Rust f32::acosh

Percentage Accurate: 52.9% → 99.3%

Time: 4.7s

Alternatives: 2

Speedup: 1.2×

Specification

\[x \geq 1\]

Math

\[\begin{array}{l} \\ \cosh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (acosh x))

C

float code(float x) {
	return acoshf(x);
}

Julia

function code(x)
	return acosh(x)
end

MATLAB

function tmp = code(x)
	tmp = acosh(x);
end

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (log (+ x (sqrt (- (* x x) 1.0)))))

C

float code(float x) {
	return logf((x + sqrtf(((x * x) - 1.0f))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + sqrt(((x * x) - 1.0e0))))
end function

Julia

function code(x)
	return log(Float32(x + sqrt(Float32(Float32(x * x) - Float32(1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - single(1.0)))));
end

Alternative 1: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \cosh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (acosh x))

C

float code(float x) {
	return acoshf(x);
}

Julia

function code(x)
	return acosh(x)
end

MATLAB

function tmp = code(x)
	tmp = acosh(x);
end

Derivation

1. Initial program 51.0%

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

   1. lift-log.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. acosh-def-rev N/A

      \[\leadsto \color{blue}{\cosh^{-1} x} \]

   7. lower-acosh.f32 99.3

      \[\leadsto \color{blue}{\cosh^{-1} x} \]

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\cosh^{-1} x} \]
5. Add Preprocessing

Alternative 2: 5.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.25}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (/ -0.25 (* x x)))

C

float code(float x) {
	return -0.25f / (x * x);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (-0.25e0) / (x * x)
end function

Julia

function code(x)
	return Float32(Float32(-0.25) / Float32(x * x))
end

MATLAB

function tmp = code(x)
	tmp = single(-0.25) / (x * x);
end

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in x around inf

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

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

   5. log-rec N/A

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

   6. remove-double-neg N/A

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

   7. lower-log.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-1}{4}}{\color{blue}{{x}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 5.4%

   \[\leadsto \frac{-0.25}{\color{blue}{x \cdot x}} \]
9. Add Preprocessing

Developer Target 1: 99.3% 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 binary32
 (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0))))))

C

float code(float x) {
	return logf((x + (sqrtf((x - 1.0f)) * sqrtf((x + 1.0f)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + (sqrt((x - 1.0e0)) * sqrt((x + 1.0e0)))))
end function

Julia

function code(x)
	return log(Float32(x + Float32(sqrt(Float32(x - Float32(1.0))) * sqrt(Float32(x + Float32(1.0))))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + (sqrt((x - single(1.0))) * sqrt((x + single(1.0))))));
end

Reproduce

herbie shell --seed 2024340 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :alt
  (! :herbie-platform default (log (+ x (* (sqrt (- x 1)) (sqrt (+ x 1))))))

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