Rust f32::acosh

Percentage Accurate: 53.0% → 98.1%

Time: 5.5s

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: 53.0% 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: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log 2 + \left(\log x + \frac{-0.25}{{x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (+ (log 2.0) (+ (log x) (/ -0.25 (pow x 2.0)))))

C

float code(float x) {
	return logf(2.0f) + (logf(x) + (-0.25f / powf(x, 2.0f)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(2.0e0) + (log(x) + ((-0.25e0) / (x ** 2.0e0)))
end function

Julia

function code(x)
	return Float32(log(Float32(2.0)) + Float32(log(x) + Float32(Float32(-0.25) / (x ^ Float32(2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = log(single(2.0)) + (log(x) + (single(-0.25) / (x ^ single(2.0))));
end

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around inf 97.5%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. associate--l+ 97.5%

      \[\leadsto \color{blue}{\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) - 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. sub-neg 97.5%

      \[\leadsto \log 2 + \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   3. mul-1-neg 97.5%

      \[\leadsto \log 2 + \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   4. log-rec 97.5%

      \[\leadsto \log 2 + \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   5. remove-double-neg 97.5%

      \[\leadsto \log 2 + \left(\color{blue}{\log x} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   6. associate-*r/ 97.5%

      \[\leadsto \log 2 + \left(\log x + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) \]

   7. metadata-eval 97.5%

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

   8. distribute-neg-frac 97.5%

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

   9. metadata-eval 97.5%

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

5. Simplified 97.5%

   \[\leadsto \color{blue}{\log 2 + \left(\log x + \frac{-0.25}{{x}^{2}}\right)} \]
6. Add Preprocessing

Alternative 2: 97.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (- (log (/ 0.5 x))))

C

float code(float x) {
	return -logf((0.5f / x));
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 50.3%

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

   1. flip-+ 6.7%

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

   2. div-inv 6.7%

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

   3. log-prod 6.7%

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

   4. pow2 6.7%

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

   5. add-sqr-sqrt 6.7%

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

   6. fma-neg 6.7%

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

   7. metadata-eval 6.7%

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

   8. fma-neg 6.7%

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

   9. metadata-eval 6.7%

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

4. Applied egg-rr 6.7%

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

   1. log-rec 6.7%

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

   2. sub-neg 6.7%

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

   3. fma-undefine 6.7%

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

   4. unpow2 6.7%

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

   5. associate--r+ 9.2%

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

   6. +-inverses 9.2%

      \[\leadsto \log \left(\color{blue}{0} - -1\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   7. metadata-eval 9.2%

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

   8. metadata-eval 9.2%

      \[\leadsto \color{blue}{0} - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   9. neg-sub0 9.2%

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

6. Simplified 9.2%

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

7. Taylor expanded in x around inf 97.3%

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

Alternative 3: 97.0% 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 50.3%

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

3. Taylor expanded in x around inf 95.4%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
4. Add Preprocessing

Alternative 4: 3.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -2.0)

C

float code(float x) {
	return -2.0f;
}

Fortran

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

Julia

function code(x)
	return Float32(-2.0)
end

MATLAB

function tmp = code(x)
	tmp = single(-2.0);
end

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around inf 97.5%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. associate--l+ 97.5%

      \[\leadsto \color{blue}{\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) - 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. sub-neg 97.5%

      \[\leadsto \log 2 + \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

   3. mul-1-neg 97.5%

      \[\leadsto \log 2 + \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   4. log-rec 97.5%

      \[\leadsto \log 2 + \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   5. remove-double-neg 97.5%

      \[\leadsto \log 2 + \left(\color{blue}{\log x} + \left(-0.25 \cdot \frac{1}{{x}^{2}}\right)\right) \]

   6. associate-*r/ 97.5%

      \[\leadsto \log 2 + \left(\log x + \left(-\color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) \]

   7. metadata-eval 97.5%

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

   8. distribute-neg-frac 97.5%

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

   9. metadata-eval 97.5%

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

5. Simplified 97.5%

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

6. Taylor expanded in x around inf 96.4%

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

8. Simplified 3.1%

   \[\leadsto \color{blue}{-2} \]
9. Add Preprocessing

Developer target: 99.3% 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 2024096 
(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)))))