Hyperbolic arcsine

Percentage Accurate: 17.6% → 99.8%

Time: 8.6s

Alternatives: 12

Speedup: 207.0×

Specification

Math

\[\begin{array}{l} \\ \log \left(x + \sqrt{x \cdot x + 1}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))

C

double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) + 1.0d0))))
end function

Java

public static double code(double x) {
	return Math.log((x + Math.sqrt(((x * x) + 1.0))));
}

Python

def code(x):
	return math.log((x + math.sqrt(((x * x) + 1.0))))

Julia

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) + 1.0))));
end

Wolfram

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 17.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \sqrt{x \cdot x + 1}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))

C

double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) + 1.0d0))))
end function

Java

public static double code(double x) {
	return Math.log((x + Math.sqrt(((x * x) + 1.0))));
}

Python

def code(x):
	return math.log((x + math.sqrt(((x * x) + 1.0))))

Julia

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) + 1.0))));
end

Wolfram

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0255:\\ \;\;\;\;\mathsf{fma}\left(0.075, {x}^{5}, -0.044642857142857144 \cdot {x}^{7}\right) + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ x (hypot 1.0 x))))))
   (if (<= x -1.15)
     (+ (log (/ -0.5 x)) (/ -0.25 (* x x)))
     (if (<= x 0.0255)
       (+
        (fma 0.075 (pow x 5.0) (* -0.044642857142857144 (pow x 7.0)))
        (+ x (* -0.16666666666666666 (pow x 3.0))))
       (+ t_0 t_0)))))

C

double code(double x) {
	double t_0 = log(sqrt((x + hypot(1.0, x))));
	double tmp;
	if (x <= -1.15) {
		tmp = log((-0.5 / x)) + (-0.25 / (x * x));
	} else if (x <= 0.0255) {
		tmp = fma(0.075, pow(x, 5.0), (-0.044642857142857144 * pow(x, 7.0))) + (x + (-0.16666666666666666 * pow(x, 3.0)));
	} else {
		tmp = t_0 + t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = log(sqrt(Float64(x + hypot(1.0, x))))
	tmp = 0.0
	if (x <= -1.15)
		tmp = Float64(log(Float64(-0.5 / x)) + Float64(-0.25 / Float64(x * x)));
	elseif (x <= 0.0255)
		tmp = Float64(fma(0.075, (x ^ 5.0), Float64(-0.044642857142857144 * (x ^ 7.0))) + Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = Float64(t_0 + t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.15], N[(N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision] + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0255], N[(N[(0.075 * N[Power[x, 5.0], $MachinePrecision] + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 2.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 2.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 3.7%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 3.7%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Step-by-step derivation

      1. flip-+ 3.1%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      2. div-sub 2.2%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      3. hypot-udef 2.2%

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

      4. hypot-udef 2.2%

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

      5. add-sqr-sqrt 2.2%

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

      6. metadata-eval 2.2%

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

   5. Applied egg-rr 2.2%

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. div-sub 3.1%

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

      2. +-commutative 3.1%

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

      3. associate--r+ 52.2%

         \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      4. +-inverses 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

   8. Taylor expanded in x around -inf 99.7%

      \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
   9. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(\log 0.5 + \log \left(\frac{-1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      3. metadata-eval 99.7%

         \[\leadsto \left(\log 0.5 + \log \left(\frac{\color{blue}{-1}}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      4. distribute-neg-frac 99.7%

         \[\leadsto \left(\log 0.5 + \log \color{blue}{\left(-\frac{1}{x}\right)}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      5. log-prod 100.0%

         \[\leadsto \color{blue}{\log \left(0.5 \cdot \left(-\frac{1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      6. distribute-rgt-neg-in 100.0%

         \[\leadsto \log \color{blue}{\left(-0.5 \cdot \frac{1}{x}\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      7. associate-*r/ 100.0%

         \[\leadsto \log \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \log \left(-\frac{\color{blue}{0.5}}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      9. distribute-neg-frac 100.0%

         \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{\color{blue}{-0.5}}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      11. associate-*r/ 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right) \]

      13. distribute-neg-frac 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \color{blue}{\frac{-0.25}{{x}^{2}}} \]

      14. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \frac{\color{blue}{-0.25}}{{x}^{2}} \]

      15. unpow2 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{\color{blue}{x \cdot x}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}} \]

   if -1.1499999999999999 < x < 0.0254999999999999984

   1. Initial program 7.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 7.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 7.8%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 7.8%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 7.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \cdot \sqrt[3]{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}\right) \cdot \sqrt[3]{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}} \]

      2. pow3 7.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}\right)}^{3}} \]

   5. Applied egg-rr 7.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}\right)}^{3}} \]

   6. Taylor expanded in x around 0 97.9%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)}}\right)}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right) + -0.16666666666666666 \cdot {x}^{3}} \]

      3. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}\right) + x\right)} + -0.16666666666666666 \cdot {x}^{3} \]

      4. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}\right) + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)} \]

      5. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.075, {x}^{5}, -0.044642857142857144 \cdot {x}^{7}\right)} + \left(x + -0.16666666666666666 \cdot {x}^{3}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.075, {x}^{5}, -0.044642857142857144 \cdot {x}^{7}\right) + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)} \]

   if 0.0254999999999999984 < x

   1. Initial program 48.6%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 48.6%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 99.9%

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

      2. log-prod 100.0%

         \[\leadsto \color{blue}{\log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)} \]

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0255:\\ \;\;\;\;\mathsf{fma}\left(0.075, {x}^{5}, -0.044642857142857144 \cdot {x}^{7}\right) + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0255:\\ \;\;\;\;\mathsf{fma}\left(0.075, {x}^{5}, -0.044642857142857144 \cdot {x}^{7}\right) + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (+ (log (/ -0.5 x)) (/ -0.25 (* x x)))
   (if (<= x 0.0255)
     (+
      (fma 0.075 (pow x 5.0) (* -0.044642857142857144 (pow x 7.0)))
      (+ x (* -0.16666666666666666 (pow x 3.0))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = log((-0.5 / x)) + (-0.25 / (x * x));
	} else if (x <= 0.0255) {
		tmp = fma(0.075, pow(x, 5.0), (-0.044642857142857144 * pow(x, 7.0))) + (x + (-0.16666666666666666 * pow(x, 3.0)));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = Float64(log(Float64(-0.5 / x)) + Float64(-0.25 / Float64(x * x)));
	elseif (x <= 0.0255)
		tmp = Float64(fma(0.075, (x ^ 5.0), Float64(-0.044642857142857144 * (x ^ 7.0))) + Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[(N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision] + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0255], N[(N[(0.075 * N[Power[x, 5.0], $MachinePrecision] + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 2.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 2.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 3.7%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 3.7%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Step-by-step derivation

      1. flip-+ 3.1%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      2. div-sub 2.2%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      3. hypot-udef 2.2%

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

      4. hypot-udef 2.2%

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

      5. add-sqr-sqrt 2.2%

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

      6. metadata-eval 2.2%

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

   5. Applied egg-rr 2.2%

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. div-sub 3.1%

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

      2. +-commutative 3.1%

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

      3. associate--r+ 52.2%

         \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      4. +-inverses 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

   8. Taylor expanded in x around -inf 99.7%

      \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
   9. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(\log 0.5 + \log \left(\frac{-1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      3. metadata-eval 99.7%

         \[\leadsto \left(\log 0.5 + \log \left(\frac{\color{blue}{-1}}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      4. distribute-neg-frac 99.7%

         \[\leadsto \left(\log 0.5 + \log \color{blue}{\left(-\frac{1}{x}\right)}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      5. log-prod 100.0%

         \[\leadsto \color{blue}{\log \left(0.5 \cdot \left(-\frac{1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      6. distribute-rgt-neg-in 100.0%

         \[\leadsto \log \color{blue}{\left(-0.5 \cdot \frac{1}{x}\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      7. associate-*r/ 100.0%

         \[\leadsto \log \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \log \left(-\frac{\color{blue}{0.5}}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      9. distribute-neg-frac 100.0%

         \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{\color{blue}{-0.5}}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      11. associate-*r/ 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right) \]

      13. distribute-neg-frac 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \color{blue}{\frac{-0.25}{{x}^{2}}} \]

      14. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \frac{\color{blue}{-0.25}}{{x}^{2}} \]

      15. unpow2 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{\color{blue}{x \cdot x}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}} \]

   if -1.1499999999999999 < x < 0.0254999999999999984

   1. Initial program 7.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 7.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 7.8%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 7.8%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 7.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \cdot \sqrt[3]{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}\right) \cdot \sqrt[3]{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}} \]

      2. pow3 7.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}\right)}^{3}} \]

   5. Applied egg-rr 7.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}\right)}^{3}} \]

   6. Taylor expanded in x around 0 97.9%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)}}\right)}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right) + -0.16666666666666666 \cdot {x}^{3}} \]

      3. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}\right) + x\right)} + -0.16666666666666666 \cdot {x}^{3} \]

      4. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}\right) + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)} \]

      5. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.075, {x}^{5}, -0.044642857142857144 \cdot {x}^{7}\right)} + \left(x + -0.16666666666666666 \cdot {x}^{3}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.075, {x}^{5}, -0.044642857142857144 \cdot {x}^{7}\right) + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)} \]

   if 0.0254999999999999984 < x

   1. Initial program 48.6%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 48.6%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0255:\\ \;\;\;\;\mathsf{fma}\left(0.075, {x}^{5}, -0.044642857142857144 \cdot {x}^{7}\right) + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0255:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (+ (log (/ -0.5 x)) (/ -0.25 (* x x)))
   (if (<= x 0.0255)
     (+
      (* -0.16666666666666666 (pow x 3.0))
      (+ (* 0.075 (pow x 5.0)) (+ x (* -0.044642857142857144 (pow x 7.0)))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = log((-0.5 / x)) + (-0.25 / (x * x));
	} else if (x <= 0.0255) {
		tmp = (-0.16666666666666666 * pow(x, 3.0)) + ((0.075 * pow(x, 5.0)) + (x + (-0.044642857142857144 * pow(x, 7.0))));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = Math.log((-0.5 / x)) + (-0.25 / (x * x));
	} else if (x <= 0.0255) {
		tmp = (-0.16666666666666666 * Math.pow(x, 3.0)) + ((0.075 * Math.pow(x, 5.0)) + (x + (-0.044642857142857144 * Math.pow(x, 7.0))));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.15:
		tmp = math.log((-0.5 / x)) + (-0.25 / (x * x))
	elif x <= 0.0255:
		tmp = (-0.16666666666666666 * math.pow(x, 3.0)) + ((0.075 * math.pow(x, 5.0)) + (x + (-0.044642857142857144 * math.pow(x, 7.0))))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = Float64(log(Float64(-0.5 / x)) + Float64(-0.25 / Float64(x * x)));
	elseif (x <= 0.0255)
		tmp = Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(Float64(0.075 * (x ^ 5.0)) + Float64(x + Float64(-0.044642857142857144 * (x ^ 7.0)))));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.15)
		tmp = log((-0.5 / x)) + (-0.25 / (x * x));
	elseif (x <= 0.0255)
		tmp = (-0.16666666666666666 * (x ^ 3.0)) + ((0.075 * (x ^ 5.0)) + (x + (-0.044642857142857144 * (x ^ 7.0))));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[(N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision] + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0255], N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 2.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 2.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 3.7%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 3.7%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Step-by-step derivation

      1. flip-+ 3.1%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      2. div-sub 2.2%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      3. hypot-udef 2.2%

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

      4. hypot-udef 2.2%

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

      5. add-sqr-sqrt 2.2%

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

      6. metadata-eval 2.2%

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

   5. Applied egg-rr 2.2%

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. div-sub 3.1%

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

      2. +-commutative 3.1%

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

      3. associate--r+ 52.2%

         \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      4. +-inverses 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

   8. Taylor expanded in x around -inf 99.7%

      \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
   9. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(\log 0.5 + \log \left(\frac{-1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      3. metadata-eval 99.7%

         \[\leadsto \left(\log 0.5 + \log \left(\frac{\color{blue}{-1}}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      4. distribute-neg-frac 99.7%

         \[\leadsto \left(\log 0.5 + \log \color{blue}{\left(-\frac{1}{x}\right)}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      5. log-prod 100.0%

         \[\leadsto \color{blue}{\log \left(0.5 \cdot \left(-\frac{1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      6. distribute-rgt-neg-in 100.0%

         \[\leadsto \log \color{blue}{\left(-0.5 \cdot \frac{1}{x}\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      7. associate-*r/ 100.0%

         \[\leadsto \log \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \log \left(-\frac{\color{blue}{0.5}}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      9. distribute-neg-frac 100.0%

         \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{\color{blue}{-0.5}}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      11. associate-*r/ 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right) \]

      13. distribute-neg-frac 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \color{blue}{\frac{-0.25}{{x}^{2}}} \]

      14. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \frac{\color{blue}{-0.25}}{{x}^{2}} \]

      15. unpow2 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{\color{blue}{x \cdot x}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}} \]

   if -1.1499999999999999 < x < 0.0254999999999999984

   1. Initial program 7.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 7.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 7.8%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 7.8%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)} \]

   if 0.0254999999999999984 < x

   1. Initial program 48.6%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 48.6%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0255:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0009:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (+ (log (/ -0.5 x)) (/ -0.25 (* x x)))
   (if (<= x 0.0009)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = log((-0.5 / x)) + (-0.25 / (x * x));
	} else if (x <= 0.0009) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = Math.log((-0.5 / x)) + (-0.25 / (x * x));
	} else if (x <= 0.0009) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = math.log((-0.5 / x)) + (-0.25 / (x * x))
	elif x <= 0.0009:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(log(Float64(-0.5 / x)) + Float64(-0.25 / Float64(x * x)));
	elseif (x <= 0.0009)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = log((-0.5 / x)) + (-0.25 / (x * x));
	elseif (x <= 0.0009)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision] + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0009], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 2.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 2.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 3.7%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 3.7%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Step-by-step derivation

      1. flip-+ 3.1%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      2. div-sub 2.2%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      3. hypot-udef 2.2%

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

      4. hypot-udef 2.2%

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

      5. add-sqr-sqrt 2.2%

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

      6. metadata-eval 2.2%

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

   5. Applied egg-rr 2.2%

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. div-sub 3.1%

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

      2. +-commutative 3.1%

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

      3. associate--r+ 52.2%

         \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      4. +-inverses 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

   8. Taylor expanded in x around -inf 99.7%

      \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
   9. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(\log 0.5 + \log \left(\frac{-1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      3. metadata-eval 99.7%

         \[\leadsto \left(\log 0.5 + \log \left(\frac{\color{blue}{-1}}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      4. distribute-neg-frac 99.7%

         \[\leadsto \left(\log 0.5 + \log \color{blue}{\left(-\frac{1}{x}\right)}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      5. log-prod 100.0%

         \[\leadsto \color{blue}{\log \left(0.5 \cdot \left(-\frac{1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      6. distribute-rgt-neg-in 100.0%

         \[\leadsto \log \color{blue}{\left(-0.5 \cdot \frac{1}{x}\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      7. associate-*r/ 100.0%

         \[\leadsto \log \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \log \left(-\frac{\color{blue}{0.5}}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      9. distribute-neg-frac 100.0%

         \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{\color{blue}{-0.5}}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      11. associate-*r/ 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right) \]

      13. distribute-neg-frac 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \color{blue}{\frac{-0.25}{{x}^{2}}} \]

      14. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \frac{\color{blue}{-0.25}}{{x}^{2}} \]

      15. unpow2 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{\color{blue}{x \cdot x}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}} \]

   if -1 < x < 8.9999999999999998e-4

   1. Initial program 7.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 7.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 7.8%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 7.8%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + x} \]

   if 8.9999999999999998e-4 < x

   1. Initial program 48.6%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 48.6%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0009:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-6}:\\ \;\;\;\;\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6e-6)
   (log (/ -1.0 (- x (hypot 1.0 x))))
   (if (<= x 5.5e-6) x (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -6e-6) {
		tmp = log((-1.0 / (x - hypot(1.0, x))));
	} else if (x <= 5.5e-6) {
		tmp = x;
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -6e-6) {
		tmp = Math.log((-1.0 / (x - Math.hypot(1.0, x))));
	} else if (x <= 5.5e-6) {
		tmp = x;
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -6e-6:
		tmp = math.log((-1.0 / (x - math.hypot(1.0, x))))
	elif x <= 5.5e-6:
		tmp = x
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -6e-6)
		tmp = log(Float64(-1.0 / Float64(x - hypot(1.0, x))));
	elseif (x <= 5.5e-6)
		tmp = x;
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -6e-6)
		tmp = log((-1.0 / (x - hypot(1.0, x))));
	elseif (x <= 5.5e-6)
		tmp = x;
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -6e-6], N[Log[N[(-1.0 / N[(x - N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.5e-6], x, N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.0000000000000002e-6

   1. Initial program 3.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 3.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 5.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 5.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Step-by-step derivation

      1. flip-+ 4.4%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      2. div-sub 3.5%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      3. hypot-udef 3.5%

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

      4. hypot-udef 3.5%

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

      5. add-sqr-sqrt 3.5%

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

      6. metadata-eval 3.5%

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

   5. Applied egg-rr 3.5%

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. div-sub 4.4%

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

      2. +-commutative 4.4%

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

      3. associate--r+ 52.8%

         \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      4. +-inverses 99.9%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      5. metadata-eval 99.9%

         \[\leadsto \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

   7. Simplified 99.9%

      \[\leadsto \log \color{blue}{\left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

   if -6.0000000000000002e-6 < x < 5.4999999999999999e-6

   1. Initial program 7.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 7.1%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 7.1%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 7.1%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around 0 100.0%

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

   if 5.4999999999999999e-6 < x

   1. Initial program 48.6%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 48.6%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-6}:\\ \;\;\;\;\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.8e-6)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 5.5e-6) x (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -4.8e-6) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 5.5e-6) {
		tmp = x;
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.8e-6) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 5.5e-6) {
		tmp = x;
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.8e-6:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 5.5e-6:
		tmp = x
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.8e-6)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 5.5e-6)
		tmp = x;
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.8e-6)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 5.5e-6)
		tmp = x;
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.8e-6], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 5.5e-6], x, N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7999999999999998e-6

   1. Initial program 3.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 3.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 5.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 5.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Step-by-step derivation

      1. flip-+ 4.4%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      2. div-sub 3.5%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      3. hypot-udef 3.5%

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

      4. hypot-udef 3.5%

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

      5. add-sqr-sqrt 3.5%

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

      6. metadata-eval 3.5%

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

   5. Applied egg-rr 3.5%

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. div-sub 4.4%

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

      2. +-commutative 4.4%

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

      3. associate--r+ 52.8%

         \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      4. +-inverses 99.9%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      5. metadata-eval 99.9%

         \[\leadsto \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

   7. Simplified 99.9%

      \[\leadsto \log \color{blue}{\left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
   8. Step-by-step derivation

      1. frac-2neg 99.9%

         \[\leadsto \log \color{blue}{\left(\frac{--1}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)} \]

      2. metadata-eval 99.9%

         \[\leadsto \log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right) \]

      3. log-rec 99.9%

         \[\leadsto \color{blue}{-\log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]

   9. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{-\log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]

   if -4.7999999999999998e-6 < x < 5.4999999999999999e-6

   1. Initial program 7.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 7.1%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 7.1%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 7.1%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around 0 100.0%

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

   if 5.4999999999999999e-6 < x

   1. Initial program 48.6%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 48.6%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 7: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 0.95)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log(((x * 2.0) + (0.5 * (1.0 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 0.95d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log(((x * 2.0d0) + (0.5d0 * (1.0d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 * (1.0 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 0.95:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log(((x * 2.0) + (0.5 * (1.0 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.95)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(0.5 * Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 0.95)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log(((x * 2.0) + (0.5 * (1.0 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.95], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 2.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 2.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 3.7%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 3.7%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around -inf 99.9%

      \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} \]

   if -1.25 < x < 0.94999999999999996

   1. Initial program 8.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 8.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 8.5%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + x} \]

   if 0.94999999999999996 < x

   1. Initial program 47.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 47.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 8: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (+ (log (/ -0.5 x)) (/ -0.25 (* x x)))
   (if (<= x 0.95)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = log((-0.5 / x)) + (-0.25 / (x * x));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log(((x * 2.0) + (0.5 * (1.0 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = log(((-0.5d0) / x)) + ((-0.25d0) / (x * x))
    else if (x <= 0.95d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log(((x * 2.0d0) + (0.5d0 * (1.0d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = Math.log((-0.5 / x)) + (-0.25 / (x * x));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 * (1.0 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = math.log((-0.5 / x)) + (-0.25 / (x * x))
	elif x <= 0.95:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log(((x * 2.0) + (0.5 * (1.0 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(log(Float64(-0.5 / x)) + Float64(-0.25 / Float64(x * x)));
	elseif (x <= 0.95)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(0.5 * Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = log((-0.5 / x)) + (-0.25 / (x * x));
	elseif (x <= 0.95)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log(((x * 2.0) + (0.5 * (1.0 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision] + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 2.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 2.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 3.7%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 3.7%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Step-by-step derivation

      1. flip-+ 3.1%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      2. div-sub 2.2%

         \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

      3. hypot-udef 2.2%

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

      4. hypot-udef 2.2%

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

      5. add-sqr-sqrt 2.2%

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

      6. metadata-eval 2.2%

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

   5. Applied egg-rr 2.2%

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. div-sub 3.1%

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

      2. +-commutative 3.1%

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

      3. associate--r+ 52.2%

         \[\leadsto \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      4. +-inverses 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]

   8. Taylor expanded in x around -inf 99.7%

      \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
   9. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(\log 0.5 + \log \left(\frac{-1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      3. metadata-eval 99.7%

         \[\leadsto \left(\log 0.5 + \log \left(\frac{\color{blue}{-1}}{x}\right)\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      4. distribute-neg-frac 99.7%

         \[\leadsto \left(\log 0.5 + \log \color{blue}{\left(-\frac{1}{x}\right)}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      5. log-prod 100.0%

         \[\leadsto \color{blue}{\log \left(0.5 \cdot \left(-\frac{1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      6. distribute-rgt-neg-in 100.0%

         \[\leadsto \log \color{blue}{\left(-0.5 \cdot \frac{1}{x}\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      7. associate-*r/ 100.0%

         \[\leadsto \log \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \log \left(-\frac{\color{blue}{0.5}}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      9. distribute-neg-frac 100.0%

         \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{\color{blue}{-0.5}}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right) \]

      11. associate-*r/ 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \left(-\frac{\color{blue}{0.25}}{{x}^{2}}\right) \]

      13. distribute-neg-frac 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \color{blue}{\frac{-0.25}{{x}^{2}}} \]

      14. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \frac{\color{blue}{-0.25}}{{x}^{2}} \]

      15. unpow2 100.0%

          \[\leadsto \log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{\color{blue}{x \cdot x}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}} \]

   if -1 < x < 0.94999999999999996

   1. Initial program 8.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 8.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 8.5%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + x} \]

   if 0.94999999999999996 < x

   1. Initial program 47.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 47.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) + \frac{-0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 9: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 1.26) (+ x (* -0.16666666666666666 (pow x 3.0))) (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.26) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.26d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.26) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 1.26:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.26)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 1.26)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.26], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 2.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 2.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 3.7%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 3.7%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around -inf 99.9%

      \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} \]

   if -1.25 < x < 1.26000000000000001

   1. Initial program 8.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 8.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 8.5%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + x} \]

   if 1.26000000000000001 < x

   1. Initial program 47.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 47.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around inf 99.6%

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. count-2 99.6%

         \[\leadsto \log \color{blue}{\left(x + x\right)} \]

   6. Simplified 99.6%

      \[\leadsto \log \color{blue}{\left(x + x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 10: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25) (log (/ -0.5 x)) (if (<= x 1.26) x (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.26) {
		tmp = x;
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.26d0) then
        tmp = x
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.26) {
		tmp = x;
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 1.26:
		tmp = x
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.26)
		tmp = x;
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 1.26)
		tmp = x;
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.26], x, N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 2.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 2.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 3.7%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 3.7%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around -inf 99.9%

      \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} \]

   if -1.25 < x < 1.26000000000000001

   1. Initial program 8.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 8.5%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 8.5%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around 0 99.0%

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

   if 1.26000000000000001 < x

   1. Initial program 47.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 47.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around inf 99.6%

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. count-2 99.6%

         \[\leadsto \log \color{blue}{\left(x + x\right)} \]

   6. Simplified 99.6%

      \[\leadsto \log \color{blue}{\left(x + x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 11: 75.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.26:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.26) x (log (+ x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.26) {
		tmp = x;
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.26d0) then
        tmp = x
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.26) {
		tmp = x;
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.26:
		tmp = x
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.26)
		tmp = x;
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.26)
		tmp = x;
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.26], x, N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.26000000000000001

   1. Initial program 6.4%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 6.4%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 6.8%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 6.8%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around 0 66.1%

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

   if 1.26000000000000001 < x

   1. Initial program 47.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Step-by-step derivation

      1. +-commutative 47.8%

         \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

   4. Taylor expanded in x around inf 99.6%

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. count-2 99.6%

         \[\leadsto \log \color{blue}{\left(x + x\right)} \]

   6. Simplified 99.6%

      \[\leadsto \log \color{blue}{\left(x + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.26:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 12: 52.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 16.9%

   \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
2. Step-by-step derivation

   1. +-commutative 16.9%

      \[\leadsto \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]

   2. hypot-1-def 30.5%

      \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

3. Simplified 30.5%

   \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

4. Taylor expanded in x around 0 50.7%

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

5. Final simplification 50.7%

   \[\leadsto x \]

Developer target: 30.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot x + 1}\\ \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* x x) 1.0))))
   (if (< x 0.0) (log (/ -1.0 (- x t_0))) (log (+ x t_0)))))

C

double code(double x) {
	double t_0 = sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = log((-1.0 / (x - t_0)));
	} else {
		tmp = log((x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((x * x) + 1.0d0))
    if (x < 0.0d0) then
        tmp = log(((-1.0d0) / (x - t_0)))
    else
        tmp = log((x + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = Math.log((-1.0 / (x - t_0)));
	} else {
		tmp = Math.log((x + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt(((x * x) + 1.0))
	tmp = 0
	if x < 0.0:
		tmp = math.log((-1.0 / (x - t_0)))
	else:
		tmp = math.log((x + t_0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(Float64(x * x) + 1.0))
	tmp = 0.0
	if (x < 0.0)
		tmp = log(Float64(-1.0 / Float64(x - t_0)));
	else
		tmp = log(Float64(x + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt(((x * x) + 1.0));
	tmp = 0.0;
	if (x < 0.0)
		tmp = log((-1.0 / (x - t_0)));
	else
		tmp = log((x + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, If[Less[x, 0.0], N[Log[N[(-1.0 / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[N[(x + t$95$0), $MachinePrecision]], $MachinePrecision]]]

Reproduce

herbie shell --seed 2023174 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))