Rust f64::acosh

Percentage Accurate: 51.9% → 99.2%

Time: 6.4s

Alternatives: 3

Speedup: 1.2×

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: 51.9% 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: 99.2% accurate, 1.1× 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 57.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 57.6

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 57.6

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

4. Applied rewrites 57.6%

   \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + x\right)} \]
5. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. log-rec N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. clear-num N/A

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

6. Applied rewrites 57.6%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 99.1

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

9. Applied rewrites 99.1%

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

Alternative 2: 99.1% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (* 2.0 x)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return log(Float64(2.0 * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 98.7

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

5. Applied rewrites 98.7%

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

Alternative 3: 0.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log 0.0))

C

double code(double x) {
	return log(0.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(0.0d0)
end function

Java

public static double code(double x) {
	return Math.log(0.0);
}

Python

def code(x):
	return math.log(0.0)

Julia

function code(x)
	return log(0.0)
end

MATLAB

function tmp = code(x)
	tmp = log(0.0);
end

Wolfram

code[x_] := N[Log[0.0], $MachinePrecision]

Derivation

1. Initial program 57.6%

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

   1. lift-+.f64 N/A

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

   2. unpow1 N/A

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

   3. sqr-pow N/A

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

   4. lower-fma.f64 N/A

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

   5. metadata-eval N/A

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

   6. unpow1/2 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. metadata-eval N/A

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

   9. unpow1/2 N/A

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

   10. lower-sqrt.f64 57.6

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, \sqrt{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(1\right)\right)}}\right)\right) \]

   15. metadata-eval 57.6

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

4. Applied rewrites 57.6%

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

5. Taylor expanded in x around -inf

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

   1. associate-*r* N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. rem-square-sqrt N/A

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

   5. metadata-eval N/A

      \[\leadsto \log \left(\left(-1 \cdot x\right) \cdot \color{blue}{0}\right) \]

   6. mul0-rgt 0.6

      \[\leadsto \log \color{blue}{0} \]

7. Applied rewrites 0.6%

   \[\leadsto \log \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× 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 2024249 
(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)))))