Hyperbolic arcsine

Percentage Accurate: 18.5% → 100.0%

Time: 10.8s

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: 18.5% 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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -29:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \left(\frac{0.0390625}{{x}^{7}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + x \cdot \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -29.0)
   (log
    (+
     (/ 0.125 (pow x 3.0))
     (- (/ 0.0390625 (pow x 7.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0))))))
   (log1p (+ x (* x (/ x (+ 1.0 (hypot 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -29.0) {
		tmp = log(((0.125 / pow(x, 3.0)) + ((0.0390625 / pow(x, 7.0)) - ((0.5 / x) + (0.0625 / pow(x, 5.0))))));
	} else {
		tmp = log1p((x + (x * (x / (1.0 + hypot(1.0, x))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -29.0) {
		tmp = Math.log(((0.125 / Math.pow(x, 3.0)) + ((0.0390625 / Math.pow(x, 7.0)) - ((0.5 / x) + (0.0625 / Math.pow(x, 5.0))))));
	} else {
		tmp = Math.log1p((x + (x * (x / (1.0 + Math.hypot(1.0, x))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -29.0:
		tmp = math.log(((0.125 / math.pow(x, 3.0)) + ((0.0390625 / math.pow(x, 7.0)) - ((0.5 / x) + (0.0625 / math.pow(x, 5.0))))))
	else:
		tmp = math.log1p((x + (x * (x / (1.0 + math.hypot(1.0, x))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -29.0)
		tmp = log(Float64(Float64(0.125 / (x ^ 3.0)) + Float64(Float64(0.0390625 / (x ^ 7.0)) - Float64(Float64(0.5 / x) + Float64(0.0625 / (x ^ 5.0))))));
	else
		tmp = log1p(Float64(x + Float64(x * Float64(x / Float64(1.0 + hypot(1.0, x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -29.0], N[Log[N[(N[(0.125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0390625 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / x), $MachinePrecision] + N[(0.0625 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[1 + N[(x + N[(x * N[(x / N[(1.0 + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -29

   1. Initial program 4.3%

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

      1. sqr-neg 4.3%

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

      2. +-commutative 4.3%

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

      3. sqr-neg 4.3%

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

      4. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

   4. Taylor expanded in x around -inf 100.0%

      \[\leadsto \log \color{blue}{\left(\left(0.0390625 \cdot \frac{1}{{x}^{7}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \log \left(\color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} + 0.0390625 \cdot \frac{1}{{x}^{7}}\right)} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right) \]

      2. associate--l+ 100.0%

         \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} + \left(0.0390625 \cdot \frac{1}{{x}^{7}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)\right)} \]

      3. associate-*r/ 100.0%

         \[\leadsto \log \left(\color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}} + \left(0.0390625 \cdot \frac{1}{{x}^{7}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0.125}}{{x}^{3}} + \left(0.0390625 \cdot \frac{1}{{x}^{7}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)\right) \]

      5. associate-*r/ 100.0%

         \[\leadsto \log \left(\frac{0.125}{{x}^{3}} + \left(\color{blue}{\frac{0.0390625 \cdot 1}{{x}^{7}}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)\right) \]

      6. metadata-eval 100.0%

         \[\leadsto \log \left(\frac{0.125}{{x}^{3}} + \left(\frac{\color{blue}{0.0390625}}{{x}^{7}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)\right) \]

      7. +-commutative 100.0%

         \[\leadsto \log \left(\frac{0.125}{{x}^{3}} + \left(\frac{0.0390625}{{x}^{7}} - \color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right)\right) \]

      8. associate-*r/ 100.0%

         \[\leadsto \log \left(\frac{0.125}{{x}^{3}} + \left(\frac{0.0390625}{{x}^{7}} - \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \log \left(\frac{0.125}{{x}^{3}} + \left(\frac{0.0390625}{{x}^{7}} - \left(\frac{\color{blue}{0.5}}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\right) \]

      10. associate-*r/ 100.0%

          \[\leadsto \log \left(\frac{0.125}{{x}^{3}} + \left(\frac{0.0390625}{{x}^{7}} - \left(\frac{0.5}{x} + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{5}}}\right)\right)\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \log \left(\frac{0.125}{{x}^{3}} + \left(\frac{0.0390625}{{x}^{7}} - \left(\frac{0.5}{x} + \frac{\color{blue}{0.0625}}{{x}^{5}}\right)\right)\right) \]

   6. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} + \left(\frac{0.0390625}{{x}^{7}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\right)} \]

   if -29 < x

   1. Initial program 23.0%

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

      1. sqr-neg 23.0%

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

      2. +-commutative 23.0%

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

      3. sqr-neg 23.0%

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

      4. hypot-1-def 42.1%

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

   3. Simplified 42.1%

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

      1. add-cube-cbrt 41.9%

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

      2. pow3 41.9%

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

      3. log-pow 41.7%

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

   5. Applied egg-rr 41.7%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 41.7%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      2. log1p-udef 41.7%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      3. expm1-udef 41.7%

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

      4. *-commutative 41.7%

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

      5. exp-to-pow 41.9%

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

      6. pow3 41.9%

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

      7. add-cube-cbrt 42.1%

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

   7. Applied egg-rr 42.1%

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

      1. log1p-def 42.1%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)} \]

      2. associate--l+ 99.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)}\right) \]

   9. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)} \]
   10. Step-by-step derivation

       1. sub-neg 99.1%

          \[\leadsto \mathsf{log1p}\left(x + \color{blue}{\left(\mathsf{hypot}\left(1, x\right) + \left(-1\right)\right)}\right) \]

       2. metadata-eval 99.1%

          \[\leadsto \mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + \color{blue}{-1}\right)\right) \]

       3. flip-+ 80.1%

          \[\leadsto \mathsf{log1p}\left(x + \color{blue}{\frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - -1 \cdot -1}{\mathsf{hypot}\left(1, x\right) - -1}}\right) \]

       4. hypot-1-def 80.1%

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

       5. hypot-1-def 80.1%

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

       6. add-sqr-sqrt 80.1%

          \[\leadsto \mathsf{log1p}\left(x + \frac{\color{blue}{\left(1 + x \cdot x\right)} - -1 \cdot -1}{\mathsf{hypot}\left(1, x\right) - -1}\right) \]

       7. add-exp-log 80.1%

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

       8. log1p-udef 80.1%

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

       9. metadata-eval 80.1%

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

       10. expm1-udef 80.9%

           \[\leadsto \mathsf{log1p}\left(x + \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x\right)\right)}}{\mathsf{hypot}\left(1, x\right) - -1}\right) \]

       11. expm1-log1p-u 80.9%

           \[\leadsto \mathsf{log1p}\left(x + \frac{\color{blue}{x \cdot x}}{\mathsf{hypot}\left(1, x\right) - -1}\right) \]

       12. pow2 80.9%

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

   11. Applied egg-rr 80.9%

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

       1. unpow2 80.9%

          \[\leadsto \mathsf{log1p}\left(x + \frac{\color{blue}{x \cdot x}}{\mathsf{hypot}\left(1, x\right) - -1}\right) \]

       2. *-un-lft-identity 80.9%

          \[\leadsto \mathsf{log1p}\left(x + \frac{x \cdot x}{\color{blue}{1 \cdot \left(\mathsf{hypot}\left(1, x\right) - -1\right)}}\right) \]

       3. times-frac 100.0%

          \[\leadsto \mathsf{log1p}\left(x + \color{blue}{\frac{x}{1} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right) - -1}}\right) \]

       4. sub-neg 100.0%

          \[\leadsto \mathsf{log1p}\left(x + \frac{x}{1} \cdot \frac{x}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(--1\right)}}\right) \]

       5. metadata-eval 100.0%

          \[\leadsto \mathsf{log1p}\left(x + \frac{x}{1} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right) + \color{blue}{1}}\right) \]

       6. +-commutative 100.0%

          \[\leadsto \mathsf{log1p}\left(x + \frac{x}{1} \cdot \frac{x}{\color{blue}{1 + \mathsf{hypot}\left(1, x\right)}}\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto \mathsf{log1p}\left(x + \color{blue}{\frac{x}{1} \cdot \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -29:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \left(\frac{0.0390625}{{x}^{7}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + x \cdot \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + x \cdot \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -100.0)
   (- (log (- (hypot 1.0 x) x)))
   (log1p (+ x (* x (/ x (+ 1.0 (hypot 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -100.0) {
		tmp = -log((hypot(1.0, x) - x));
	} else {
		tmp = log1p((x + (x * (x / (1.0 + hypot(1.0, x))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -100.0) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else {
		tmp = Math.log1p((x + (x * (x / (1.0 + Math.hypot(1.0, x))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -100.0:
		tmp = -math.log((math.hypot(1.0, x) - x))
	else:
		tmp = math.log1p((x + (x * (x / (1.0 + math.hypot(1.0, x))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -100.0)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	else
		tmp = log1p(Float64(x + Float64(x * Float64(x / Float64(1.0 + hypot(1.0, x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -100.0], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), N[Log[1 + N[(x + N[(x * N[(x / N[(1.0 + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -100

   1. Initial program 4.3%

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

      1. sqr-neg 4.3%

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

      2. +-commutative 4.3%

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

      3. sqr-neg 4.3%

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

      4. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

      1. flip-+ 4.0%

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

      2. frac-2neg 4.0%

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

      3. log-div 4.0%

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

      4. pow2 4.0%

         \[\leadsto \log \left(-\left(\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. hypot-1-def 4.0%

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

      6. hypot-1-def 4.0%

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

      7. add-sqr-sqrt 4.5%

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

      8. +-commutative 4.5%

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

      9. fma-def 4.5%

         \[\leadsto \log \left(-\left({x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   5. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\log \left(-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 4.5%

         \[\leadsto \log \color{blue}{\left(0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      2. associate--r- 4.5%

         \[\leadsto \log \color{blue}{\left(\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      3. neg-sub0 4.5%

         \[\leadsto \log \left(\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      4. +-commutative 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. sub-neg 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - {x}^{2}\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      6. fma-udef 4.5%

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

      7. unpow2 4.5%

         \[\leadsto \log \left(\left(\color{blue}{{x}^{2}} + 1\right) - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      8. +-commutative 4.5%

         \[\leadsto \log \left(\color{blue}{\left(1 + {x}^{2}\right)} - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      9. associate--l+ 47.6%

         \[\leadsto \log \color{blue}{\left(1 + \left({x}^{2} - {x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      10. +-inverses 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-sub0 100.0%

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

      14. neg-sub0 100.0%

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

      15. associate--r- 100.0%

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

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   if -100 < x

   1. Initial program 23.0%

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

      1. sqr-neg 23.0%

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

      2. +-commutative 23.0%

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

      3. sqr-neg 23.0%

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

      4. hypot-1-def 42.1%

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

   3. Simplified 42.1%

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

      1. add-cube-cbrt 41.9%

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

      2. pow3 41.9%

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

      3. log-pow 41.7%

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

   5. Applied egg-rr 41.7%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 41.7%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      2. log1p-udef 41.7%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      3. expm1-udef 41.7%

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

      4. *-commutative 41.7%

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

      5. exp-to-pow 41.9%

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

      6. pow3 41.9%

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

      7. add-cube-cbrt 42.1%

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

   7. Applied egg-rr 42.1%

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

      1. log1p-def 42.1%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)} \]

      2. associate--l+ 99.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)}\right) \]

   9. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)} \]
   10. Step-by-step derivation

       1. sub-neg 99.1%

          \[\leadsto \mathsf{log1p}\left(x + \color{blue}{\left(\mathsf{hypot}\left(1, x\right) + \left(-1\right)\right)}\right) \]

       2. metadata-eval 99.1%

          \[\leadsto \mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + \color{blue}{-1}\right)\right) \]

       3. flip-+ 80.1%

          \[\leadsto \mathsf{log1p}\left(x + \color{blue}{\frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - -1 \cdot -1}{\mathsf{hypot}\left(1, x\right) - -1}}\right) \]

       4. hypot-1-def 80.1%

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

       5. hypot-1-def 80.1%

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

       6. add-sqr-sqrt 80.1%

          \[\leadsto \mathsf{log1p}\left(x + \frac{\color{blue}{\left(1 + x \cdot x\right)} - -1 \cdot -1}{\mathsf{hypot}\left(1, x\right) - -1}\right) \]

       7. add-exp-log 80.1%

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

       8. log1p-udef 80.1%

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

       9. metadata-eval 80.1%

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

       10. expm1-udef 80.9%

           \[\leadsto \mathsf{log1p}\left(x + \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x\right)\right)}}{\mathsf{hypot}\left(1, x\right) - -1}\right) \]

       11. expm1-log1p-u 80.9%

           \[\leadsto \mathsf{log1p}\left(x + \frac{\color{blue}{x \cdot x}}{\mathsf{hypot}\left(1, x\right) - -1}\right) \]

       12. pow2 80.9%

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

   11. Applied egg-rr 80.9%

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

       1. unpow2 80.9%

          \[\leadsto \mathsf{log1p}\left(x + \frac{\color{blue}{x \cdot x}}{\mathsf{hypot}\left(1, x\right) - -1}\right) \]

       2. *-un-lft-identity 80.9%

          \[\leadsto \mathsf{log1p}\left(x + \frac{x \cdot x}{\color{blue}{1 \cdot \left(\mathsf{hypot}\left(1, x\right) - -1\right)}}\right) \]

       3. times-frac 100.0%

          \[\leadsto \mathsf{log1p}\left(x + \color{blue}{\frac{x}{1} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right) - -1}}\right) \]

       4. sub-neg 100.0%

          \[\leadsto \mathsf{log1p}\left(x + \frac{x}{1} \cdot \frac{x}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(--1\right)}}\right) \]

       5. metadata-eval 100.0%

          \[\leadsto \mathsf{log1p}\left(x + \frac{x}{1} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right) + \color{blue}{1}}\right) \]

       6. +-commutative 100.0%

          \[\leadsto \mathsf{log1p}\left(x + \frac{x}{1} \cdot \frac{x}{\color{blue}{1 + \mathsf{hypot}\left(1, x\right)}}\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto \mathsf{log1p}\left(x + \color{blue}{\frac{x}{1} \cdot \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + x \cdot \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00092:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.96)
   (- (log (- (* x -2.0) (/ 0.5 x))))
   (if (<= x 0.00092)
     (+ x (* (pow x 3.0) -0.16666666666666666))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = -log(((x * -2.0) - (0.5 / x)));
	} else if (x <= 0.00092) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = -Math.log(((x * -2.0) - (0.5 / x)));
	} else if (x <= 0.00092) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.96:
		tmp = -math.log(((x * -2.0) - (0.5 / x)))
	elif x <= 0.00092:
		tmp = x + (math.pow(x, 3.0) * -0.16666666666666666)
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.96)
		tmp = Float64(-log(Float64(Float64(x * -2.0) - Float64(0.5 / x))));
	elseif (x <= 0.00092)
		tmp = Float64(x + Float64((x ^ 3.0) * -0.16666666666666666));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.96)
		tmp = -log(((x * -2.0) - (0.5 / x)));
	elseif (x <= 0.00092)
		tmp = x + ((x ^ 3.0) * -0.16666666666666666);
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.96], (-N[Log[N[(N[(x * -2.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.00092], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $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 < -0.95999999999999996

   1. Initial program 4.3%

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

      1. sqr-neg 4.3%

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

      2. +-commutative 4.3%

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

      3. sqr-neg 4.3%

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

      4. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

      1. flip-+ 4.0%

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

      2. frac-2neg 4.0%

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

      3. log-div 4.0%

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

      4. pow2 4.0%

         \[\leadsto \log \left(-\left(\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. hypot-1-def 4.0%

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

      6. hypot-1-def 4.0%

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

      7. add-sqr-sqrt 4.5%

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

      8. +-commutative 4.5%

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

      9. fma-def 4.5%

         \[\leadsto \log \left(-\left({x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   5. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\log \left(-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 4.5%

         \[\leadsto \log \color{blue}{\left(0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      2. associate--r- 4.5%

         \[\leadsto \log \color{blue}{\left(\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      3. neg-sub0 4.5%

         \[\leadsto \log \left(\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      4. +-commutative 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. sub-neg 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - {x}^{2}\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      6. fma-udef 4.5%

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

      7. unpow2 4.5%

         \[\leadsto \log \left(\left(\color{blue}{{x}^{2}} + 1\right) - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      8. +-commutative 4.5%

         \[\leadsto \log \left(\color{blue}{\left(1 + {x}^{2}\right)} - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      9. associate--l+ 47.6%

         \[\leadsto \log \color{blue}{\left(1 + \left({x}^{2} - {x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      10. +-inverses 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-sub0 100.0%

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

      14. neg-sub0 100.0%

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

      15. associate--r- 100.0%

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

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around -inf 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

   10. Simplified 99.2%

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

   if -0.95999999999999996 < x < 9.2000000000000003e-4

   1. Initial program 8.5%

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

      1. sqr-neg 8.5%

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

      2. +-commutative 8.5%

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

      3. sqr-neg 8.5%

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

      4. 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 99.8%

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

   if 9.2000000000000003e-4 < x

   1. Initial program 48.0%

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

      1. sqr-neg 48.0%

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

      2. +-commutative 48.0%

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

      3. sqr-neg 48.0%

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

      4. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00092:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.00092:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.00105)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.00092)
     (+ x (* (pow x 3.0) -0.16666666666666666))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.00105) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.00092) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.00105) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.00092) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.00105:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.00092:
		tmp = x + (math.pow(x, 3.0) * -0.16666666666666666)
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.00105)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.00092)
		tmp = Float64(x + Float64((x ^ 3.0) * -0.16666666666666666));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.00105)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.00092)
		tmp = x + ((x ^ 3.0) * -0.16666666666666666);
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.00105], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.00092], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $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 < -0.00104999999999999994

   1. Initial program 4.3%

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

      1. sqr-neg 4.3%

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

      2. +-commutative 4.3%

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

      3. sqr-neg 4.3%

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

      4. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

      1. flip-+ 4.0%

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

      2. frac-2neg 4.0%

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

      3. log-div 4.0%

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

      4. pow2 4.0%

         \[\leadsto \log \left(-\left(\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. hypot-1-def 4.0%

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

      6. hypot-1-def 4.0%

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

      7. add-sqr-sqrt 4.5%

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

      8. +-commutative 4.5%

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

      9. fma-def 4.5%

         \[\leadsto \log \left(-\left({x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   5. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\log \left(-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 4.5%

         \[\leadsto \log \color{blue}{\left(0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      2. associate--r- 4.5%

         \[\leadsto \log \color{blue}{\left(\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      3. neg-sub0 4.5%

         \[\leadsto \log \left(\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      4. +-commutative 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. sub-neg 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - {x}^{2}\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      6. fma-udef 4.5%

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

      7. unpow2 4.5%

         \[\leadsto \log \left(\left(\color{blue}{{x}^{2}} + 1\right) - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      8. +-commutative 4.5%

         \[\leadsto \log \left(\color{blue}{\left(1 + {x}^{2}\right)} - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      9. associate--l+ 47.6%

         \[\leadsto \log \color{blue}{\left(1 + \left({x}^{2} - {x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      10. +-inverses 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-sub0 100.0%

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

      14. neg-sub0 100.0%

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

      15. associate--r- 100.0%

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

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   if -0.00104999999999999994 < x < 9.2000000000000003e-4

   1. Initial program 8.5%

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

      1. sqr-neg 8.5%

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

      2. +-commutative 8.5%

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

      3. sqr-neg 8.5%

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

      4. 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 99.8%

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

   if 9.2000000000000003e-4 < x

   1. Initial program 48.0%

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

      1. sqr-neg 48.0%

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

      2. +-commutative 48.0%

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

      3. sqr-neg 48.0%

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

      4. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.00092:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.2)
   (- (log (- (hypot 1.0 x) x)))
   (log1p (+ x (+ (hypot 1.0 x) -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.2) {
		tmp = -log((hypot(1.0, x) - x));
	} else {
		tmp = log1p((x + (hypot(1.0, x) + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.2) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else {
		tmp = Math.log1p((x + (Math.hypot(1.0, x) + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.2:
		tmp = -math.log((math.hypot(1.0, x) - x))
	else:
		tmp = math.log1p((x + (math.hypot(1.0, x) + -1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.2)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	else
		tmp = log1p(Float64(x + Float64(hypot(1.0, x) + -1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.2], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), N[Log[1 + N[(x + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.20000000000000001

   1. Initial program 4.3%

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

      1. sqr-neg 4.3%

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

      2. +-commutative 4.3%

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

      3. sqr-neg 4.3%

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

      4. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

      1. flip-+ 4.0%

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

      2. frac-2neg 4.0%

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

      3. log-div 4.0%

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

      4. pow2 4.0%

         \[\leadsto \log \left(-\left(\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. hypot-1-def 4.0%

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

      6. hypot-1-def 4.0%

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

      7. add-sqr-sqrt 4.5%

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

      8. +-commutative 4.5%

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

      9. fma-def 4.5%

         \[\leadsto \log \left(-\left({x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   5. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\log \left(-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 4.5%

         \[\leadsto \log \color{blue}{\left(0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      2. associate--r- 4.5%

         \[\leadsto \log \color{blue}{\left(\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      3. neg-sub0 4.5%

         \[\leadsto \log \left(\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      4. +-commutative 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. sub-neg 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - {x}^{2}\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      6. fma-udef 4.5%

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

      7. unpow2 4.5%

         \[\leadsto \log \left(\left(\color{blue}{{x}^{2}} + 1\right) - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      8. +-commutative 4.5%

         \[\leadsto \log \left(\color{blue}{\left(1 + {x}^{2}\right)} - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      9. associate--l+ 47.6%

         \[\leadsto \log \color{blue}{\left(1 + \left({x}^{2} - {x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      10. +-inverses 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-sub0 100.0%

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

      14. neg-sub0 100.0%

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

      15. associate--r- 100.0%

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

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   if -0.20000000000000001 < x

   1. Initial program 23.0%

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

      1. sqr-neg 23.0%

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

      2. +-commutative 23.0%

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

      3. sqr-neg 23.0%

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

      4. hypot-1-def 42.1%

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

   3. Simplified 42.1%

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

      1. add-cube-cbrt 41.9%

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

      2. pow3 41.9%

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

      3. log-pow 41.7%

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

   5. Applied egg-rr 41.7%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 41.7%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      2. log1p-udef 41.7%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      3. expm1-udef 41.7%

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

      4. *-commutative 41.7%

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

      5. exp-to-pow 41.9%

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

      6. pow3 41.9%

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

      7. add-cube-cbrt 42.1%

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

   7. Applied egg-rr 42.1%

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

      1. log1p-def 42.1%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)} \]

      2. associate--l+ 99.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)}\right) \]

   9. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \]

Alternative 6: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.96)
   (- (log (- (* x -2.0) (/ 0.5 x))))
   (if (<= x 0.96)
     (+ x (* (pow x 3.0) -0.16666666666666666))
     (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = -log(((x * -2.0) - (0.5 / x)));
	} else if (x <= 0.96) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} 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 <= (-0.96d0)) then
        tmp = -log(((x * (-2.0d0)) - (0.5d0 / x)))
    else if (x <= 0.96d0) then
        tmp = x + ((x ** 3.0d0) * (-0.16666666666666666d0))
    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 <= -0.96) {
		tmp = -Math.log(((x * -2.0) - (0.5 / x)));
	} else if (x <= 0.96) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 * (1.0 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.96:
		tmp = -math.log(((x * -2.0) - (0.5 / x)))
	elif x <= 0.96:
		tmp = x + (math.pow(x, 3.0) * -0.16666666666666666)
	else:
		tmp = math.log(((x * 2.0) + (0.5 * (1.0 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.96)
		tmp = Float64(-log(Float64(Float64(x * -2.0) - Float64(0.5 / x))));
	elseif (x <= 0.96)
		tmp = Float64(x + Float64((x ^ 3.0) * -0.16666666666666666));
	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 <= -0.96)
		tmp = -log(((x * -2.0) - (0.5 / x)));
	elseif (x <= 0.96)
		tmp = x + ((x ^ 3.0) * -0.16666666666666666);
	else
		tmp = log(((x * 2.0) + (0.5 * (1.0 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.96], (-N[Log[N[(N[(x * -2.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.96], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $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 < -0.95999999999999996

   1. Initial program 4.3%

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

      1. sqr-neg 4.3%

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

      2. +-commutative 4.3%

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

      3. sqr-neg 4.3%

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

      4. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

      1. flip-+ 4.0%

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

      2. frac-2neg 4.0%

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

      3. log-div 4.0%

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

      4. pow2 4.0%

         \[\leadsto \log \left(-\left(\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. hypot-1-def 4.0%

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

      6. hypot-1-def 4.0%

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

      7. add-sqr-sqrt 4.5%

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

      8. +-commutative 4.5%

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

      9. fma-def 4.5%

         \[\leadsto \log \left(-\left({x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   5. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\log \left(-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 4.5%

         \[\leadsto \log \color{blue}{\left(0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      2. associate--r- 4.5%

         \[\leadsto \log \color{blue}{\left(\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      3. neg-sub0 4.5%

         \[\leadsto \log \left(\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      4. +-commutative 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. sub-neg 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - {x}^{2}\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      6. fma-udef 4.5%

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

      7. unpow2 4.5%

         \[\leadsto \log \left(\left(\color{blue}{{x}^{2}} + 1\right) - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      8. +-commutative 4.5%

         \[\leadsto \log \left(\color{blue}{\left(1 + {x}^{2}\right)} - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      9. associate--l+ 47.6%

         \[\leadsto \log \color{blue}{\left(1 + \left({x}^{2} - {x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      10. +-inverses 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-sub0 100.0%

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

      14. neg-sub0 100.0%

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

      15. associate--r- 100.0%

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

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around -inf 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

   10. Simplified 99.2%

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

   if -0.95999999999999996 < x < 0.95999999999999996

   1. Initial program 9.8%

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

      1. sqr-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. sqr-neg 9.8%

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

      4. hypot-1-def 9.8%

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

   3. Simplified 9.8%

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

   4. Taylor expanded in x around 0 99.2%

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

   if 0.95999999999999996 < x

   1. Initial program 46.7%

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

      1. sqr-neg 46.7%

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

      2. +-commutative 46.7%

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

      3. sqr-neg 46.7%

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

      4. 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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 7: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log((x * 2.0));
	}
	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 <= 1.25d0) then
        tmp = x + ((x ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = log((x * 2.0d0))
    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 <= 1.25) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 4.3%

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

      1. sqr-neg 4.3%

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

      2. +-commutative 4.3%

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

      3. sqr-neg 4.3%

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

      4. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

   4. Taylor expanded in x around -inf 98.4%

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

   if -1.25 < x < 1.25

   1. Initial program 9.8%

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

      1. sqr-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. sqr-neg 9.8%

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

      4. hypot-1-def 9.8%

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

   3. Simplified 9.8%

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

   4. Taylor expanded in x around 0 99.2%

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

   if 1.25 < x

   1. Initial program 46.7%

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

      1. sqr-neg 46.7%

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

      2. +-commutative 46.7%

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

      3. sqr-neg 46.7%

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

      4. 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 99.0%

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

      1. *-commutative 99.0%

         \[\leadsto \log \color{blue}{\left(x \cdot 2\right)} \]

   6. Simplified 99.0%

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

4. Final simplification 98.9%

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

Alternative 8: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.96)
   (- (log (- (* x -2.0) (/ 0.5 x))))
   (if (<= x 1.25)
     (+ x (* (pow x 3.0) -0.16666666666666666))
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = -log(((x * -2.0) - (0.5 / x)));
	} else if (x <= 1.25) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.96d0)) then
        tmp = -log(((x * (-2.0d0)) - (0.5d0 / x)))
    else if (x <= 1.25d0) then
        tmp = x + ((x ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = -Math.log(((x * -2.0) - (0.5 / x)));
	} else if (x <= 1.25) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.96:
		tmp = -math.log(((x * -2.0) - (0.5 / x)))
	elif x <= 1.25:
		tmp = x + (math.pow(x, 3.0) * -0.16666666666666666)
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.96)
		tmp = Float64(-log(Float64(Float64(x * -2.0) - Float64(0.5 / x))));
	elseif (x <= 1.25)
		tmp = Float64(x + Float64((x ^ 3.0) * -0.16666666666666666));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.96)
		tmp = -log(((x * -2.0) - (0.5 / x)));
	elseif (x <= 1.25)
		tmp = x + ((x ^ 3.0) * -0.16666666666666666);
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.96], (-N[Log[N[(N[(x * -2.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.25], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.95999999999999996

   1. Initial program 4.3%

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

      1. sqr-neg 4.3%

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

      2. +-commutative 4.3%

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

      3. sqr-neg 4.3%

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

      4. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

      1. flip-+ 4.0%

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

      2. frac-2neg 4.0%

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

      3. log-div 4.0%

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

      4. pow2 4.0%

         \[\leadsto \log \left(-\left(\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. hypot-1-def 4.0%

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

      6. hypot-1-def 4.0%

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

      7. add-sqr-sqrt 4.5%

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

      8. +-commutative 4.5%

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

      9. fma-def 4.5%

         \[\leadsto \log \left(-\left({x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   5. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\log \left(-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 4.5%

         \[\leadsto \log \color{blue}{\left(0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      2. associate--r- 4.5%

         \[\leadsto \log \color{blue}{\left(\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      3. neg-sub0 4.5%

         \[\leadsto \log \left(\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      4. +-commutative 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      5. sub-neg 4.5%

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - {x}^{2}\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      6. fma-udef 4.5%

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

      7. unpow2 4.5%

         \[\leadsto \log \left(\left(\color{blue}{{x}^{2}} + 1\right) - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      8. +-commutative 4.5%

         \[\leadsto \log \left(\color{blue}{\left(1 + {x}^{2}\right)} - {x}^{2}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      9. associate--l+ 47.6%

         \[\leadsto \log \color{blue}{\left(1 + \left({x}^{2} - {x}^{2}\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      10. +-inverses 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-sub0 100.0%

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

      14. neg-sub0 100.0%

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

      15. associate--r- 100.0%

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

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around -inf 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

   10. Simplified 99.2%

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

   if -0.95999999999999996 < x < 1.25

   1. Initial program 9.8%

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

      1. sqr-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. sqr-neg 9.8%

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

      4. hypot-1-def 9.8%

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

   3. Simplified 9.8%

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

   4. Taylor expanded in x around 0 99.2%

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

   if 1.25 < x

   1. Initial program 46.7%

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

      1. sqr-neg 46.7%

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

      2. +-commutative 46.7%

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

      3. sqr-neg 46.7%

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

      4. 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 99.0%

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

      1. *-commutative 99.0%

         \[\leadsto \log \color{blue}{\left(x \cdot 2\right)} \]

   6. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 9: 99.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25) (log (/ -0.5 x)) (if (<= x 1.25) x (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = log((x * 2.0));
	}
	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 <= 1.25d0) then
        tmp = x
    else
        tmp = log((x * 2.0d0))
    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 <= 1.25) {
		tmp = x;
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 1.25:
		tmp = x
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.25)
		tmp = x;
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = x;
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], x, N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 4.3%

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

      1. sqr-neg 4.3%

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

      2. +-commutative 4.3%

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

      3. sqr-neg 4.3%

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

      4. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

   4. Taylor expanded in x around -inf 98.4%

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

   if -1.25 < x < 1.25

   1. Initial program 9.8%

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

      1. sqr-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. sqr-neg 9.8%

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

      4. hypot-1-def 9.8%

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

   3. Simplified 9.8%

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

   4. Taylor expanded in x around 0 98.2%

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

   if 1.25 < x

   1. Initial program 46.7%

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

      1. sqr-neg 46.7%

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

      2. +-commutative 46.7%

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

      3. sqr-neg 46.7%

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

      4. 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 99.0%

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

      1. *-commutative 99.0%

         \[\leadsto \log \color{blue}{\left(x \cdot 2\right)} \]

   6. Simplified 99.0%

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

4. Final simplification 98.5%

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

Alternative 10: 75.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.25) x (log (* x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = x
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = x
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = x;
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = x;
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], x, N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 8.0%

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

      1. sqr-neg 8.0%

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

      2. +-commutative 8.0%

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

      3. sqr-neg 8.0%

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

      4. hypot-1-def 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in x around 0 66.9%

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

   if 1.25 < x

   1. Initial program 46.7%

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

      1. sqr-neg 46.7%

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

      2. +-commutative 46.7%

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

      3. sqr-neg 46.7%

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

      4. 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 99.0%

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

      1. *-commutative 99.0%

         \[\leadsto \log \color{blue}{\left(x \cdot 2\right)} \]

   6. Simplified 99.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 11: 58.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.56) x (log1p x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.56) {
		tmp = x;
	} else {
		tmp = log1p(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.56) {
		tmp = x;
	} else {
		tmp = Math.log1p(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.56:
		tmp = x
	else:
		tmp = math.log1p(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.56)
		tmp = x;
	else
		tmp = log1p(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.56], x, N[Log[1 + x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5600000000000001

   1. Initial program 8.0%

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

      1. sqr-neg 8.0%

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

      2. +-commutative 8.0%

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

      3. sqr-neg 8.0%

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

      4. hypot-1-def 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in x around 0 66.9%

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

   if 1.5600000000000001 < x

   1. Initial program 46.7%

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

      1. sqr-neg 46.7%

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

      2. +-commutative 46.7%

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

      3. sqr-neg 46.7%

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

      4. 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-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. log-pow 99.6%

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

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 99.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      2. log1p-udef 99.6%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot \log \left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      3. expm1-udef 99.6%

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

      4. *-commutative 99.6%

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

      5. exp-to-pow 100.0%

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

      6. pow3 100.0%

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

      7. add-cube-cbrt 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. log1p-def 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)} \]

      2. associate--l+ 100.0%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)}\right) \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)} \]

   10. Taylor expanded in x around 0 31.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \end{array} \]

Alternative 12: 52.0% 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 18.4%

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

   1. sqr-neg 18.4%

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

   2. +-commutative 18.4%

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

   3. sqr-neg 18.4%

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

   4. hypot-1-def 33.1%

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

3. Simplified 33.1%

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

4. Taylor expanded in x around 0 50.3%

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

5. Final simplification 50.3%

   \[\leadsto x \]

Developer target: 31.0% 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 2023338 
(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)))))