Rust f32::acosh

Percentage Accurate: 53.4% → 98.2%

Time: 6.9s

Alternatives: 5

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: 53.4% 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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 49.9%

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

2. Taylor expanded in x around inf 98.2%

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

   1. *-commutative 98.2%

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

   2. associate-*r/ 98.2%

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

   3. metadata-eval 98.2%

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

4. Simplified 98.2%

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

5. Final simplification 98.2%

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

Alternative 2: 97.5% 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 49.9%

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

   1. flip-+ 8.7%

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

   2. log-div 8.7%

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

   3. add-sqr-sqrt 8.3%

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

   4. associate--r- 10.5%

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

   5. +-commutative 10.5%

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

   6. *-un-lft-identity 10.5%

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

   7. *-un-lft-identity 10.5%

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

   8. distribute-rgt-out-- 10.5%

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

   9. pow2 10.5%

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

   10. metadata-eval 10.5%

       \[\leadsto \log \left(1 + {x}^{2} \cdot \color{blue}{0}\right) - \log \left(x - \sqrt{x \cdot x - 1}\right) \]

   11. fma-neg 10.4%

       \[\leadsto \log \left(1 + {x}^{2} \cdot 0\right) - \log \left(x - \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   12. metadata-eval 10.4%

       \[\leadsto \log \left(1 + {x}^{2} \cdot 0\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right) \]

3. Applied egg-rr 10.4%

   \[\leadsto \color{blue}{\log \left(1 + {x}^{2} \cdot 0\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)} \]
4. Step-by-step derivation

   1. mul0-rgt 10.4%

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

   2. metadata-eval 10.4%

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

   3. metadata-eval 10.4%

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

   4. sub0-neg 10.4%

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

5. Simplified 10.4%

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

6. Taylor expanded in x around inf 97.4%

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

7. Final simplification 97.4%

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

Alternative 3: 96.8% 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 49.9%

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

2. Taylor expanded in x around inf 96.7%

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

3. Final simplification 96.7%

   \[\leadsto \log \left(x + x\right) \]

Alternative 4: 44.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log x \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

function code(x)
	return log(x)
end

MATLAB

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

Derivation

1. Initial program 49.9%

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

   1. add-exp-log 49.8%

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

   2. expm1-def 49.8%

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

   3. pow2 49.8%

      \[\leadsto \log \left(x + \sqrt{\mathsf{expm1}\left(\log \color{blue}{\left({x}^{2}\right)}\right)}\right) \]

   4. log-pow 49.8%

      \[\leadsto \log \left(x + \sqrt{\mathsf{expm1}\left(\color{blue}{2 \cdot \log x}\right)}\right) \]

   5. *-commutative 49.8%

      \[\leadsto \log \left(x + \sqrt{\mathsf{expm1}\left(\color{blue}{\log x \cdot 2}\right)}\right) \]

3. Applied egg-rr 49.8%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{\mathsf{expm1}\left(\log x \cdot 2\right)}}\right) \]

4. Taylor expanded in x around inf 44.1%

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

   1. mul-1-neg 44.1%

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

   2. log-rec 44.1%

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

   3. remove-double-neg 44.1%

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

6. Simplified 44.1%

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

7. Final simplification 44.1%

   \[\leadsto \log x \]

Alternative 5: 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 49.9%

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

2. Taylor expanded in x around inf 98.2%

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

   1. *-commutative 98.2%

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

   2. associate-*r/ 98.2%

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

   3. metadata-eval 98.2%

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

4. Simplified 98.2%

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

6. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\log 0 - \log 0} \]
7. Step-by-step derivation

   1. +-inverses 6.1%

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

8. Simplified 6.1%

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

9. Final simplification 6.1%

   \[\leadsto 0 \]

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 2023301 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

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