Given's Rotation SVD example

Percentage Accurate: 79.3% → 99.8%

Time: 11.9s

Alternatives: 5

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: 79.3% 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

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{p\_m \cdot \mathsf{fma}\left({\left(\frac{p\_m}{x}\right)}^{2}, -1.5, 1\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p\_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.5)
   (/ (* p_m (fma (pow (/ p_m x) 2.0) -1.5 1.0)) (- 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 / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.5) {
		tmp = (p_m * fma(pow((p_m / x), 2.0), -1.5, 1.0)) / -x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((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)))) <= -0.5)
		tmp = Float64(Float64(p_m * fma((Float64(p_m / x) ^ 2.0), -1.5, 1.0)) / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p_m * 2.0), x)))));
	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], -0.5], N[(N[(p$95$m * N[(N[Power[N[(p$95$m / x), $MachinePrecision], 2.0], $MachinePrecision] * -1.5 + 1.0), $MachinePrecision]), $MachinePrecision] / (-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 (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.5

   1. Initial program 14.1%

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

      1. add-log-exp 14.1%

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

      2. +-commutative 14.1%

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

      3. distribute-rgt-in 14.1%

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

      4. metadata-eval 14.1%

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

      5. fma-define 14.1%

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

   4. Applied egg-rr 14.1%

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

   5. Taylor expanded in x around -inf 38.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 38.2%

         \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]

      2. distribute-rgt-out 38.2%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-16 + 4\right)}}{p \cdot {x}^{2}}}{x} \]

      3. metadata-eval 38.2%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot \color{blue}{-12}}{p \cdot {x}^{2}}}{x} \]

   7. Simplified 38.2%

      \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}{x}} \]

   8. Taylor expanded in x around inf 50.6%

      \[\leadsto -\color{blue}{\frac{p + -1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{x}} \]

   9. Taylor expanded in p around 0 52.0%

      \[\leadsto -\frac{\color{blue}{p \cdot \left(1 + -1.5 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}}{x} \]
   10. Step-by-step derivation

       1. +-commutative 52.0%

          \[\leadsto -\frac{p \cdot \color{blue}{\left(-1.5 \cdot \frac{{p}^{2}}{{x}^{2}} + 1\right)}}{x} \]

       2. *-commutative 52.0%

          \[\leadsto -\frac{p \cdot \left(\color{blue}{\frac{{p}^{2}}{{x}^{2}} \cdot -1.5} + 1\right)}{x} \]

       3. fma-define 52.0%

          \[\leadsto -\frac{p \cdot \color{blue}{\mathsf{fma}\left(\frac{{p}^{2}}{{x}^{2}}, -1.5, 1\right)}}{x} \]

       4. unpow2 52.0%

          \[\leadsto -\frac{p \cdot \mathsf{fma}\left(\frac{\color{blue}{p \cdot p}}{{x}^{2}}, -1.5, 1\right)}{x} \]

       5. unpow2 52.0%

          \[\leadsto -\frac{p \cdot \mathsf{fma}\left(\frac{p \cdot p}{\color{blue}{x \cdot x}}, -1.5, 1\right)}{x} \]

       6. times-frac 52.0%

          \[\leadsto -\frac{p \cdot \mathsf{fma}\left(\color{blue}{\frac{p}{x} \cdot \frac{p}{x}}, -1.5, 1\right)}{x} \]

       7. unpow2 52.0%

          \[\leadsto -\frac{p \cdot \mathsf{fma}\left(\color{blue}{{\left(\frac{p}{x}\right)}^{2}}, -1.5, 1\right)}{x} \]

   11. Simplified 52.0%

       \[\leadsto -\frac{\color{blue}{p \cdot \mathsf{fma}\left({\left(\frac{p}{x}\right)}^{2}, -1.5, 1\right)}}{x} \]

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 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. add-sqr-sqrt 100.0%

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

      2. hypot-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 47.6%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{p \cdot \mathsf{fma}\left({\left(\frac{p}{x}\right)}^{2}, -1.5, 1\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p\_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -4.4e-8)
   (/ 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 <= -4.4e-8) {
		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 <= -4.4e-8) {
		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 <= -4.4e-8:
		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 <= -4.4e-8)
		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 <= -4.4e-8)
		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[LessEqual[x, -4.4e-8], 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 < -4.3999999999999997e-8

   1. Initial program 41.8%

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

      1. add-log-exp 41.8%

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

      2. +-commutative 41.8%

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

      3. distribute-rgt-in 41.8%

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

      4. metadata-eval 41.8%

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

      5. fma-define 41.8%

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

   4. Applied egg-rr 41.8%

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

   5. Taylor expanded in x around -inf 46.2%

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

      1. mul-1-neg 46.2%

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

      2. distribute-neg-frac2 46.2%

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

   7. Simplified 46.2%

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

   if -4.3999999999999997e-8 < x

   1. Initial program 87.5%

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

      1. add-sqr-sqrt 87.5%

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

      2. hypot-define 87.5%

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

      3. associate-*l* 87.5%

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

      4. sqrt-prod 87.5%

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

      5. metadata-eval 87.5%

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

      6. sqrt-unprod 41.7%

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

      7. add-sqr-sqrt 87.5%

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

   4. Applied egg-rr 87.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq 5200000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -2.9e-9)
   (/ p_m (- x))
   (if (<= x 1.8e-43)
     (sqrt 0.5)
     (if (<= x 5200000000000.0) 1.0 (if (<= x 6.6e+41) (sqrt 0.5) 1.0)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2.9e-9) {
		tmp = p_m / -x;
	} else if (x <= 1.8e-43) {
		tmp = sqrt(0.5);
	} else if (x <= 5200000000000.0) {
		tmp = 1.0;
	} else if (x <= 6.6e+41) {
		tmp = sqrt(0.5);
	} 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 <= (-2.9d-9)) then
        tmp = p_m / -x
    else if (x <= 1.8d-43) then
        tmp = sqrt(0.5d0)
    else if (x <= 5200000000000.0d0) then
        tmp = 1.0d0
    else if (x <= 6.6d+41) then
        tmp = sqrt(0.5d0)
    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 <= -2.9e-9) {
		tmp = p_m / -x;
	} else if (x <= 1.8e-43) {
		tmp = Math.sqrt(0.5);
	} else if (x <= 5200000000000.0) {
		tmp = 1.0;
	} else if (x <= 6.6e+41) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -2.9e-9:
		tmp = p_m / -x
	elif x <= 1.8e-43:
		tmp = math.sqrt(0.5)
	elif x <= 5200000000000.0:
		tmp = 1.0
	elif x <= 6.6e+41:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -2.9e-9)
		tmp = Float64(p_m / Float64(-x));
	elseif (x <= 1.8e-43)
		tmp = sqrt(0.5);
	elseif (x <= 5200000000000.0)
		tmp = 1.0;
	elseif (x <= 6.6e+41)
		tmp = sqrt(0.5);
	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 <= -2.9e-9)
		tmp = p_m / -x;
	elseif (x <= 1.8e-43)
		tmp = sqrt(0.5);
	elseif (x <= 5200000000000.0)
		tmp = 1.0;
	elseif (x <= 6.6e+41)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -2.9e-9], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[x, 1.8e-43], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[x, 5200000000000.0], 1.0, If[LessEqual[x, 6.6e+41], N[Sqrt[0.5], $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.89999999999999991e-9

   1. Initial program 41.8%

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

      1. add-log-exp 41.8%

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

      2. +-commutative 41.8%

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

      3. distribute-rgt-in 41.8%

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

      4. metadata-eval 41.8%

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

      5. fma-define 41.8%

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

   4. Applied egg-rr 41.8%

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

   5. Taylor expanded in x around -inf 46.2%

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

      1. mul-1-neg 46.2%

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

      2. distribute-neg-frac2 46.2%

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

   7. Simplified 46.2%

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

   if -2.89999999999999991e-9 < x < 1.7999999999999999e-43 or 5.2e12 < x < 6.6000000000000001e41

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0 72.1%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]

   if 1.7999999999999999e-43 < x < 5.2e12 or 6.6000000000000001e41 < 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. add-log-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 78.0%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq 5200000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.5% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-228}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -3e-228) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3e-228) {
		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 <= (-3d-228)) 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 <= -3e-228) {
		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 <= -3e-228:
		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 <= -3e-228)
		tmp = Float64(p_m / Float64(-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 <= -3e-228)
		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, -3e-228], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3e-228

   1. Initial program 54.6%

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

      1. add-log-exp 54.6%

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

      2. +-commutative 54.6%

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

      3. distribute-rgt-in 54.6%

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

      4. metadata-eval 54.6%

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

      5. fma-define 54.6%

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

   4. Applied egg-rr 54.6%

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

   5. Taylor expanded in x around -inf 29.0%

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

      1. mul-1-neg 29.0%

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

      2. distribute-neg-frac2 29.0%

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

   7. Simplified 29.0%

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

   if -3e-228 < 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. add-log-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 58.9%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-228}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.8% 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 77.5%

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

   1. add-log-exp 77.5%

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

   2. +-commutative 77.5%

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

   3. distribute-rgt-in 77.5%

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

   4. metadata-eval 77.5%

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

   5. fma-define 77.5%

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

4. Applied egg-rr 77.5%

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

5. Taylor expanded in x around inf 35.8%

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

6. Final simplification 35.8%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 79.3% 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 2024096 
(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))))))))