Given's Rotation SVD example

Average Error: 12.4 → 5.4

Time: 5.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(x, p \cdot 2\right)\\ \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.95:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{\mathsf{fma}\left(4, \frac{p}{x} \cdot \frac{p}{x}, \frac{-16}{{\left(\frac{x}{p}\right)}^{4}}\right)}{1 - \frac{x}{t_0}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(x, \sqrt{{t_0}^{-2}}, 1\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (hypot x (* p 2.0))))
   (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.95)
     (sqrt
      (*
       0.5
       (/
        (fma 4.0 (* (/ p x) (/ p x)) (/ -16.0 (pow (/ x p) 4.0)))
        (- 1.0 (/ x t_0)))))
     (sqrt (* 0.5 (fma x (sqrt (pow t_0 -2.0)) 1.0))))))

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, (p * 2.0));
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.95) {
		tmp = sqrt((0.5 * (fma(4.0, ((p / x) * (p / x)), (-16.0 / pow((x / p), 4.0))) / (1.0 - (x / t_0)))));
	} else {
		tmp = sqrt((0.5 * fma(x, sqrt(pow(t_0, -2.0)), 1.0)));
	}
	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(p * 2.0))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.95)
		tmp = sqrt(Float64(0.5 * Float64(fma(4.0, Float64(Float64(p / x) * Float64(p / x)), Float64(-16.0 / (Float64(x / p) ^ 4.0))) / Float64(1.0 - Float64(x / t_0)))));
	else
		tmp = sqrt(Float64(0.5 * fma(x, sqrt((t_0 ^ -2.0)), 1.0)));
	end
	return 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[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.95], N[Sqrt[N[(0.5 * N[(N[(4.0 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision] + N[(-16.0 / N[Power[N[(x / p), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(x * N[Sqrt[N[Power[t$95$0, -2.0], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Error

Bits error versus p

Bits error versus x

Target

Original	12.4
Target	12.4
Herbie	5.4

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.94999999999999996

   1. Initial program 52.3

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

   2. Applied egg-rr 52.3

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

   3. Taylor expanded in x around inf 34.5

      \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{4 \cdot \frac{{p}^{2}}{{x}^{2}} - 16 \cdot \frac{{p}^{4}}{{x}^{4}}}}{1 - \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   4. Simplified 22.7

      \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(4, \frac{p}{x} \cdot \frac{p}{x}, \frac{-16}{{\left(\frac{x}{p}\right)}^{4}}\right)}}{1 - \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   if -0.94999999999999996 < (/.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

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

   3. Applied egg-rr 0.0

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

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.95:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{\mathsf{fma}\left(4, \frac{p}{x} \cdot \frac{p}{x}, \frac{-16}{{\left(\frac{x}{p}\right)}^{4}}\right)}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(x, \sqrt{{\left(\mathsf{hypot}\left(x, p \cdot 2\right)\right)}^{-2}}, 1\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))))))