Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.8%

Time: 9.0s

Alternatives: 7

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

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)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

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

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
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]

Initial Program: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

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)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

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

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
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]

Alternative 1: 99.8% accurate, 0.4× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ (- p_m) x)
   (exp (* 0.5 (log (fma x (/ 0.5 (hypot x (* p_m 2.0))) 0.5))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = -p_m / x;
	} else {
		tmp = exp((0.5 * log(fma(x, (0.5 / hypot(x, (p_m * 2.0))), 0.5))));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = exp(Float64(0.5 * log(fma(x, Float64(0.5 / hypot(x, Float64(p_m * 2.0))), 0.5))));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p$95$m) / x), $MachinePrecision], N[Exp[N[(0.5 * N[Log[N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

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 19.1%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 19.1%

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

      2. metadata-eval 19.1%

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

      3. associate-*r/ 19.1%

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

      4. +-commutative 19.1%

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

      5. add-sqr-sqrt 19.1%

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

      6. hypot-define 19.1%

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

      7. associate-*l* 19.1%

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

      8. sqrt-prod 19.1%

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

      9. metadata-eval 19.1%

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

      10. sqrt-unprod 12.5%

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

      11. add-sqr-sqrt 19.1%

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

   4. Applied egg-rr 19.1%

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

      1. associate-/l* 19.1%

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

   6. Simplified 19.1%

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

   7. Taylor expanded in x around -inf 63.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   8. Step-by-step derivation

      1. associate-*r/ 63.6%

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

      2. neg-mul-1 63.6%

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

   9. Simplified 63.6%

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

   if -1 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. log-prod 99.8%

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

      4. metadata-eval 99.8%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right) \]

      5. add-log-exp 99.8%

         \[\leadsto 0 + \color{blue}{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      6. +-commutative 99.8%

         \[\leadsto 0 + \sqrt{0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      7. distribute-lft-in 99.8%

         \[\leadsto 0 + \sqrt{\color{blue}{0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 0.5 \cdot 1}} \]

      8. metadata-eval 99.8%

         \[\leadsto 0 + \sqrt{0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + \color{blue}{0.5}} \]

      9. fma-define 99.8%

         \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5\right)}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 + \sqrt{\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5\right)}} \]
   5. Step-by-step derivation

      1. +-lft-identity 99.8%

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

      2. fma-undefine 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. associate-*r/ 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. *-commutative 99.8%

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

      9. fma-undefine 99.8%

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

      10. associate-*l/ 99.8%

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

      11. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

      1. pow1/2 99.8%

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

      2. pow-to-exp 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 89.5%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ (- p_m) x)
   (sqrt (fma x (/ 0.5 (hypot x (* p_m 2.0))) 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(fma(x, (0.5 / hypot(x, (p_m * 2.0))), 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(x, Float64(0.5 / hypot(x, Float64(p_m * 2.0))), 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

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 19.1%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 19.1%

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

      2. metadata-eval 19.1%

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

      3. associate-*r/ 19.1%

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

      4. +-commutative 19.1%

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

      5. add-sqr-sqrt 19.1%

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

      6. hypot-define 19.1%

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

      7. associate-*l* 19.1%

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

      8. sqrt-prod 19.1%

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

      9. metadata-eval 19.1%

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

      10. sqrt-unprod 12.5%

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

      11. add-sqr-sqrt 19.1%

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

   4. Applied egg-rr 19.1%

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

      1. associate-/l* 19.1%

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

   6. Simplified 19.1%

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

   7. Taylor expanded in x around -inf 63.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   8. Step-by-step derivation

      1. associate-*r/ 63.6%

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

      2. neg-mul-1 63.6%

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

   9. Simplified 63.6%

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

   if -1 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. log-prod 99.8%

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

      4. metadata-eval 99.8%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right) \]

      5. add-log-exp 99.8%

         \[\leadsto 0 + \color{blue}{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      6. +-commutative 99.8%

         \[\leadsto 0 + \sqrt{0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      7. distribute-lft-in 99.8%

         \[\leadsto 0 + \sqrt{\color{blue}{0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 0.5 \cdot 1}} \]

      8. metadata-eval 99.8%

         \[\leadsto 0 + \sqrt{0.5 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + \color{blue}{0.5}} \]

      9. fma-define 99.8%

         \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5\right)}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 + \sqrt{\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5\right)}} \]
   5. Step-by-step derivation

      1. +-lft-identity 99.8%

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

      2. fma-undefine 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. associate-*r/ 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. *-commutative 99.8%

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

      9. fma-undefine 99.8%

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

      10. associate-*l/ 99.8%

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

      11. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 89.5%

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

Alternative 3: 99.8% accurate, 0.7× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ (- p_m) x)
   (sqrt (+ 0.5 (/ 0.5 (/ (hypot x (* p_m 2.0)) x))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt((0.5 + (0.5 / (hypot(x, (p_m * 2.0)) / x))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = -p_m / x;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / (Math.hypot(x, (p_m * 2.0)) / x))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0:
		tmp = -p_m / x
	else:
		tmp = math.sqrt((0.5 + (0.5 / (math.hypot(x, (p_m * 2.0)) / x))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(hypot(x, Float64(p_m * 2.0)) / x))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0)
		tmp = -p_m / x;
	else
		tmp = sqrt((0.5 + (0.5 / (hypot(x, (p_m * 2.0)) / x))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[(N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

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 19.1%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 19.1%

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

      2. metadata-eval 19.1%

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

      3. associate-*r/ 19.1%

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

      4. +-commutative 19.1%

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

      5. add-sqr-sqrt 19.1%

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

      6. hypot-define 19.1%

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

      7. associate-*l* 19.1%

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

      8. sqrt-prod 19.1%

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

      9. metadata-eval 19.1%

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

      10. sqrt-unprod 12.5%

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

      11. add-sqr-sqrt 19.1%

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

   4. Applied egg-rr 19.1%

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

      1. associate-/l* 19.1%

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

   6. Simplified 19.1%

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

   7. Taylor expanded in x around -inf 63.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   8. Step-by-step derivation

      1. associate-*r/ 63.6%

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

      2. neg-mul-1 63.6%

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

   9. Simplified 63.6%

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

   if -1 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*r/ 99.8%

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

      4. +-commutative 99.8%

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

      5. add-sqr-sqrt 99.8%

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

      6. hypot-define 99.8%

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

      7. associate-*l* 99.8%

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

      8. sqrt-prod 99.8%

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

      9. metadata-eval 99.8%

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

      10. sqrt-unprod 41.9%

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

      11. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 89.5%

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

Alternative 4: 68.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p\_m}{x}\\ \mathbf{if}\;p\_m \leq 9 \cdot 10^{-198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.5 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 2.6 \cdot 10^{-74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1250000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= p_m 9e-198)
     t_0
     (if (<= p_m 1.5e-100)
       1.0
       (if (<= p_m 2.6e-74) t_0 (if (<= p_m 1250000000.0) 1.0 (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 9e-198) {
		tmp = t_0;
	} else if (p_m <= 1.5e-100) {
		tmp = 1.0;
	} else if (p_m <= 2.6e-74) {
		tmp = t_0;
	} else if (p_m <= 1250000000.0) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p_m / x
    if (p_m <= 9d-198) then
        tmp = t_0
    else if (p_m <= 1.5d-100) then
        tmp = 1.0d0
    else if (p_m <= 2.6d-74) then
        tmp = t_0
    else if (p_m <= 1250000000.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 9e-198) {
		tmp = t_0;
	} else if (p_m <= 1.5e-100) {
		tmp = 1.0;
	} else if (p_m <= 2.6e-74) {
		tmp = t_0;
	} else if (p_m <= 1250000000.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	tmp = 0
	if p_m <= 9e-198:
		tmp = t_0
	elif p_m <= 1.5e-100:
		tmp = 1.0
	elif p_m <= 2.6e-74:
		tmp = t_0
	elif p_m <= 1250000000.0:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (p_m <= 9e-198)
		tmp = t_0;
	elseif (p_m <= 1.5e-100)
		tmp = 1.0;
	elseif (p_m <= 2.6e-74)
		tmp = t_0;
	elseif (p_m <= 1250000000.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	tmp = 0.0;
	if (p_m <= 9e-198)
		tmp = t_0;
	elseif (p_m <= 1.5e-100)
		tmp = 1.0;
	elseif (p_m <= 2.6e-74)
		tmp = t_0;
	elseif (p_m <= 1250000000.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, If[LessEqual[p$95$m, 9e-198], t$95$0, If[LessEqual[p$95$m, 1.5e-100], 1.0, If[LessEqual[p$95$m, 2.6e-74], t$95$0, If[LessEqual[p$95$m, 1250000000.0], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 8.9999999999999996e-198 or 1.5e-100 < p < 2.6000000000000001e-74

   1. Initial program 73.1%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 73.1%

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

      2. metadata-eval 73.1%

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

      3. associate-*r/ 73.1%

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

      4. +-commutative 73.1%

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

      5. add-sqr-sqrt 73.1%

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

      6. hypot-define 73.1%

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

      7. associate-*l* 73.1%

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

      8. sqrt-prod 73.1%

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

      9. metadata-eval 73.1%

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

      10. sqrt-unprod 10.1%

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

      11. add-sqr-sqrt 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. associate-/l* 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in x around -inf 21.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   8. Step-by-step derivation

      1. associate-*r/ 21.5%

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

      2. neg-mul-1 21.5%

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

   9. Simplified 21.5%

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

   if 8.9999999999999996e-198 < p < 1.5e-100 or 2.6000000000000001e-74 < p < 1.25e9

   1. Initial program 76.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 76.5%

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

      2. metadata-eval 76.5%

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

      3. associate-*r/ 76.5%

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

      4. +-commutative 76.5%

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

      5. add-sqr-sqrt 76.5%

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

      6. hypot-define 76.5%

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

      7. associate-*l* 76.5%

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

      8. sqrt-prod 76.5%

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

      9. metadata-eval 76.5%

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

      10. sqrt-unprod 76.5%

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

      11. add-sqr-sqrt 76.5%

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

   4. Applied egg-rr 76.5%

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

      1. associate-/l* 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in x around inf 57.9%

      \[\leadsto \color{blue}{1} \]

   if 1.25e9 < p

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0 85.6%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 9 \cdot 10^{-198}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.5 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1250000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.2% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -4.2e-135) (/ (- p_m) x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -4.2e-135) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d-135)) then
        tmp = -p_m / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -4.2e-135) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -4.2e-135:
		tmp = -p_m / x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -4.2e-135)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -4.2e-135)
		tmp = -p_m / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -4.2e-135], N[((-p$95$m) / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e-135

   1. Initial program 54.3%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 54.3%

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

      2. metadata-eval 54.3%

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

      3. associate-*r/ 54.3%

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

      4. +-commutative 54.3%

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

      5. add-sqr-sqrt 54.3%

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

      6. hypot-define 54.3%

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

      7. associate-*l* 54.3%

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

      8. sqrt-prod 54.3%

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

      9. metadata-eval 54.3%

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

      10. sqrt-unprod 25.9%

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

      11. add-sqr-sqrt 54.3%

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

   4. Applied egg-rr 54.3%

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

      1. associate-/l* 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in x around -inf 37.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   8. Step-by-step derivation

      1. associate-*r/ 37.5%

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

      2. neg-mul-1 37.5%

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

   9. Simplified 37.5%

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

   if -4.2e-135 < x

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. hypot-define 100.0%

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

      7. associate-*l* 100.0%

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

      8. sqrt-prod 100.0%

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

      9. metadata-eval 100.0%

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

      10. sqrt-unprod 41.3%

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

      11. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 57.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.2% accurate, 26.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -1e+36) (/ p_m x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1e+36) {
		tmp = p_m / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d+36)) then
        tmp = p_m / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -1e+36) {
		tmp = p_m / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1e+36:
		tmp = p_m / x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1e+36)
		tmp = Float64(p_m / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1e+36)
		tmp = p_m / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1e+36], N[(p$95$m / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000004e36

   1. Initial program 54.4%

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

   3. Taylor expanded in x around -inf 45.7%

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

   4. Taylor expanded in p around 0 47.2%

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

   if -1.00000000000000004e36 < x

   1. Initial program 83.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-in 83.0%

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

      2. metadata-eval 83.0%

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

      3. associate-*r/ 83.0%

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

      4. +-commutative 83.0%

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

      5. add-sqr-sqrt 83.0%

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

      6. hypot-define 83.0%

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

      7. associate-*l* 83.0%

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

      8. sqrt-prod 83.0%

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

      9. metadata-eval 83.0%

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

      10. sqrt-unprod 34.1%

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

      11. add-sqr-sqrt 83.0%

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

   4. Applied egg-rr 83.0%

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

      1. associate-/l* 83.0%

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

   6. Simplified 83.0%

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

   7. Taylor expanded in x around inf 40.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.9% accurate, 215.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ 1 \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 1.0)

C

p_m = fabs(p);
double code(double p_m, double x) {
	return 1.0;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return 1.0;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 1.0

Julia

p_m = abs(p)
function code(p_m, x)
	return 1.0
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = 1.0;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 1.0

Derivation

1. Initial program 76.8%

   \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-lft-in 76.8%

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

   2. metadata-eval 76.8%

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

   3. associate-*r/ 76.8%

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

   4. +-commutative 76.8%

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

   5. add-sqr-sqrt 76.8%

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

   6. hypot-define 76.8%

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

   7. associate-*l* 76.8%

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

   8. sqrt-prod 76.8%

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

   9. metadata-eval 76.8%

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

   10. sqrt-unprod 33.5%

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

   11. add-sqr-sqrt 76.8%

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

4. Applied egg-rr 76.8%

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

   1. associate-/l* 76.8%

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

6. Simplified 76.8%

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

7. Taylor expanded in x around inf 33.9%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 33.9%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer target: 79.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x))))))

C

double code(double p, double x) {
	return sqrt((0.5 + (copysign(0.5, x) / hypot(1.0, ((2.0 * p) / x)))));
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 + (Math.copySign(0.5, x) / Math.hypot(1.0, ((2.0 * p) / x)))));
}

Python

def code(p, x):
	return math.sqrt((0.5 + (math.copysign(0.5, x) / math.hypot(1.0, ((2.0 * p) / x)))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 + Float64(copysign(0.5, x) / hypot(1.0, Float64(Float64(2.0 * p) / x)))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 + ((sign(x) * abs(0.5)) / hypot(1.0, ((2.0 * p) / x)))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 + N[(N[With[{TMP1 = Abs[0.5], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * p), $MachinePrecision] / x), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2024036 
(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))))))))