Rust f64::acosh

Percentage Accurate: 52.1% → 99.6%

Time: 6.4s

Alternatives: 4

Speedup: 2.0×

Specification

\[x \geq 1\]

Math

\[\begin{array}{l} \\ \cosh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acosh x))

C

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

Python

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

Julia

function code(x)
	return acosh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 52.1% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot \left(2 - \frac{0.5 + \frac{0.125}{{x}^{2}}}{{x}^{2}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (* x (- 2.0 (/ (+ 0.5 (/ 0.125 (pow x 2.0))) (pow x 2.0))))))

C

double code(double x) {
	return log((x * (2.0 - ((0.5 + (0.125 / pow(x, 2.0))) / pow(x, 2.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x * (2.0d0 - ((0.5d0 + (0.125d0 / (x ** 2.0d0))) / (x ** 2.0d0)))))
end function

Java

public static double code(double x) {
	return Math.log((x * (2.0 - ((0.5 + (0.125 / Math.pow(x, 2.0))) / Math.pow(x, 2.0)))));
}

Python

def code(x):
	return math.log((x * (2.0 - ((0.5 + (0.125 / math.pow(x, 2.0))) / math.pow(x, 2.0)))))

Julia

function code(x)
	return log(Float64(x * Float64(2.0 - Float64(Float64(0.5 + Float64(0.125 / (x ^ 2.0))) / (x ^ 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = log((x * (2.0 - ((0.5 + (0.125 / (x ^ 2.0))) / (x ^ 2.0)))));
end

Wolfram

code[x_] := N[Log[N[(x * N[(2.0 - N[(N[(0.5 + N[(0.125 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in x around inf 99.6%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 + -1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg 99.6%

      \[\leadsto \log \left(x \cdot \left(2 + \color{blue}{\left(-\frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}\right)\right) \]

   2. unsub-neg 99.6%

      \[\leadsto \log \left(x \cdot \color{blue}{\left(2 - \frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}\right) \]

   3. associate-*r/ 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{0.5 + \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}}{{x}^{2}}\right)\right) \]

   4. metadata-eval 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{0.5 + \frac{\color{blue}{0.125}}{{x}^{2}}}{{x}^{2}}\right)\right) \]

5. Simplified 99.6%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 - \frac{0.5 + \frac{0.125}{{x}^{2}}}{{x}^{2}}\right)\right)} \]
6. Add Preprocessing

Alternative 2: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot \left(2 - \frac{\frac{0.152587890625}{x}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (* x (- 2.0 (/ (/ 0.152587890625 x) x)))))

C

double code(double x) {
	return log((x * (2.0 - ((0.152587890625 / x) / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x * (2.0d0 - ((0.152587890625d0 / x) / x))))
end function

Java

public static double code(double x) {
	return Math.log((x * (2.0 - ((0.152587890625 / x) / x))));
}

Python

def code(x):
	return math.log((x * (2.0 - ((0.152587890625 / x) / x))))

Julia

function code(x)
	return log(Float64(x * Float64(2.0 - Float64(Float64(0.152587890625 / x) / x))))
end

MATLAB

function tmp = code(x)
	tmp = log((x * (2.0 - ((0.152587890625 / x) / x))));
end

Wolfram

code[x_] := N[Log[N[(x * N[(2.0 - N[(N[(0.152587890625 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in x around inf 99.6%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 + -1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg 99.6%

      \[\leadsto \log \left(x \cdot \left(2 + \color{blue}{\left(-\frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}\right)\right) \]

   2. unsub-neg 99.6%

      \[\leadsto \log \left(x \cdot \color{blue}{\left(2 - \frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}\right) \]

   3. associate-*r/ 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{0.5 + \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}}{{x}^{2}}\right)\right) \]

   4. metadata-eval 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{0.5 + \frac{\color{blue}{0.125}}{{x}^{2}}}{{x}^{2}}\right)\right) \]

5. Simplified 99.6%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 - \frac{0.5 + \frac{0.125}{{x}^{2}}}{{x}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{\color{blue}{\left(\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}} \cdot \sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}\right) \cdot \sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}}}{{x}^{2}}\right)\right) \]

   2. unpow2 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{\left(\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}} \cdot \sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}\right) \cdot \sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}}{\color{blue}{x \cdot x}}\right)\right) \]

   3. times-frac 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \color{blue}{\frac{\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}} \cdot \sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}}{x}}\right)\right) \]

   4. pow2 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{\color{blue}{{\left(\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}\right)}^{2}}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}}{x}\right)\right) \]

   5. +-commutative 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{{\left(\sqrt[3]{\color{blue}{\frac{0.125}{{x}^{2}} + 0.5}}\right)}^{2}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}}{x}\right)\right) \]

   6. div-inv 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{{\left(\sqrt[3]{\color{blue}{0.125 \cdot \frac{1}{{x}^{2}}} + 0.5}\right)}^{2}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}}{x}\right)\right) \]

   7. fma-define 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(0.125, \frac{1}{{x}^{2}}, 0.5\right)}}\right)}^{2}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}}{x}\right)\right) \]

   8. pow-flip 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{{\left(\sqrt[3]{\mathsf{fma}\left(0.125, \color{blue}{{x}^{\left(-2\right)}}, 0.5\right)}\right)}^{2}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}}{x}\right)\right) \]

   9. metadata-eval 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{{\left(\sqrt[3]{\mathsf{fma}\left(0.125, {x}^{\color{blue}{-2}}, 0.5\right)}\right)}^{2}}{x} \cdot \frac{\sqrt[3]{0.5 + \frac{0.125}{{x}^{2}}}}{x}\right)\right) \]

7. Applied egg-rr 99.6%

   \[\leadsto \log \left(x \cdot \left(2 - \color{blue}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(0.125, {x}^{-2}, 0.5\right)}\right)}^{2}}{x} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(0.125, {x}^{-2}, 0.5\right)}}{x}}\right)\right) \]

9. Simplified 98.5%

   \[\leadsto \log \left(x \cdot \left(2 - \color{blue}{\frac{0.390625}{x} \cdot \frac{0.390625}{x}}\right)\right) \]
10. Step-by-step derivation

    1. associate-*l/ 98.5%

       \[\leadsto \log \left(x \cdot \left(2 - \color{blue}{\frac{0.390625 \cdot \frac{0.390625}{x}}{x}}\right)\right) \]

    2. associate-*r/ 98.5%

       \[\leadsto \log \left(x \cdot \left(2 - \frac{\color{blue}{\frac{0.390625 \cdot 0.390625}{x}}}{x}\right)\right) \]

    3. metadata-eval 98.5%

       \[\leadsto \log \left(x \cdot \left(2 - \frac{\frac{\color{blue}{0.152587890625}}{x}}{x}\right)\right) \]

11. Applied egg-rr 98.5%

    \[\leadsto \log \left(x \cdot \left(2 - \color{blue}{\frac{\frac{0.152587890625}{x}}{x}}\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.1% 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 53.8%

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

3. Taylor expanded in x around inf 98.4%

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

Alternative 4: 1.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log 0.390625 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log 0.390625))

C

double code(double x) {
	return log(0.390625);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(0.390625d0)
end function

Java

public static double code(double x) {
	return Math.log(0.390625);
}

Python

def code(x):
	return math.log(0.390625)

Julia

function code(x)
	return log(0.390625)
end

MATLAB

function tmp = code(x)
	tmp = log(0.390625);
end

Wolfram

code[x_] := N[Log[0.390625], $MachinePrecision]

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in x around inf 99.6%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 + -1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg 99.6%

      \[\leadsto \log \left(x \cdot \left(2 + \color{blue}{\left(-\frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}\right)\right) \]

   2. unsub-neg 99.6%

      \[\leadsto \log \left(x \cdot \color{blue}{\left(2 - \frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}\right) \]

   3. associate-*r/ 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{0.5 + \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}}{{x}^{2}}\right)\right) \]

   4. metadata-eval 99.6%

      \[\leadsto \log \left(x \cdot \left(2 - \frac{0.5 + \frac{\color{blue}{0.125}}{{x}^{2}}}{{x}^{2}}\right)\right) \]

5. Simplified 99.6%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 - \frac{0.5 + \frac{0.125}{{x}^{2}}}{{x}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. *-un-lft-identity 99.6%

      \[\leadsto \log \color{blue}{\left(1 \cdot \left(x \cdot \left(2 - \frac{0.5 + \frac{0.125}{{x}^{2}}}{{x}^{2}}\right)\right)\right)} \]

   2. log-prod 99.6%

      \[\leadsto \color{blue}{\log 1 + \log \left(x \cdot \left(2 - \frac{0.5 + \frac{0.125}{{x}^{2}}}{{x}^{2}}\right)\right)} \]

   3. metadata-eval 99.6%

      \[\leadsto \color{blue}{0} + \log \left(x \cdot \left(2 - \frac{0.5 + \frac{0.125}{{x}^{2}}}{{x}^{2}}\right)\right) \]

   4. sub-neg 99.6%

      \[\leadsto 0 + \log \left(x \cdot \color{blue}{\left(2 + \left(-\frac{0.5 + \frac{0.125}{{x}^{2}}}{{x}^{2}}\right)\right)}\right) \]

   5. +-commutative 99.6%

      \[\leadsto 0 + \log \left(x \cdot \color{blue}{\left(\left(-\frac{0.5 + \frac{0.125}{{x}^{2}}}{{x}^{2}}\right) + 2\right)}\right) \]

   6. div-inv 99.6%

      \[\leadsto 0 + \log \left(x \cdot \left(\left(-\color{blue}{\left(0.5 + \frac{0.125}{{x}^{2}}\right) \cdot \frac{1}{{x}^{2}}}\right) + 2\right)\right) \]

   7. distribute-lft-neg-in 99.6%

      \[\leadsto 0 + \log \left(x \cdot \left(\color{blue}{\left(-\left(0.5 + \frac{0.125}{{x}^{2}}\right)\right) \cdot \frac{1}{{x}^{2}}} + 2\right)\right) \]

   8. fma-define 99.6%

      \[\leadsto 0 + \log \left(x \cdot \color{blue}{\mathsf{fma}\left(-\left(0.5 + \frac{0.125}{{x}^{2}}\right), \frac{1}{{x}^{2}}, 2\right)}\right) \]

   9. +-commutative 99.6%

      \[\leadsto 0 + \log \left(x \cdot \mathsf{fma}\left(-\color{blue}{\left(\frac{0.125}{{x}^{2}} + 0.5\right)}, \frac{1}{{x}^{2}}, 2\right)\right) \]

   10. div-inv 99.6%

       \[\leadsto 0 + \log \left(x \cdot \mathsf{fma}\left(-\left(\color{blue}{0.125 \cdot \frac{1}{{x}^{2}}} + 0.5\right), \frac{1}{{x}^{2}}, 2\right)\right) \]

   11. fma-define 99.6%

       \[\leadsto 0 + \log \left(x \cdot \mathsf{fma}\left(-\color{blue}{\mathsf{fma}\left(0.125, \frac{1}{{x}^{2}}, 0.5\right)}, \frac{1}{{x}^{2}}, 2\right)\right) \]

   12. pow-flip 99.6%

       \[\leadsto 0 + \log \left(x \cdot \mathsf{fma}\left(-\mathsf{fma}\left(0.125, \color{blue}{{x}^{\left(-2\right)}}, 0.5\right), \frac{1}{{x}^{2}}, 2\right)\right) \]

   13. metadata-eval 99.6%

       \[\leadsto 0 + \log \left(x \cdot \mathsf{fma}\left(-\mathsf{fma}\left(0.125, {x}^{\color{blue}{-2}}, 0.5\right), \frac{1}{{x}^{2}}, 2\right)\right) \]

   14. pow-flip 99.6%

       \[\leadsto 0 + \log \left(x \cdot \mathsf{fma}\left(-\mathsf{fma}\left(0.125, {x}^{-2}, 0.5\right), \color{blue}{{x}^{\left(-2\right)}}, 2\right)\right) \]

   15. metadata-eval 99.6%

       \[\leadsto 0 + \log \left(x \cdot \mathsf{fma}\left(-\mathsf{fma}\left(0.125, {x}^{-2}, 0.5\right), {x}^{\color{blue}{-2}}, 2\right)\right) \]

7. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \log \left(x \cdot \mathsf{fma}\left(-\mathsf{fma}\left(0.125, {x}^{-2}, 0.5\right), {x}^{-2}, 2\right)\right)} \]

9. Simplified 1.6%

   \[\leadsto \color{blue}{\log 0.390625} \]
10. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024147 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

  :alt
  (! :herbie-platform default (log (+ x (* (sqrt (- x 1)) (sqrt (+ x 1))))))

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