Average Error: 13.1 → 5.7
Time: 5.8s
Precision: binary64
\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, p + p\right)}, 0.5\right)\right)\right)}\\ \end{array} \]
(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)))) -1.0)
   (sqrt (* (/ p x) (/ p x)))
   (sqrt (expm1 (log1p (fma 0.5 (/ x (hypot x (+ p p))) 0.5))))))
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)))) <= -1.0) {
		tmp = sqrt(((p / x) * (p / x)));
	} else {
		tmp = sqrt(expm1(log1p(fma(0.5, (x / hypot(x, (p + p))), 0.5))));
	}
	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)))) <= -1.0)
		tmp = sqrt(Float64(Float64(p / x) * Float64(p / x)));
	else
		tmp = sqrt(expm1(log1p(fma(0.5, Float64(x / hypot(x, Float64(p + p))), 0.5))));
	end
	return 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], -1.0], N[Sqrt[N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(Exp[N[Log[1 + N[(0.5 * N[(x / N[Sqrt[x ^ 2 + N[(p + p), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $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 -1:\\
\;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, p + p\right)}, 0.5\right)\right)\right)}\\


\end{array}

Error

Bits error versus p

Bits error versus x

Target

Original13.1
Target13.1
Herbie5.7
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -1

    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. Simplified53.2

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}, 0.5\right)}} \]

    3. Applied egg-rr57.8

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

    4. Taylor expanded in x around -inf 30.6

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

    5. Simplified22.8

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

    if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

    1. Initial program 0.2

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

    2. Simplified0.2

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}, 0.5\right)}} \]

    3. Applied egg-rr0.3

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

    4. Applied egg-rr0.2

      \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, p + p\right)}, 0.5\right)\right)\right)}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, p + p\right)}, 0.5\right)\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(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))))))))