Given's Rotation SVD example

Percentage Accurate: 79.5% → 99.9%

Time: 11.7s

Precision: binary64

Cost: 52996

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, p + p\right)}}\\ \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99999998:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left({t_0}^{2}, t_0, 1\right)}\\ \end{array} \]

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 (cbrt (/ x (hypot x (+ p p))))))
   (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.99999998)
     (fabs (/ p x))
     (sqrt (* 0.5 (fma (pow t_0 2.0) t_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 = cbrt((x / hypot(x, (p + p))));
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.99999998) {
		tmp = fabs((p / x));
	} else {
		tmp = sqrt((0.5 * fma(pow(t_0, 2.0), t_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 = cbrt(Float64(x / hypot(x, Float64(p + p))))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.99999998)
		tmp = abs(Float64(p / x));
	else
		tmp = sqrt(Float64(0.5 * fma((t_0 ^ 2.0), t_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[Power[N[(x / N[Sqrt[x ^ 2 + N[(p + p), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.99999998], N[Abs[N[(p / x), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	79.5%
Target	79.5%
Herbie	99.9%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.999999980000000011

   1. Initial program 11.5%

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

   2. Applied egg-rr 11.5%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
      Step-by-step derivation

      [Start] 11.5	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      add-exp-log [=>] 11.5	\[ \sqrt{0.5 \cdot \color{blue}{e^{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]
      log1p-udef [<=] 11.5	\[ \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]
      +-commutative [=>] 11.5	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]
      add-sqr-sqrt [=>] 11.5	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]
      hypot-def [=>] 11.5	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]
      associate-*l* [=>] 11.5	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]
      sqrt-prod [=>] 11.5	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]
      metadata-eval [=>] 11.5	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]
      sqrt-unprod [<=] 6.8	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]
      add-sqr-sqrt [<=] 11.5	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   3. Taylor expanded in x around -inf 46.6%

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

   4. Simplified 49.4%

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

      [Start] 46.6	\[ \sqrt{0.5} \cdot \sqrt{e^{2 \cdot \log \left(\frac{-1}{x}\right) + \log \left(--2 \cdot {p}^{2}\right)}} \]
      exp-sum [=>] 46.7	\[ \sqrt{0.5} \cdot \sqrt{\color{blue}{e^{2 \cdot \log \left(\frac{-1}{x}\right)} \cdot e^{\log \left(--2 \cdot {p}^{2}\right)}}} \]
      *-commutative [=>] 46.7	\[ \sqrt{0.5} \cdot \sqrt{e^{\color{blue}{\log \left(\frac{-1}{x}\right) \cdot 2}} \cdot e^{\log \left(--2 \cdot {p}^{2}\right)}} \]
      exp-to-pow [=>] 46.9	\[ \sqrt{0.5} \cdot \sqrt{\color{blue}{{\left(\frac{-1}{x}\right)}^{2}} \cdot e^{\log \left(--2 \cdot {p}^{2}\right)}} \]
      rem-exp-log [=>] 49.4	\[ \sqrt{0.5} \cdot \sqrt{{\left(\frac{-1}{x}\right)}^{2} \cdot \color{blue}{\left(--2 \cdot {p}^{2}\right)}} \]
      distribute-lft-neg-in [=>] 49.4	\[ \sqrt{0.5} \cdot \sqrt{{\left(\frac{-1}{x}\right)}^{2} \cdot \color{blue}{\left(\left(--2\right) \cdot {p}^{2}\right)}} \]
      metadata-eval [=>] 49.4	\[ \sqrt{0.5} \cdot \sqrt{{\left(\frac{-1}{x}\right)}^{2} \cdot \left(\color{blue}{2} \cdot {p}^{2}\right)} \]
      unpow2 [=>] 49.4	\[ \sqrt{0.5} \cdot \sqrt{{\left(\frac{-1}{x}\right)}^{2} \cdot \left(2 \cdot \color{blue}{\left(p \cdot p\right)}\right)} \]

   5. Applied egg-rr 46.2%

      \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot \left(\left(p \cdot p\right) \cdot {x}^{-2}\right)\right) \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3}} \]
      Step-by-step derivation

      [Start] 49.4	\[ \sqrt{0.5} \cdot \sqrt{{\left(\frac{-1}{x}\right)}^{2} \cdot \left(2 \cdot \left(p \cdot p\right)\right)} \]
      add-cube-cbrt [=>] 49.0	\[ \color{blue}{\left(\sqrt[3]{\sqrt{0.5} \cdot \sqrt{{\left(\frac{-1}{x}\right)}^{2} \cdot \left(2 \cdot \left(p \cdot p\right)\right)}} \cdot \sqrt[3]{\sqrt{0.5} \cdot \sqrt{{\left(\frac{-1}{x}\right)}^{2} \cdot \left(2 \cdot \left(p \cdot p\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{0.5} \cdot \sqrt{{\left(\frac{-1}{x}\right)}^{2} \cdot \left(2 \cdot \left(p \cdot p\right)\right)}}} \]
      pow3 [=>] 49.0	\[ \color{blue}{{\left(\sqrt[3]{\sqrt{0.5} \cdot \sqrt{{\left(\frac{-1}{x}\right)}^{2} \cdot \left(2 \cdot \left(p \cdot p\right)\right)}}\right)}^{3}} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]
      Step-by-step derivation

      [Start] 46.2	\[ {\left({\left(\left(2 \cdot \left(\left(p \cdot p\right) \cdot {x}^{-2}\right)\right) \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3} \]
      exp-to-pow [<=] 46.2	\[ \color{blue}{e^{\log \left({\left(\left(2 \cdot \left(\left(p \cdot p\right) \cdot {x}^{-2}\right)\right) \cdot 0.5\right)}^{0.16666666666666666}\right) \cdot 3}} \]
      *-commutative [=>] 46.2	\[ e^{\color{blue}{3 \cdot \log \left({\left(\left(2 \cdot \left(\left(p \cdot p\right) \cdot {x}^{-2}\right)\right) \cdot 0.5\right)}^{0.16666666666666666}\right)}} \]
      log-pow [=>] 46.4	\[ e^{3 \cdot \color{blue}{\left(0.16666666666666666 \cdot \log \left(\left(2 \cdot \left(\left(p \cdot p\right) \cdot {x}^{-2}\right)\right) \cdot 0.5\right)\right)}} \]
      associate-*r* [=>] 46.8	\[ e^{\color{blue}{\left(3 \cdot 0.16666666666666666\right) \cdot \log \left(\left(2 \cdot \left(\left(p \cdot p\right) \cdot {x}^{-2}\right)\right) \cdot 0.5\right)}} \]
      metadata-eval [=>] 46.8	\[ e^{\color{blue}{0.5} \cdot \log \left(\left(2 \cdot \left(\left(p \cdot p\right) \cdot {x}^{-2}\right)\right) \cdot 0.5\right)} \]
      *-commutative [<=] 46.8	\[ e^{\color{blue}{\log \left(\left(2 \cdot \left(\left(p \cdot p\right) \cdot {x}^{-2}\right)\right) \cdot 0.5\right) \cdot 0.5}} \]
      exp-to-pow [=>] 49.5	\[ \color{blue}{{\left(\left(2 \cdot \left(\left(p \cdot p\right) \cdot {x}^{-2}\right)\right) \cdot 0.5\right)}^{0.5}} \]
      unpow1/2 [=>] 49.5	\[ \color{blue}{\sqrt{\left(2 \cdot \left(\left(p \cdot p\right) \cdot {x}^{-2}\right)\right) \cdot 0.5}} \]
      *-commutative [=>] 49.5	\[ \sqrt{\color{blue}{\left(\left(\left(p \cdot p\right) \cdot {x}^{-2}\right) \cdot 2\right)} \cdot 0.5} \]
      associate-*l* [=>] 49.5	\[ \sqrt{\color{blue}{\left(\left(p \cdot p\right) \cdot {x}^{-2}\right) \cdot \left(2 \cdot 0.5\right)}} \]
      unpow2 [<=] 49.5	\[ \sqrt{\left(\color{blue}{{p}^{2}} \cdot {x}^{-2}\right) \cdot \left(2 \cdot 0.5\right)} \]
      metadata-eval [=>] 49.5	\[ \sqrt{\left({p}^{2} \cdot {x}^{-2}\right) \cdot \color{blue}{1}} \]
      associate-*r* [<=] 49.5	\[ \sqrt{\color{blue}{{p}^{2} \cdot \left({x}^{-2} \cdot 1\right)}} \]
      unpow2 [=>] 49.5	\[ \sqrt{\color{blue}{\left(p \cdot p\right)} \cdot \left({x}^{-2} \cdot 1\right)} \]
      *-rgt-identity [=>] 49.5	\[ \sqrt{\left(p \cdot p\right) \cdot \color{blue}{{x}^{-2}}} \]
      metadata-eval [<=] 49.5	\[ \sqrt{\left(p \cdot p\right) \cdot {x}^{\color{blue}{\left(2 \cdot -1\right)}}} \]
      pow-sqr [<=] 49.3	\[ \sqrt{\left(p \cdot p\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)}} \]

   if -0.999999980000000011 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.8%

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

   2. Applied egg-rr 99.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
      Step-by-step derivation

      [Start] 99.8	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      add-exp-log [=>] 99.8	\[ \sqrt{0.5 \cdot \color{blue}{e^{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]
      log1p-udef [<=] 99.8	\[ \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]
      +-commutative [=>] 99.8	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]
      add-sqr-sqrt [=>] 99.8	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]
      hypot-def [=>] 99.8	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]
      associate-*l* [=>] 99.8	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]
      sqrt-prod [=>] 99.8	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]
      metadata-eval [=>] 99.8	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]
      sqrt-unprod [<=] 48.5	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]
      add-sqr-sqrt [<=] 99.8	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, p + p\right)}}\right)}^{2}, \sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, p + p\right)}}, 1\right)}} \]
      Step-by-step derivation

      [Start] 99.8	\[ \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}} \]
      log1p-udef [=>] 99.8	\[ \sqrt{0.5 \cdot e^{\color{blue}{\log \left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
      add-exp-log [<=] 99.8	\[ \sqrt{0.5 \cdot \color{blue}{\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}} \]
      +-commutative [=>] 99.8	\[ \sqrt{0.5 \cdot \color{blue}{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 1\right)}} \]
      add-cube-cbrt [=>] 99.8	\[ \sqrt{0.5 \cdot \left(\color{blue}{\left(\sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} \cdot \sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right) \cdot \sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} + 1\right)} \]
      fma-def [=>] 99.8	\[ \sqrt{0.5 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} \cdot \sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}, \sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}, 1\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99999998:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, p + p\right)}}\right)}^{2}, \sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, p + p\right)}}, 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	33284

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99999998:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	20612

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99999998:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]

Alternative 3
Accuracy	69.2%
Cost	7256

\[\begin{array}{l} t_0 := \left|\frac{p}{x}\right|\\ \mathbf{if}\;p \leq -2.6 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1 \cdot 10^{-266}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7.5 \cdot 10^{-254}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 9.8 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Accuracy	69.0%
Cost	7256

\[\begin{array}{l} t_0 := \left|\frac{p}{x}\right|\\ \mathbf{if}\;p \leq -3.5 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{0.5 + -0.25 \cdot \frac{x}{p}}\\ \mathbf{elif}\;p \leq -1.9 \cdot 10^{-266}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.8 \cdot 10^{-248}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.8 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.9 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Accuracy	68.8%
Cost	6992

\[\begin{array}{l} \mathbf{if}\;p \leq -8.5 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.6 \cdot 10^{-144}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Accuracy	43.7%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-46}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	36.1%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023160 
(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))))))))