Given's Rotation SVD example

Percentage Accurate: 78.8% → 99.8%

Time: 10.6s

Alternatives: 6

Speedup: 1.0×

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: 78.8% 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, 0.7× speedup

Math

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

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))
   (sqrt (* 0.5 (+ -1.0 (+ 2.0 (/ x (hypot x (* p_m 2.0)))))))))

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 * (-1.0 + (2.0 + (x / hypot(x, (p_m * 2.0)))))));
	}
	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 * (-1.0 + (2.0 + (x / Math.hypot(x, (p_m * 2.0)))))));
	}
	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 * (-1.0 + (2.0 + (x / math.hypot(x, (p_m * 2.0)))))))
	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(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(-1.0 + Float64(2.0 + Float64(x / hypot(x, Float64(p_m * 2.0)))))));
	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 * (-1.0 + (2.0 + (x / hypot(x, (p_m * 2.0)))))));
	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[(-1.0 + N[(2.0 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -1

   1. Initial program 17.5%

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

      1. expm1-log1p-u 17.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}} \]

      2. expm1-undefine 17.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}} \]

      3. +-commutative 17.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)} - 1\right)} \]

      4. add-sqr-sqrt 17.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)} - 1\right)} \]

      5. hypot-define 17.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)} - 1\right)} \]

      6. associate-*l* 17.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)} - 1\right)} \]

      7. sqrt-prod 17.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)} - 1\right)} \]

      8. metadata-eval 17.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)} - 1\right)} \]

      9. sqrt-unprod 10.8%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)} - 1\right)} \]

      10. add-sqr-sqrt 17.5%

          \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)} - 1\right)} \]

   4. Applied egg-rr 17.5%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} - 1\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 17.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \left(-1\right)\right)}} \]

      2. metadata-eval 17.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \color{blue}{-1}\right)} \]

      3. +-commutative 17.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}} \]

      4. log1p-undefine 17.5%

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

      5. rem-exp-log 17.5%

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

      6. associate-+r+ 17.5%

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

      7. metadata-eval 17.5%

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

   6. Simplified 17.5%

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

      1. sqrt-prod 17.5%

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

      2. +-commutative 17.5%

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

      3. +-commutative 17.5%

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

      4. associate-+l+ 17.5%

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

      5. clear-num 17.5%

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

      6. associate-/r/ 14.3%

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

      7. metadata-eval 14.3%

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

      8. fma-undefine 2.4%

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

      9. *-un-lft-identity 2.4%

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

      10. sqrt-prod 2.4%

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

      11. *-commutative 2.4%

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

   8. Applied egg-rr 17.5%

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

      1. *-rgt-identity 17.5%

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

      2. *-commutative 17.5%

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

   10. Simplified 17.5%

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

   11. Taylor expanded in x around -inf 68.9%

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

       1. neg-mul-1 68.9%

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

       2. distribute-neg-frac2 68.9%

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

   13. Simplified 68.9%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x 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. expm1-log1p-u 99.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}} \]

      2. expm1-undefine 99.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}} \]

      3. +-commutative 99.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)} - 1\right)} \]

      4. add-sqr-sqrt 99.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)} - 1\right)} \]

      5. hypot-define 99.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)} - 1\right)} \]

      6. associate-*l* 99.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)} - 1\right)} \]

      7. sqrt-prod 99.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)} - 1\right)} \]

      8. metadata-eval 99.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)} - 1\right)} \]

      9. sqrt-unprod 52.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)} - 1\right)} \]

      10. add-sqr-sqrt 99.5%

          \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)} - 1\right)} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} - 1\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 99.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \left(-1\right)\right)}} \]

      2. metadata-eval 99.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \color{blue}{-1}\right)} \]

      3. +-commutative 99.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}} \]

      4. log1p-undefine 100.0%

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

      5. rem-exp-log 100.0%

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

      6. associate-+r+ 100.0%

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

      7. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 92.8%

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

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+122} \lor \neg \left(x \leq -3.1 \cdot 10^{+98}\right) \land x \leq -2.65 \cdot 10^{+69}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p\_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (or (<= x -1.3e+122) (and (not (<= x -3.1e+98)) (<= x -2.65e+69)))
   (/ p_m (- x))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p_m 2.0) x)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x <= -1.3e+122) || (!(x <= -3.1e+98) && (x <= -2.65e+69))) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((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 <= -1.3e+122) || (!(x <= -3.1e+98) && (x <= -2.65e+69))) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p_m * 2.0), x)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x <= -1.3e+122) or (not (x <= -3.1e+98) and (x <= -2.65e+69)):
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p_m * 2.0), x)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if ((x <= -1.3e+122) || (!(x <= -3.1e+98) && (x <= -2.65e+69)))
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(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 <= -1.3e+122) || (~((x <= -3.1e+98)) && (x <= -2.65e+69)))
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p_m * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[Or[LessEqual[x, -1.3e+122], And[N[Not[LessEqual[x, -3.1e+98]], $MachinePrecision], LessEqual[x, -2.65e+69]]], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p$95$m * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004e122 or -3.10000000000000019e98 < x < -2.65e69

   1. Initial program 41.4%

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

      1. expm1-log1p-u 41.4%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}} \]

      2. expm1-undefine 41.4%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}} \]

      3. +-commutative 41.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)} - 1\right)} \]

      4. add-sqr-sqrt 41.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)} - 1\right)} \]

      5. hypot-define 41.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)} - 1\right)} \]

      6. associate-*l* 41.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)} - 1\right)} \]

      7. sqrt-prod 41.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)} - 1\right)} \]

      8. metadata-eval 41.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)} - 1\right)} \]

      9. sqrt-unprod 18.2%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)} - 1\right)} \]

      10. add-sqr-sqrt 41.4%

          \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)} - 1\right)} \]

   4. Applied egg-rr 41.4%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} - 1\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 41.4%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \left(-1\right)\right)}} \]

      2. metadata-eval 41.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \color{blue}{-1}\right)} \]

      3. +-commutative 41.4%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}} \]

      4. log1p-undefine 41.4%

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

      5. rem-exp-log 41.4%

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

      6. associate-+r+ 41.4%

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

      7. metadata-eval 41.4%

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

   6. Simplified 41.4%

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

      1. sqrt-prod 41.4%

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

      2. +-commutative 41.4%

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

      3. +-commutative 41.4%

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

      4. associate-+l+ 41.4%

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

      5. clear-num 41.5%

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

      6. associate-/r/ 37.8%

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

      7. metadata-eval 37.8%

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

      8. fma-undefine 20.1%

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

      9. *-un-lft-identity 20.1%

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

      10. sqrt-prod 20.1%

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

      11. *-commutative 20.1%

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

   8. Applied egg-rr 41.4%

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

      1. *-rgt-identity 41.4%

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

      2. *-commutative 41.4%

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

   10. Simplified 41.4%

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

   11. Taylor expanded in x around -inf 64.8%

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

       1. neg-mul-1 64.8%

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

       2. distribute-neg-frac2 64.8%

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

   13. Simplified 64.8%

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

   if -1.30000000000000004e122 < x < -3.10000000000000019e98 or -2.65e69 < x

   1. Initial program 85.6%

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

      1. add-sqr-sqrt 85.6%

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

      2. hypot-define 85.6%

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

      3. associate-*l* 85.6%

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

      4. sqrt-prod 85.6%

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

      5. metadata-eval 85.6%

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

      6. sqrt-unprod 46.0%

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

      7. add-sqr-sqrt 85.6%

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

   4. Applied egg-rr 85.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+122} \lor \neg \left(x \leq -3.1 \cdot 10^{+98}\right) \land x \leq -2.65 \cdot 10^{+69}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 26000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= p_m 26000000.0) 1.0 (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 26000000.0) {
		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) :: tmp
    if (p_m <= 26000000.0d0) 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 tmp;
	if (p_m <= 26000000.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 26000000.0:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 26000000.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 26000000.0)
		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_] := If[LessEqual[p$95$m, 26000000.0], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 2.6e7

   1. Initial program 75.3%

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

      1. expm1-log1p-u 74.8%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}} \]

      2. expm1-undefine 74.8%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}} \]

      3. +-commutative 74.8%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)} - 1\right)} \]

      4. add-sqr-sqrt 74.8%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)} - 1\right)} \]

      5. hypot-define 74.8%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)} - 1\right)} \]

      6. associate-*l* 74.8%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)} - 1\right)} \]

      7. sqrt-prod 74.8%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)} - 1\right)} \]

      8. metadata-eval 74.8%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)} - 1\right)} \]

      9. sqrt-unprod 22.6%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)} - 1\right)} \]

      10. add-sqr-sqrt 74.8%

          \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)} - 1\right)} \]

   4. Applied egg-rr 74.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} - 1\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 74.8%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \left(-1\right)\right)}} \]

      2. metadata-eval 74.8%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \color{blue}{-1}\right)} \]

      3. +-commutative 74.8%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}} \]

      4. log1p-undefine 75.3%

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

      5. rem-exp-log 75.3%

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

      6. associate-+r+ 75.3%

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

      7. metadata-eval 75.3%

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

   6. Simplified 75.3%

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

      1. sqrt-prod 74.7%

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

      2. +-commutative 74.7%

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

      3. +-commutative 74.7%

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

      4. associate-+l+ 74.7%

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

      5. clear-num 74.7%

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

      6. associate-/r/ 73.7%

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

      7. metadata-eval 73.7%

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

      8. fma-undefine 69.9%

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

      9. *-un-lft-identity 69.9%

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

      10. sqrt-prod 70.4%

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

      11. *-commutative 70.4%

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

   8. Applied egg-rr 75.3%

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

      1. *-rgt-identity 75.3%

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

      2. *-commutative 75.3%

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

   10. Simplified 75.3%

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

   11. Taylor expanded in x around inf 40.1%

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

   if 2.6e7 < p

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0 91.2%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 26000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.4e-48) (/ p_m (- x)) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.4e-48) {
		tmp = p_m / -x;
	} 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) :: tmp
    if (p_m <= 2.4d-48) then
        tmp = p_m / -x
    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 tmp;
	if (p_m <= 2.4e-48) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 2.4e-48:
		tmp = p_m / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 2.4e-48)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 2.4e-48)
		tmp = p_m / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 2.4e-48], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 2.4e-48

   1. Initial program 74.4%

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

      1. expm1-log1p-u 73.9%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}} \]

      2. expm1-undefine 73.9%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}} \]

      3. +-commutative 73.9%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)} - 1\right)} \]

      4. add-sqr-sqrt 73.9%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)} - 1\right)} \]

      5. hypot-define 73.9%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)} - 1\right)} \]

      6. associate-*l* 73.9%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)} - 1\right)} \]

      7. sqrt-prod 73.9%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)} - 1\right)} \]

      8. metadata-eval 73.9%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)} - 1\right)} \]

      9. sqrt-unprod 18.8%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)} - 1\right)} \]

      10. add-sqr-sqrt 73.9%

          \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)} - 1\right)} \]

   4. Applied egg-rr 73.9%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} - 1\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 73.9%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \left(-1\right)\right)}} \]

      2. metadata-eval 73.9%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \color{blue}{-1}\right)} \]

      3. +-commutative 73.9%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}} \]

      4. log1p-undefine 74.4%

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

      5. rem-exp-log 74.4%

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

      6. associate-+r+ 74.4%

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

      7. metadata-eval 74.4%

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

   6. Simplified 74.4%

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

      1. sqrt-prod 73.9%

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

      2. +-commutative 73.9%

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

      3. +-commutative 73.9%

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

      4. associate-+l+ 73.9%

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

      5. clear-num 73.9%

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

      6. associate-/r/ 72.8%

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

      7. metadata-eval 72.8%

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

      8. fma-undefine 68.8%

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

      9. *-un-lft-identity 68.8%

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

      10. sqrt-prod 69.3%

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

      11. *-commutative 69.3%

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

   8. Applied egg-rr 74.4%

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

      1. *-rgt-identity 74.4%

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

      2. *-commutative 74.4%

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

   10. Simplified 74.4%

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

   11. Taylor expanded in x around -inf 23.4%

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

       1. neg-mul-1 23.4%

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

       2. distribute-neg-frac2 23.4%

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

   13. Simplified 23.4%

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

   if 2.4e-48 < p

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0 84.9%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 29.5% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p\_m}{x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -1e-311) (/ p_m (- x)) (/ p_m x)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1e-311) {
		tmp = p_m / -x;
	} else {
		tmp = p_m / x;
	}
	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-311)) then
        tmp = p_m / -x
    else
        tmp = p_m / x
    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-311) {
		tmp = p_m / -x;
	} else {
		tmp = p_m / x;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1e-311:
		tmp = p_m / -x
	else:
		tmp = p_m / x
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1e-311)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = Float64(p_m / x);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1e-311)
		tmp = p_m / -x;
	else
		tmp = p_m / x;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1e-311], N[(p$95$m / (-x)), $MachinePrecision], N[(p$95$m / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999948e-312

   1. Initial program 60.1%

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

      1. expm1-log1p-u 60.1%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}} \]

      2. expm1-undefine 60.1%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}} \]

      3. +-commutative 60.1%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)} - 1\right)} \]

      4. add-sqr-sqrt 60.1%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)} - 1\right)} \]

      5. hypot-define 60.1%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)} - 1\right)} \]

      6. associate-*l* 60.1%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)} - 1\right)} \]

      7. sqrt-prod 60.1%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)} - 1\right)} \]

      8. metadata-eval 60.1%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)} - 1\right)} \]

      9. sqrt-unprod 32.2%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)} - 1\right)} \]

      10. add-sqr-sqrt 60.1%

          \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)} - 1\right)} \]

   4. Applied egg-rr 60.1%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} - 1\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 60.1%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \left(-1\right)\right)}} \]

      2. metadata-eval 60.1%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \color{blue}{-1}\right)} \]

      3. +-commutative 60.1%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}} \]

      4. log1p-undefine 60.1%

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

      5. rem-exp-log 60.1%

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

      6. associate-+r+ 60.1%

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

      7. metadata-eval 60.1%

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

   6. Simplified 60.1%

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

      1. sqrt-prod 60.0%

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

      2. +-commutative 60.0%

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

      3. +-commutative 60.0%

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

      4. associate-+l+ 60.0%

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

      5. clear-num 60.1%

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

      6. associate-/r/ 58.5%

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

      7. metadata-eval 58.5%

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

      8. fma-undefine 52.7%

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

      9. *-un-lft-identity 52.7%

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

      10. sqrt-prod 52.7%

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

      11. *-commutative 52.7%

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

   8. Applied egg-rr 60.1%

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

      1. *-rgt-identity 60.1%

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

      2. *-commutative 60.1%

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

   10. Simplified 60.1%

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

   11. Taylor expanded in x around -inf 35.4%

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

       1. neg-mul-1 35.4%

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

       2. distribute-neg-frac2 35.4%

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

   13. Simplified 35.4%

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

   if -9.99999999999948e-312 < 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. expm1-log1p-u 99.3%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}} \]

      2. expm1-undefine 99.3%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}} \]

      3. +-commutative 99.3%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)} - 1\right)} \]

      4. add-sqr-sqrt 99.3%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)} - 1\right)} \]

      5. hypot-define 99.3%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)} - 1\right)} \]

      6. associate-*l* 99.3%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)} - 1\right)} \]

      7. sqrt-prod 99.3%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)} - 1\right)} \]

      8. metadata-eval 99.3%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)} - 1\right)} \]

      9. sqrt-unprod 52.6%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)} - 1\right)} \]

      10. add-sqr-sqrt 99.3%

          \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)} - 1\right)} \]

   4. Applied egg-rr 99.3%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} - 1\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 99.3%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \left(-1\right)\right)}} \]

      2. metadata-eval 99.3%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \color{blue}{-1}\right)} \]

      3. +-commutative 99.3%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}} \]

      4. log1p-undefine 100.0%

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

      5. rem-exp-log 100.0%

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

      6. associate-+r+ 100.0%

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

      7. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around -inf 4.7%

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

      1. associate-*r/ 4.7%

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

      2. *-commutative 4.7%

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

      3. associate-/l* 4.7%

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

   9. Simplified 4.7%

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

   10. Taylor expanded in p around 0 3.8%

       \[\leadsto \color{blue}{\frac{p}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 6.8% accurate, 71.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \frac{p\_m}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

p_m = abs(p)
function code(p_m, x)
	return Float64(p_m / x)
end

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := N[(p$95$m / x), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

   1. expm1-log1p-u 80.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}} \]

   2. expm1-undefine 80.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}} \]

   3. +-commutative 80.6%

      \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)} - 1\right)} \]

   4. add-sqr-sqrt 80.6%

      \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)} - 1\right)} \]

   5. hypot-define 80.6%

      \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)} - 1\right)} \]

   6. associate-*l* 80.6%

      \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)} - 1\right)} \]

   7. sqrt-prod 80.6%

      \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)} - 1\right)} \]

   8. metadata-eval 80.6%

      \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)} - 1\right)} \]

   9. sqrt-unprod 42.9%

      \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)} - 1\right)} \]

   10. add-sqr-sqrt 80.6%

       \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)} - 1\right)} \]

4. Applied egg-rr 80.6%

   \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} - 1\right)}} \]
5. Step-by-step derivation

   1. sub-neg 80.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \left(-1\right)\right)}} \]

   2. metadata-eval 80.6%

      \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \color{blue}{-1}\right)} \]

   3. +-commutative 80.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}} \]

   4. log1p-undefine 81.0%

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

   5. rem-exp-log 81.0%

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

   6. associate-+r+ 81.0%

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

   7. metadata-eval 81.0%

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

6. Simplified 81.0%

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

7. Taylor expanded in x around -inf 15.3%

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

   1. associate-*r/ 15.3%

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

   2. *-commutative 15.3%

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

   3. associate-/l* 15.3%

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

9. Simplified 15.3%

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

10. Taylor expanded in p around 0 13.6%

    \[\leadsto \color{blue}{\frac{p}{x}} \]
11. Add Preprocessing

Developer target: 78.8% 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 2024090 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (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))))))))