	else
		tmp = sqrt(Float64(0.5 * Float64(Float64(t_0 + x) / t_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)
	t_0 = hypot(x, (2.0 * p));
	tmp = 0.0;
	if ((x / sqrt((((4.0 * p) * p) + (x * x)))) <= -0.9999995)
		tmp = -(sqrt((p * p)) / x);
	else
		tmp = sqrt((0.5 * ((t_0 + x) / t_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_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(2.0 * p), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9999995], (-N[(N[Sqrt[N[(p * p), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), N[Sqrt[N[(0.5 * N[(N[(t$95$0 + x), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	13.0
Target	13.0
Herbie	5.8

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.999999500000000041

   1. Initial program 53.5

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

   2. Taylor expanded in x around -inf 23.8

      \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt{{p}^{2}}}{x}} \]

   3. Simplified 23.8

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

   if -0.999999500000000041 < (/.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 0.1

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

Alternatives

Alternative 1
Error	5.8
Cost	20740

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \leq -0.9999995:\\ \;\;\;\;-\frac{\sqrt{p \cdot p}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x\right)}\\ \end{array} \]

Alternative 2
Error	12.5
Cost	13572

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+31}:\\ \;\;\;\;-\frac{\sqrt{p \cdot p}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 1\right)}\\ \end{array} \]

Alternative 3
Error	22.9
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+30}:\\ \;\;\;\;-\frac{\sqrt{p \cdot p}}{x}\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot 2}\\ \end{array} \]

Alternative 4
Error	19.8
Cost	6856

\[\begin{array}{l} \mathbf{if}\;p \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 8.5 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5 \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Error	20.5
Cost	6728

\[\begin{array}{l} \mathbf{if}\;p \leq -8.5 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-91}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	46.8
Cost	704

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

Reproduce

herbie shell --seed 2023010 
(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))))))))