Rust f32::acosh

Percentage Accurate: 53.2% → 98.2%

Time: 6.7s

Alternatives: 3

Speedup: 2.0×

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: 53.2% 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: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log1p (* x (- 2.0 (/ (+ 1.0 (/ 0.5 x)) x)))))

C

float code(float x) {
	return log1pf((x * (2.0f - ((1.0f + (0.5f / x)) / x))));
}

Julia

function code(x)
	return log1p(Float32(x * Float32(Float32(2.0) - Float32(Float32(Float32(1.0) + Float32(Float32(0.5) / x)) / x))))
end

Derivation

1. Initial program 48.8%

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

   1. log1p-expm1-u 48.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]

   2. expm1-undefine 48.8%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(x + \sqrt{x \cdot x - 1}\right)} - 1}\right) \]

   3. add-exp-log 48.8%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x + \sqrt{x \cdot x - 1}\right)} - 1\right) \]

   4. fmm-def 48.8%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right) - 1\right) \]

   5. metadata-eval 48.8%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right) - 1\right) \]

4. Applied egg-rr 48.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - 1\right)} \]

5. Taylor expanded in x around inf 98.6%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + -1 \cdot \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]

6. Taylor expanded in x around -inf 98.6%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{-1 \cdot \left(x \cdot \left(\left(\frac{0.5}{{x}^{2}} + \frac{1}{x}\right) - 2\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate-*r* 98.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(\frac{0.5}{{x}^{2}} + \frac{1}{x}\right) - 2\right)}\right) \]

   2. neg-mul-1 98.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(-x\right)} \cdot \left(\left(\frac{0.5}{{x}^{2}} + \frac{1}{x}\right) - 2\right)\right) \]

   3. unpow2 98.6%

      \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1}{x}\right) - 2\right)\right) \]

   4. associate-/r* 98.6%

      \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\left(\color{blue}{\frac{\frac{0.5}{x}}{x}} + \frac{1}{x}\right) - 2\right)\right) \]

   5. metadata-eval 98.6%

      \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\left(\frac{\frac{\color{blue}{1 \cdot 0.5}}{x}}{x} + \frac{1}{x}\right) - 2\right)\right) \]

   6. associate-*l/ 98.6%

      \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\left(\frac{\color{blue}{\frac{1}{x} \cdot 0.5}}{x} + \frac{1}{x}\right) - 2\right)\right) \]

   7. associate-*r/ 98.6%

      \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\left(\color{blue}{\frac{1}{x} \cdot \frac{0.5}{x}} + \frac{1}{x}\right) - 2\right)\right) \]

   8. *-rgt-identity 98.6%

      \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\left(\frac{1}{x} \cdot \frac{0.5}{x} + \color{blue}{\frac{1}{x} \cdot 1}\right) - 2\right)\right) \]

   9. distribute-lft-in 98.6%

      \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\color{blue}{\frac{1}{x} \cdot \left(\frac{0.5}{x} + 1\right)} - 2\right)\right) \]

   10. +-commutative 98.6%

       \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\frac{1}{x} \cdot \color{blue}{\left(1 + \frac{0.5}{x}\right)} - 2\right)\right) \]

   11. associate-*l/ 98.6%

       \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\color{blue}{\frac{1 \cdot \left(1 + \frac{0.5}{x}\right)}{x}} - 2\right)\right) \]

   12. *-lft-identity 98.6%

       \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\frac{\color{blue}{1 + \frac{0.5}{x}}}{x} - 2\right)\right) \]

   13. sub-neg 98.6%

       \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \color{blue}{\left(\frac{1 + \frac{0.5}{x}}{x} + \left(-2\right)\right)}\right) \]

   14. metadata-eval 98.6%

       \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \left(\frac{1 + \frac{0.5}{x}}{x} + \color{blue}{-2}\right)\right) \]

   15. +-commutative 98.6%

       \[\leadsto \mathsf{log1p}\left(\left(-x\right) \cdot \color{blue}{\left(-2 + \frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]

8. Simplified 98.6%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(-x\right) \cdot \left(-2 + \frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]

9. Taylor expanded in x around inf 98.6%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + -1 \cdot \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]
10. Step-by-step derivation

    1. mul-1-neg 98.6%

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + \color{blue}{\left(-\frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right)\right) \]

    2. associate-*r/ 98.6%

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + \left(-\frac{1 + \color{blue}{\frac{0.5 \cdot 1}{x}}}{x}\right)\right)\right) \]

    3. metadata-eval 98.6%

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + \left(-\frac{1 + \frac{\color{blue}{0.5}}{x}}{x}\right)\right)\right) \]

    4. unsub-neg 98.6%

       \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]

11. Simplified 98.6%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]

12. Final simplification 98.6%

    \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)\right) \]
13. Add Preprocessing

Alternative 2: 96.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (log (+ x x)))

C

float code(float x) {
	return logf((x + x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + x))
end function

Julia

function code(x)
	return log(Float32(x + x))
end

MATLAB

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

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in x around inf 97.2%

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

4. Final simplification 97.2%

   \[\leadsto \log \left(x + x\right) \]
5. Add Preprocessing

Alternative 3: 6.1% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 0.0)

C

float code(float x) {
	return 0.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.0e0
end function

Julia

function code(x)
	return Float32(0.0)
end

MATLAB

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

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in x around inf 97.2%

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

   1. add-sqr-sqrt 97.2%

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

   2. log-prod 97.2%

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

5. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\log \left(\frac{0}{0}\right) + \log \left(\frac{0}{0}\right)} \]

7. Simplified 6.1%

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

8. Final simplification 6.1%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer target: 99.2% 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 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 2024131 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

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

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