Given's Rotation SVD example

?

Percentage Accurate: 79.5% → 89.3%
Time: 19.0s
Precision: binary64
Cost: 27140

?

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99995:\\ \;\;\;\;{2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(\frac{p}{x} \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]
(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)))) -0.99995)
   (* (pow 2.0 0.25) (* (pow 2.0 0.25) (* (/ p x) (sqrt 0.5))))
   (pow
    (pow (+ 0.5 (/ 0.5 (/ (hypot x (* p 2.0)) x))) 1.5)
    0.3333333333333333)))
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)))) <= -0.99995) {
		tmp = pow(2.0, 0.25) * (pow(2.0, 0.25) * ((p / x) * sqrt(0.5)));
	} else {
		tmp = pow(pow((0.5 + (0.5 / (hypot(x, (p * 2.0)) / x))), 1.5), 0.3333333333333333);
	}
	return tmp;
}

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)))) <= -0.99995) {
		tmp = Math.pow(2.0, 0.25) * (Math.pow(2.0, 0.25) * ((p / x) * Math.sqrt(0.5)));
	} else {
		tmp = Math.pow(Math.pow((0.5 + (0.5 / (Math.hypot(x, (p * 2.0)) / x))), 1.5), 0.3333333333333333);
	}
	return tmp;
}

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)))) <= -0.99995:
		tmp = math.pow(2.0, 0.25) * (math.pow(2.0, 0.25) * ((p / x) * math.sqrt(0.5)))
	else:
		tmp = math.pow(math.pow((0.5 + (0.5 / (math.hypot(x, (p * 2.0)) / x))), 1.5), 0.3333333333333333)
	return tmp

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)))) <= -0.99995)
		tmp = Float64((2.0 ^ 0.25) * Float64((2.0 ^ 0.25) * Float64(Float64(p / x) * sqrt(0.5))));
	else
		tmp = (Float64(0.5 + Float64(0.5 / Float64(hypot(x, Float64(p * 2.0)) / x))) ^ 1.5) ^ 0.3333333333333333;
	end
	return tmp
end

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)))) <= -0.99995)
		tmp = (2.0 ^ 0.25) * ((2.0 ^ 0.25) * ((p / x) * sqrt(0.5)));
	else
		tmp = ((0.5 + (0.5 / (hypot(x, (p * 2.0)) / x))) ^ 1.5) ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

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], -0.99995], N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(N[(p / x), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(0.5 + N[(0.5 / N[(N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]

\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 -0.99995:\\
\;\;\;\;{2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(\frac{p}{x} \cdot \sqrt{0.5}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;{\left({\left(0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}}\right)}^{1.5}\right)}^{0.3333333333333333}\\


\end{array}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 9 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original79.5%
Target79.5%
Herbie89.3%
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \]

Derivation?

  1. Split input into 2 regimes

  2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.999950000000000006

    1. Initial program 23.6%

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

    2. Taylor expanded in x around -inf 51.8%

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

    3. Simplified51.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{2}}{\frac{x}{p \cdot \sqrt{0.5}}}} \]
      Step-by-step derivation

      [Start]51.8%

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

      mul-1-neg [=>]51.8%

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

      associate-/l* [=>]51.8%

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

      distribute-neg-frac [=>]51.8%

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

      *-commutative [=>]51.8%

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

    4. Applied egg-rr67.9%

      \[\leadsto \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(\frac{p}{x} \cdot \sqrt{0.5}\right)\right)} \]

    if -0.999950000000000006 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

    1. Initial program 99.9%

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

    2. Applied egg-rr99.9%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
      Step-by-step derivation

      [Start]99.9%

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

      add-cbrt-cube [=>]99.9%

      \[ \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      pow1/3 [=>]99.9%

      \[ \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}^{0.3333333333333333}} \]

    3. Applied egg-rr99.9%

      \[\leadsto {\left({\color{blue}{\left(\frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}} + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification91.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99995:\\ \;\;\;\;{2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(\frac{p}{x} \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy89.3%
Cost27140
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99995:\\ \;\;\;\;{2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(\frac{p}{x} \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]
Alternative 2
Accuracy89.3%
Cost27076
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99995:\\ \;\;\;\;{2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(\frac{p}{x} \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
Alternative 3
Accuracy89.2%
Cost20612
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99995:\\ \;\;\;\;\sqrt{0.5} \cdot \left(p \cdot \frac{\sqrt{2}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
Alternative 4
Accuracy68.0%
Cost13512
\[\begin{array}{l} \mathbf{if}\;p \leq -4.8 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-282}:\\ \;\;\;\;\left(\frac{p}{x} \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\\ \mathbf{elif}\;p \leq 8.5 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
Alternative 5
Accuracy67.9%
Cost13512
\[\begin{array}{l} \mathbf{if}\;p \leq -1.5 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.5 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(p \cdot \frac{\sqrt{2}}{x}\right)\\ \mathbf{elif}\;p \leq 8 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
Alternative 6
Accuracy69.2%
Cost7492
\[\begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq -1.45 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\frac{p \cdot p}{\frac{x}{-2}} - x}\right)}\\ \mathbf{elif}\;p \leq -4.5 \cdot 10^{-308}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8.2 \cdot 10^{-275}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 3.4 \cdot 10^{-209}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.5 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 6.8 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
Alternative 7
Accuracy69.4%
Cost7388
\[\begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq -7.8 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -7.5 \cdot 10^{-308}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8 \cdot 10^{-275}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.1 \cdot 10^{-220}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.2 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.85 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
Alternative 8
Accuracy44.7%
Cost388
\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-97}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 9
Accuracy36.7%
Cost64
\[1 \]

Reproduce?

herbie shell --seed 2023277 
(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))))))))