Rust f64::acosh

Percentage Accurate: 52.5% → 99.9%

Time: 4.9s

Alternatives: 4

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: 52.5% 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: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.6%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. difference-of-sqr-1 N/A

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

   8. *-commutative N/A

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

   9. unpow-prod-down N/A

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

   10. lower-fma.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower--.f64 N/A

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

   14. pow1/2 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. unsub-neg N/A

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

   6. remove-double-neg N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. distribute-lft-neg-out N/A

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

   9. mul-1-neg N/A

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

   10. *-commutative N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. mul-1-neg N/A

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

   14. distribute-lft-neg-out N/A

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

   15. *-commutative N/A

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

   16. distribute-rgt-neg-in N/A

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

   17. metadata-eval N/A

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 3: 99.1% 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 51.6%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. mul-1-neg N/A

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

   4. log-rec N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \log 2 \]

   5. remove-double-neg N/A

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

   6. lower-log.f64 N/A

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

   7. lower-log.f64 98.9

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

5. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\log x + \log 2} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

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

Alternative 4: 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 51.6%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 98.8

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

5. Applied rewrites 98.8%

   \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
6. 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 2024331 
(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)))))