Rust f64::acosh

Percentage Accurate: 51.4% → 99.6%

Time: 3.2s

Alternatives: 6

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: 51.4% 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, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

2. Taylor expanded in x around inf 99.3%

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

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

4. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 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 50.4%

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

2. Taylor expanded in x around inf 98.9%

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

3. Final simplification 98.9%

   \[\leadsto \log \left(x + x\right) \]

Alternative 3: 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 50.4%

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

   1. pow1/2 50.4%

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

   2. add-cube-cbrt 50.4%

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

   3. pow3 50.4%

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

   4. pow-pow 50.4%

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

   5. fma-neg 50.4%

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

   6. metadata-eval 50.4%

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

   7. metadata-eval 50.4%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{1.5}}\right) \]

3. Applied egg-rr 50.4%

   \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}}\right) \]

4. Taylor expanded in x around inf 31.4%

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

5. Final simplification 31.4%

   \[\leadsto \log x \]

Alternative 4: 4.2% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \frac{1.96875}{x} \cdot \frac{1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ 1.96875 x) (/ 1.0 x)))

C

double code(double x) {
	return (1.96875 / x) * (1.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.96875d0 / x) * (1.0d0 / x)
end function

Java

public static double code(double x) {
	return (1.96875 / x) * (1.0 / x);
}

Python

def code(x):
	return (1.96875 / x) * (1.0 / x)

Julia

function code(x)
	return Float64(Float64(1.96875 / x) * Float64(1.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = (1.96875 / x) * (1.0 / x);
end

Wolfram

code[x_] := N[(N[(1.96875 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

2. Taylor expanded in x around inf 99.3%

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

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

4. Simplified 99.3%

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

5. Taylor expanded in x around inf 99.4%

   \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) - \left(0.03125 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

7. Simplified 1.6%

   \[\leadsto \color{blue}{-2 - \left(\frac{-2}{x \cdot x} + \frac{0.03125}{x \cdot x}\right)} \]

8. Taylor expanded in x around 0 4.2%

   \[\leadsto \color{blue}{\frac{1.96875}{{x}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 4.2%

      \[\leadsto \frac{1.96875}{\color{blue}{x \cdot x}} \]

10. Simplified 4.2%

    \[\leadsto \color{blue}{\frac{1.96875}{x \cdot x}} \]
11. Step-by-step derivation

    1. associate-/r* 4.2%

       \[\leadsto \color{blue}{\frac{\frac{1.96875}{x}}{x}} \]

    2. div-inv 4.2%

       \[\leadsto \color{blue}{\frac{1.96875}{x} \cdot \frac{1}{x}} \]

12. Applied egg-rr 4.2%

    \[\leadsto \color{blue}{\frac{1.96875}{x} \cdot \frac{1}{x}} \]

13. Final simplification 4.2%

    \[\leadsto \frac{1.96875}{x} \cdot \frac{1}{x} \]

Alternative 5: 4.2% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ \frac{1.96875}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.96875 (* x x)))

C

double code(double x) {
	return 1.96875 / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.96875d0 / (x * x)
end function

Java

public static double code(double x) {
	return 1.96875 / (x * x);
}

Python

def code(x):
	return 1.96875 / (x * x)

Julia

function code(x)
	return Float64(1.96875 / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 1.96875 / (x * x);
end

Wolfram

code[x_] := N[(1.96875 / N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

2. Taylor expanded in x around inf 99.3%

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

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

4. Simplified 99.3%

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

5. Taylor expanded in x around inf 99.4%

   \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) - \left(0.03125 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

7. Simplified 1.6%

   \[\leadsto \color{blue}{-2 - \left(\frac{-2}{x \cdot x} + \frac{0.03125}{x \cdot x}\right)} \]

8. Taylor expanded in x around 0 4.2%

   \[\leadsto \color{blue}{\frac{1.96875}{{x}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 4.2%

      \[\leadsto \frac{1.96875}{\color{blue}{x \cdot x}} \]

10. Simplified 4.2%

    \[\leadsto \color{blue}{\frac{1.96875}{x \cdot x}} \]

11. Final simplification 4.2%

    \[\leadsto \frac{1.96875}{x \cdot x} \]

Alternative 6: 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 50.4%

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

2. Taylor expanded in x around inf 98.9%

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

3. Taylor expanded in x around inf 98.9%

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

5. Simplified 1.6%

   \[\leadsto \color{blue}{-2} \]

6. Final simplification 1.6%

   \[\leadsto -2 \]

Developer target: 100.0% 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 2023189 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

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