Hyperbolic arc-cosine

Percentage Accurate: 51.9% → 99.4%

Time: 7.7s

Alternatives: 7

Speedup: 2.0×

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: 51.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{0.5}{x}}{x}\\ \mathsf{log1p}\left(x \cdot \frac{4 + \frac{-1 - \frac{0.5}{x}}{x} \cdot t\_0}{2 + t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ 0.5 x)) x)))
   (log1p (* x (/ (+ 4.0 (* (/ (- -1.0 (/ 0.5 x)) x) t_0)) (+ 2.0 t_0))))))

C

double code(double x) {
	double t_0 = (1.0 + (0.5 / x)) / x;
	return log1p((x * ((4.0 + (((-1.0 - (0.5 / x)) / x) * t_0)) / (2.0 + t_0))));
}

Java

public static double code(double x) {
	double t_0 = (1.0 + (0.5 / x)) / x;
	return Math.log1p((x * ((4.0 + (((-1.0 - (0.5 / x)) / x) * t_0)) / (2.0 + t_0))));
}

Python

def code(x):
	t_0 = (1.0 + (0.5 / x)) / x
	return math.log1p((x * ((4.0 + (((-1.0 - (0.5 / x)) / x) * t_0)) / (2.0 + t_0))))

Julia

function code(x)
	t_0 = Float64(Float64(1.0 + Float64(0.5 / x)) / x)
	return log1p(Float64(x * Float64(Float64(4.0 + Float64(Float64(Float64(-1.0 - Float64(0.5 / x)) / x) * t_0)) / Float64(2.0 + t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[Log[1 + N[(x * N[(N[(4.0 + N[(N[(N[(-1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 45.9%

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

   1. log1p-expm1-u 45.9%

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

   2. expm1-undefine 45.9%

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

   3. add-exp-log 45.9%

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

   4. fmm-def 45.9%

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

   5. metadata-eval 45.9%

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

4. Applied egg-rr 45.9%

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

5. Taylor expanded in x around inf 99.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + -1 \cdot \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + \color{blue}{\left(-\frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right)\right) \]

   2. unsub-neg 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 - \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]

   3. *-un-lft-identity 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{\color{blue}{1 \cdot \left(1 + 0.5 \cdot \frac{1}{x}\right)}}{x}\right)\right) \]

   4. *-un-lft-identity 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{\color{blue}{1 + 0.5 \cdot \frac{1}{x}}}{x}\right)\right) \]

   5. un-div-inv 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \color{blue}{\frac{0.5}{x}}}{x}\right)\right) \]

7. Applied egg-rr 99.7%

   \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]
8. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 + \left(-\frac{1 + \frac{0.5}{x}}{x}\right)\right)}\right) \]

   2. flip-+ 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\frac{2 \cdot 2 - \left(-\frac{1 + \frac{0.5}{x}}{x}\right) \cdot \left(-\frac{1 + \frac{0.5}{x}}{x}\right)}{2 - \left(-\frac{1 + \frac{0.5}{x}}{x}\right)}}\right) \]

   3. metadata-eval 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \frac{\color{blue}{4} - \left(-\frac{1 + \frac{0.5}{x}}{x}\right) \cdot \left(-\frac{1 + \frac{0.5}{x}}{x}\right)}{2 - \left(-\frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]

   4. distribute-neg-frac2 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \frac{4 - \color{blue}{\frac{1 + \frac{0.5}{x}}{-x}} \cdot \left(-\frac{1 + \frac{0.5}{x}}{x}\right)}{2 - \left(-\frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]

   5. distribute-neg-frac2 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \frac{4 - \frac{1 + \frac{0.5}{x}}{-x} \cdot \color{blue}{\frac{1 + \frac{0.5}{x}}{-x}}}{2 - \left(-\frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]

   6. distribute-neg-frac2 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \frac{4 - \frac{1 + \frac{0.5}{x}}{-x} \cdot \frac{1 + \frac{0.5}{x}}{-x}}{2 - \color{blue}{\frac{1 + \frac{0.5}{x}}{-x}}}\right) \]

9. Applied egg-rr 99.7%

   \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\frac{4 - \frac{1 + \frac{0.5}{x}}{-x} \cdot \frac{1 + \frac{0.5}{x}}{-x}}{2 - \frac{1 + \frac{0.5}{x}}{-x}}}\right) \]

10. Final simplification 99.7%

    \[\leadsto \mathsf{log1p}\left(x \cdot \frac{4 + \frac{-1 - \frac{0.5}{x}}{x} \cdot \frac{1 + \frac{0.5}{x}}{x}}{2 + \frac{1 + \frac{0.5}{x}}{x}}\right) \]
11. Add Preprocessing

Alternative 2: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\left(-1 + \frac{-0.5}{x}\right) + \frac{x}{0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p (+ (+ -1.0 (/ -0.5 x)) (/ x 0.5))))

C

double code(double x) {
	return log1p(((-1.0 + (-0.5 / x)) + (x / 0.5)));
}

Java

public static double code(double x) {
	return Math.log1p(((-1.0 + (-0.5 / x)) + (x / 0.5)));
}

Python

def code(x):
	return math.log1p(((-1.0 + (-0.5 / x)) + (x / 0.5)))

Julia

function code(x)
	return log1p(Float64(Float64(-1.0 + Float64(-0.5 / x)) + Float64(x / 0.5)))
end

Wolfram

code[x_] := N[Log[1 + N[(N[(-1.0 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[(x / 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 45.9%

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

   1. log1p-expm1-u 45.9%

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

   2. expm1-undefine 45.9%

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

   3. add-exp-log 45.9%

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

   4. fmm-def 45.9%

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

   5. metadata-eval 45.9%

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

4. Applied egg-rr 45.9%

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

5. Taylor expanded in x around inf 99.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + -1 \cdot \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + \color{blue}{\left(-\frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right)\right) \]

   2. unsub-neg 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 - \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]

   3. *-un-lft-identity 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{\color{blue}{1 \cdot \left(1 + 0.5 \cdot \frac{1}{x}\right)}}{x}\right)\right) \]

   4. *-un-lft-identity 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{\color{blue}{1 + 0.5 \cdot \frac{1}{x}}}{x}\right)\right) \]

   5. un-div-inv 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \color{blue}{\frac{0.5}{x}}}{x}\right)\right) \]

7. Applied egg-rr 99.7%

   \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]
8. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 + \left(-\frac{1 + \frac{0.5}{x}}{x}\right)\right)}\right) \]

   2. distribute-lft-in 99.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot 2 + x \cdot \left(-\frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]

   3. metadata-eval 99.7%

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

   4. div-inv 99.7%

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

   5. add-cube-cbrt 99.7%

      \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{0.5} + x \cdot \left(-\frac{1 + \frac{0.5}{x}}{x}\right)\right) \]

   6. associate-/l* 99.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{0.5}} + x \cdot \left(-\frac{1 + \frac{0.5}{x}}{x}\right)\right) \]

   7. fma-define 99.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(-\frac{1 + \frac{0.5}{x}}{x}\right)\right)}\right) \]

   8. pow2 99.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(-\frac{1 + \frac{0.5}{x}}{x}\right)\right)\right) \]

   9. distribute-neg-frac2 99.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \color{blue}{\frac{1 + \frac{0.5}{x}}{-x}}\right)\right) \]

9. Applied egg-rr 99.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \frac{1 + \frac{0.5}{x}}{-x}\right)}\right) \]
10. Step-by-step derivation

    1. fma-undefine 99.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5} + x \cdot \frac{1 + \frac{0.5}{x}}{-x}}\right) \]

    2. +-commutative 99.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \frac{1 + \frac{0.5}{x}}{-x} + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}}\right) \]

    3. *-commutative 99.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1 + \frac{0.5}{x}}{-x} \cdot x} + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    4. neg-mul-1 99.7%

       \[\leadsto \mathsf{log1p}\left(\frac{1 + \frac{0.5}{x}}{\color{blue}{-1 \cdot x}} \cdot x + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    5. associate-*l/ 99.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{\left(1 + \frac{0.5}{x}\right) \cdot x}{-1 \cdot x}} + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    6. associate-/l* 99.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(1 + \frac{0.5}{x}\right) \cdot \frac{x}{-1 \cdot x}} + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    7. neg-mul-1 99.7%

       \[\leadsto \mathsf{log1p}\left(\left(1 + \frac{0.5}{x}\right) \cdot \frac{x}{\color{blue}{-x}} + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    8. distribute-frac-neg2 99.7%

       \[\leadsto \mathsf{log1p}\left(\left(1 + \frac{0.5}{x}\right) \cdot \color{blue}{\left(-\frac{x}{x}\right)} + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    9. *-inverses 99.7%

       \[\leadsto \mathsf{log1p}\left(\left(1 + \frac{0.5}{x}\right) \cdot \left(-\color{blue}{1}\right) + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    10. distribute-rgt-neg-in 99.7%

        \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(-\left(1 + \frac{0.5}{x}\right) \cdot 1\right)} + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    11. *-rgt-identity 99.7%

        \[\leadsto \mathsf{log1p}\left(\left(-\color{blue}{\left(1 + \frac{0.5}{x}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    12. distribute-neg-in 99.7%

        \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\left(-1\right) + \left(-\frac{0.5}{x}\right)\right)} + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    13. metadata-eval 99.7%

        \[\leadsto \mathsf{log1p}\left(\left(\color{blue}{-1} + \left(-\frac{0.5}{x}\right)\right) + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    14. distribute-neg-frac 99.7%

        \[\leadsto \mathsf{log1p}\left(\left(-1 + \color{blue}{\frac{-0.5}{x}}\right) + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    15. metadata-eval 99.7%

        \[\leadsto \mathsf{log1p}\left(\left(-1 + \frac{\color{blue}{-0.5}}{x}\right) + {\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{0.5}\right) \]

    16. associate-*r/ 99.7%

        \[\leadsto \mathsf{log1p}\left(\left(-1 + \frac{-0.5}{x}\right) + \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{0.5}}\right) \]

11. Simplified 99.7%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(-1 + \frac{-0.5}{x}\right) + \frac{x}{0.5}}\right) \]
12. Add Preprocessing

Alternative 3: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{-0.5}{x} + x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ (/ -0.5 x) (* x 2.0))))

C

double code(double x) {
	return log(((-0.5 / x) + (x * 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((((-0.5d0) / x) + (x * 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.log(((-0.5 / x) + (x * 2.0)));
}

Python

def code(x):
	return math.log(((-0.5 / x) + (x * 2.0)))

Julia

function code(x)
	return log(Float64(Float64(-0.5 / x) + Float64(x * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = log(((-0.5 / x) + (x * 2.0)));
end

Wolfram

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

Derivation

1. Initial program 45.9%

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

   1. log1p-expm1-u 45.9%

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

   2. expm1-undefine 45.9%

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

   3. add-exp-log 45.9%

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

   4. fmm-def 45.9%

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

   5. metadata-eval 45.9%

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

4. Applied egg-rr 45.9%

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

5. Taylor expanded in x around inf 99.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + -1 \cdot \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + \color{blue}{\left(-\frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right)\right) \]

   2. unsub-neg 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 - \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]

   3. *-un-lft-identity 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{\color{blue}{1 \cdot \left(1 + 0.5 \cdot \frac{1}{x}\right)}}{x}\right)\right) \]

   4. *-un-lft-identity 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{\color{blue}{1 + 0.5 \cdot \frac{1}{x}}}{x}\right)\right) \]

   5. un-div-inv 99.7%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \color{blue}{\frac{0.5}{x}}}{x}\right)\right) \]

7. Applied egg-rr 99.7%

   \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]
8. Step-by-step derivation

   1. add-log-exp 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. log-prod 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto \color{blue}{0} + \log \left(e^{\mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)\right)}\right) \]

   5. add-log-exp 99.7%

      \[\leadsto 0 + \color{blue}{\mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)\right)} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{0 + \mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)\right)} \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{\log \left(\frac{-0.5}{x} + x \cdot 2\right)} \]
12. Add Preprocessing

Alternative 4: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(-1 + \left(x + x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p (+ -1.0 (+ x x))))

C

double code(double x) {
	return log1p((-1.0 + (x + x)));
}

Java

public static double code(double x) {
	return Math.log1p((-1.0 + (x + x)));
}

Python

def code(x):
	return math.log1p((-1.0 + (x + x)))

Julia

function code(x)
	return log1p(Float64(-1.0 + Float64(x + x)))
end

Wolfram

code[x_] := N[Log[1 + N[(-1.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 45.9%

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

   1. log1p-expm1-u 45.9%

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

   2. expm1-undefine 45.9%

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

   3. add-exp-log 45.9%

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

   4. fmm-def 45.9%

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

   5. metadata-eval 45.9%

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

4. Applied egg-rr 45.9%

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

5. Taylor expanded in x around inf 99.3%

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

6. Final simplification 99.3%

   \[\leadsto \mathsf{log1p}\left(-1 + \left(x + x\right)\right) \]
7. Add Preprocessing

Alternative 5: 98.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ x x)))

C

double code(double x) {
	return log((x + x));
}

Fortran

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

Java

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

Python

def code(x):
	return math.log((x + x))

Julia

function code(x)
	return log(Float64(x + x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.9%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.3%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
4. Add Preprocessing

Alternative 6: 31.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log x))

C

double code(double x) {
	return log(x);
}

Fortran

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

Java

public static double code(double x) {
	return Math.log(x);
}

Python

def code(x):
	return math.log(x)

Julia

function code(x)
	return log(x)
end

MATLAB

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

Wolfram

code[x_] := N[Log[x], $MachinePrecision]

Derivation

1. Initial program 45.9%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.3%

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

4. Taylor expanded in x around 0 99.1%

   \[\leadsto \color{blue}{\log 2 + \log x} \]

6. Simplified 31.6%

   \[\leadsto \color{blue}{\log x} \]
7. Add Preprocessing

Alternative 7: 1.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 45.9%

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

   1. log1p-expm1-u 45.9%

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

   2. expm1-undefine 45.9%

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

   3. add-exp-log 45.9%

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

   4. fmm-def 45.9%

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

   5. metadata-eval 45.9%

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

4. Applied egg-rr 45.9%

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

5. Taylor expanded in x around 0 0.0%

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

7. Simplified 1.6%

   \[\leadsto \color{blue}{-2} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024172 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))