Rust f64::acosh

Percentage Accurate: 51.2% → 99.4%

Time: 3.6s

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: 51.2% 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, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r/ 99.3%

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

   3. metadata-eval 99.3%

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

4. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 98.9%

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

   1. count-2 98.9%

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

   2. *-commutative 98.9%

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

   3. metadata-eval 98.9%

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

   4. div-inv 98.9%

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

   5. add-sqr-sqrt 98.9%

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

   6. associate-/l* 98.9%

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

   7. log-div 99.0%

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

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{x}\right) - \log \left(\frac{0.5}{\sqrt{x}}\right)} \]
5. Step-by-step derivation

   1. diff-log 98.9%

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

   2. clear-num 98.9%

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

   3. log-rec 99.2%

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

   4. add-sqr-sqrt 99.2%

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

   5. sqrt-unprod 99.2%

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

   6. frac-times 99.2%

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

   7. metadata-eval 99.2%

      \[\leadsto -\log \left(\frac{\sqrt{\frac{\color{blue}{0.25}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x}}\right) \]

   8. add-sqr-sqrt 99.2%

      \[\leadsto -\log \left(\frac{\sqrt{\frac{0.25}{\color{blue}{x}}}}{\sqrt{x}}\right) \]

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{-\log \left(\frac{\sqrt{\frac{0.25}{x}}}{\sqrt{x}}\right)} \]

7. Taylor expanded in x around 0 99.3%

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

8. Final simplification 99.3%

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

Alternative 3: 99.0% 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 59.1%

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

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 4: 1.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 98.9%

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

3. Taylor expanded in x around 0 98.9%

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

5. Simplified 1.6%

   \[\leadsto \color{blue}{-1} \]

6. Final simplification 1.6%

   \[\leadsto -1 \]

Developer target: 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 2023193 
(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)))))