Rust f64::acosh

Percentage Accurate: 52.3% → 99.4%

Time: 5.1s

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: 52.3% 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, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\\ \log \left(\frac{x \cdot \left(4 - t\_0 \cdot t\_0\right)}{t\_0 + 2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (+ 0.5 (/ 0.125 (* x x))) (* x x))))
   (log (/ (* x (- 4.0 (* t_0 t_0))) (+ t_0 2.0)))))

C

double code(double x) {
	double t_0 = (0.5 + (0.125 / (x * x))) / (x * x);
	return log(((x * (4.0 - (t_0 * t_0))) / (t_0 + 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (0.5d0 + (0.125d0 / (x * x))) / (x * x)
    code = log(((x * (4.0d0 - (t_0 * t_0))) / (t_0 + 2.0d0)))
end function

Java

public static double code(double x) {
	double t_0 = (0.5 + (0.125 / (x * x))) / (x * x);
	return Math.log(((x * (4.0 - (t_0 * t_0))) / (t_0 + 2.0)));
}

Python

def code(x):
	t_0 = (0.5 + (0.125 / (x * x))) / (x * x)
	return math.log(((x * (4.0 - (t_0 * t_0))) / (t_0 + 2.0)))

Julia

function code(x)
	t_0 = Float64(Float64(0.5 + Float64(0.125 / Float64(x * x))) / Float64(x * x))
	return log(Float64(Float64(x * Float64(4.0 - Float64(t_0 * t_0))) / Float64(t_0 + 2.0)))
end

MATLAB

function tmp = code(x)
	t_0 = (0.5 + (0.125 / (x * x))) / (x * x);
	tmp = log(((x * (4.0 - (t_0 * t_0))) / (t_0 + 2.0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 + N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[Log[N[(N[(x * N[(4.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in x around inf 99.3%

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

   1. mul-1-neg 99.3%

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

   2. unsub-neg 99.3%

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

   3. unpow2 99.3%

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

   4. associate-*r/ 99.3%

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

   5. metadata-eval 99.3%

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

   6. unpow2 99.3%

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

5. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. flip-- 99.3%

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

   3. associate-*l/ 99.3%

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

   4. metadata-eval 99.3%

      \[\leadsto \log \left(\frac{\left(\color{blue}{4} - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x} \cdot \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right) \cdot x}{2 + \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}}\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto \log \color{blue}{\left(\frac{\left(4 - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x} \cdot \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right) \cdot x}{2 + \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}}\right)} \]

8. Final simplification 99.3%

   \[\leadsto \log \left(\frac{x \cdot \left(4 - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x} \cdot \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right)}{\frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x} + 2}\right) \]
9. Add Preprocessing

Alternative 2: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in x around inf 99.3%

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

   1. mul-1-neg 99.3%

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

   2. unsub-neg 99.3%

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

   3. unpow2 99.3%

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

   4. associate-*r/ 99.3%

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

   5. metadata-eval 99.3%

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

   6. unpow2 99.3%

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

5. Simplified 99.3%

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

Alternative 3: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in x around inf 99.1%

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

   1. sub-neg 99.1%

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

   2. unpow2 99.1%

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

   3. associate-*r/ 99.1%

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

   4. metadata-eval 99.1%

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

   5. distribute-neg-frac 99.1%

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

   6. metadata-eval 99.1%

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

5. Simplified 99.1%

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

Alternative 4: 98.8% 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 52.7%

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

3. Taylor expanded in x around inf 98.6%

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

Developer target: 99.8% 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 2024097 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

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

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