Rust f32::acosh

Percentage Accurate: 53.7% → 98.0%

Time: 5.1s

Alternatives: 4

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: 53.7% 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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (log (* x (- 2.0 (/ 0.5 (pow x 2.0))))))

C

float code(float x) {
	return logf((x * (2.0f - (0.5f / powf(x, 2.0f)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x * (2.0e0 - (0.5e0 / (x ** 2.0e0)))))
end function

Julia

function code(x)
	return log(Float32(x * Float32(Float32(2.0) - Float32(Float32(0.5) / (x ^ Float32(2.0))))))
end

MATLAB

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around inf 98.5%

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

   1. associate-*r/ 98.5%

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

   2. metadata-eval 98.5%

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

5. Simplified 98.5%

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

6. Final simplification 98.5%

   \[\leadsto \log \left(x \cdot \left(2 - \frac{0.5}{{x}^{2}}\right)\right) \]
7. Add Preprocessing

Alternative 2: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{fma}\left(x, 2, -1\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log1p (fma x 2.0 -1.0)))

C

float code(float x) {
	return log1pf(fmaf(x, 2.0f, -1.0f));
}

Julia

function code(x)
	return log1p(fma(x, Float32(2.0), Float32(-1.0)))
end

Derivation

1. Initial program 56.5%

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

   1. add-cbrt-cube 40.8%

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

   2. pow1/3 40.4%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 40.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   4. pow3 40.4%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(x + \sqrt{x \cdot x - 1}\right)}^{3}\right)} \]

   5. log-pow 56.0%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   6. fma-neg 56.0%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right) \]

   7. metadata-eval 56.0%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right) \]

4. Applied egg-rr 56.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

5. Taylor expanded in x around inf 95.7%

   \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{x}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* 96.5%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(x + x\right)} \]

   2. metadata-eval 96.5%

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

   3. *-un-lft-identity 96.5%

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

   4. flip-+ -0.0%

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

   5. +-inverses -0.0%

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

   6. metadata-eval -0.0%

      \[\leadsto \log \left(\frac{\color{blue}{{0}^{3}}}{x - x}\right) \]

   7. +-inverses -0.0%

      \[\leadsto \log \left(\frac{{\color{blue}{\left(x \cdot x - x \cdot x\right)}}^{3}}{x - x}\right) \]

   8. +-inverses -0.0%

      \[\leadsto \log \left(\frac{{\left(x \cdot x - x \cdot x\right)}^{3}}{\color{blue}{0}}\right) \]

   9. metadata-eval -0.0%

      \[\leadsto \log \left(\frac{{\left(x \cdot x - x \cdot x\right)}^{3}}{\color{blue}{{0}^{3}}}\right) \]

   10. +-inverses -0.0%

       \[\leadsto \log \left(\frac{{\left(x \cdot x - x \cdot x\right)}^{3}}{{\color{blue}{\left(x \cdot x - x \cdot x\right)}}^{3}}\right) \]

   11. cube-div -0.0%

       \[\leadsto \log \color{blue}{\left({\left(\frac{x \cdot x - x \cdot x}{x \cdot x - x \cdot x}\right)}^{3}\right)} \]

   12. +-inverses -0.0%

       \[\leadsto \log \left({\left(\frac{x \cdot x - x \cdot x}{\color{blue}{0}}\right)}^{3}\right) \]

   13. +-inverses -0.0%

       \[\leadsto \log \left({\left(\frac{x \cdot x - x \cdot x}{\color{blue}{x - x}}\right)}^{3}\right) \]

   14. flip-+ 13.6%

       \[\leadsto \log \left({\color{blue}{\left(x + x\right)}}^{3}\right) \]

   15. log-pow 26.0%

       \[\leadsto \color{blue}{3 \cdot \log \left(x + x\right)} \]

   16. log1p-expm1-u 13.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \log \left(x + x\right)\right)\right)} \]

   17. expm1-undefine 13.6%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{3 \cdot \log \left(x + x\right)} - 1}\right) \]

7. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(x, 2, -1\right)\right)} \]

8. Final simplification 96.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{fma}\left(x, 2, -1\right)\right) \]
9. Add Preprocessing

Alternative 3: 96.6% 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 56.5%

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

3. Taylor expanded in x around inf 96.5%

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

4. Final simplification 96.5%

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

Alternative 4: 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 56.5%

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

   1. add-cbrt-cube 40.8%

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

   2. pow1/3 40.4%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 40.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   4. pow3 40.4%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(x + \sqrt{x \cdot x - 1}\right)}^{3}\right)} \]

   5. log-pow 56.0%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   6. fma-neg 56.0%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right) \]

   7. metadata-eval 56.0%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right) \]

4. Applied egg-rr 56.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

5. Taylor expanded in x around inf 95.7%

   \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{x}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* 96.5%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(x + x\right)} \]

   2. metadata-eval 96.5%

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

   3. *-un-lft-identity 96.5%

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

   4. flip-+ -0.0%

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

   5. difference-of-squares -0.0%

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

   6. +-inverses -0.0%

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

   7. +-inverses -0.0%

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

   8. +-inverses -0.0%

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

   9. +-inverses -0.0%

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

   10. associate-*r/ -0.0%

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

   11. +-inverses -0.0%

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

   12. +-inverses -0.0%

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

   13. flip-+ 17.9%

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

   14. sum-log 28.0%

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

7. Applied egg-rr -0.0%

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

9. Simplified 6.1%

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

10. Final simplification 6.1%

    \[\leadsto 0 \]
11. 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 2024080 
(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)))))