Rust f32::acosh

Percentage Accurate: 53.0% → 98.3%

Time: 10.6s

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: 53.0% 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: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

   1. log1p-expm1-u 54.3%

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

   2. expm1-undefine 54.3%

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

   3. add-exp-log 54.3%

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

   4. fma-neg 54.3%

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

   5. metadata-eval 54.3%

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

4. Applied egg-rr 54.3%

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

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. 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. unsub-neg 98.6%

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

   3. associate-*r/ 98.6%

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

   4. metadata-eval 98.6%

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

7. Simplified 98.6%

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

   1. sub-neg 98.6%

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

   2. distribute-lft-in 98.6%

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

   3. distribute-neg-frac2 98.6%

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

9. Applied egg-rr 98.6%

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

10. Taylor expanded in x around 0 98.6%

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

    1. div-sub 98.6%

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

    2. sub-neg 98.6%

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

    3. associate-/l* 98.6%

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

    4. *-inverses 98.6%

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

    5. metadata-eval 98.6%

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

    6. distribute-neg-frac 98.6%

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

    7. metadata-eval 98.6%

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

12. Simplified 98.6%

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

    1. add-log-exp 98.6%

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

    2. *-un-lft-identity 98.6%

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

    3. log-prod 98.6%

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

    4. metadata-eval 98.6%

       \[\leadsto \color{blue}{0} + \log \left(e^{\mathsf{log1p}\left(x \cdot 2 + \left(-1 + \frac{-0.5}{x}\right)\right)}\right) \]

    5. add-log-exp 98.6%

       \[\leadsto 0 + \color{blue}{\mathsf{log1p}\left(x \cdot 2 + \left(-1 + \frac{-0.5}{x}\right)\right)} \]

    6. +-commutative 98.6%

       \[\leadsto 0 + \mathsf{log1p}\left(\color{blue}{\left(-1 + \frac{-0.5}{x}\right) + x \cdot 2}\right) \]

    7. +-commutative 98.6%

       \[\leadsto 0 + \mathsf{log1p}\left(\color{blue}{\left(\frac{-0.5}{x} + -1\right)} + x \cdot 2\right) \]

    8. associate-+l+ 98.6%

       \[\leadsto 0 + \mathsf{log1p}\left(\color{blue}{\frac{-0.5}{x} + \left(-1 + x \cdot 2\right)}\right) \]

    9. +-commutative 98.6%

       \[\leadsto 0 + \mathsf{log1p}\left(\frac{-0.5}{x} + \color{blue}{\left(x \cdot 2 + -1\right)}\right) \]

    10. fma-define 98.6%

        \[\leadsto 0 + \mathsf{log1p}\left(\frac{-0.5}{x} + \color{blue}{\mathsf{fma}\left(x, 2, -1\right)}\right) \]

14. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{0 + \mathsf{log1p}\left(\frac{-0.5}{x} + \mathsf{fma}\left(x, 2, -1\right)\right)} \]
15. Step-by-step derivation

    1. +-lft-identity 98.6%

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

    2. log1p-undefine 98.7%

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

    3. +-commutative 98.7%

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

    4. associate-+l+ 98.7%

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

    5. fma-undefine 98.7%

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

    6. associate-+l+ 98.7%

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

    7. metadata-eval 98.7%

       \[\leadsto \log \left(\frac{-0.5}{x} + \left(x \cdot 2 + \color{blue}{0}\right)\right) \]

    8. +-rgt-identity 98.7%

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

16. Simplified 98.7%

    \[\leadsto \color{blue}{\log \left(\frac{-0.5}{x} + x \cdot 2\right)} \]
17. 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 54.3%

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

   1. fma-neg 54.3%

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

   2. metadata-eval 54.3%

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

   3. flip-+ 9.5%

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

   4. metadata-eval 9.5%

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

   5. fma-neg 9.5%

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

   6. div-inv 9.5%

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

   7. log-prod 9.5%

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

   8. pow2 9.5%

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

   9. add-sqr-sqrt 9.3%

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

   10. fma-neg 9.3%

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

   11. metadata-eval 9.3%

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

4. Applied egg-rr 9.3%

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

   1. log-rec 9.3%

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

   2. sub-neg 9.3%

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

   3. fma-undefine 9.2%

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

   4. unpow2 9.2%

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

   5. associate--r+ 11.7%

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

   6. +-inverses 11.7%

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

   7. metadata-eval 11.7%

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

   8. metadata-eval 11.7%

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

   9. neg-sub0 11.7%

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

6. Simplified 11.7%

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

7. Taylor expanded in x around inf 96.5%

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

Alternative 3: 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 54.3%

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

3. Taylor expanded in x around inf 96.5%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
4. 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 2024107 
(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)))))