Rust f32::acosh

Percentage Accurate: 52.7% → 96.7%

Time: 6.1s

Alternatives: 6

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: 52.7% 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: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log 2 + \log x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (+ (log 2.0) (log x)))

C

float code(float x) {
	return logf(2.0f) + logf(x);
}

Fortran

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

Julia

function code(x)
	return Float32(log(Float32(2.0)) + log(x))
end

MATLAB

function tmp = code(x)
	tmp = log(single(2.0)) + log(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 97.6%

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

   1. mul-1-neg 97.6%

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

   2. log-rec 97.6%

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

   3. remove-double-neg 97.6%

      \[\leadsto \log 2 + \color{blue}{\log x} \]

5. Simplified 97.6%

   \[\leadsto \color{blue}{\log 2 + \log x} \]

6. Final simplification 97.6%

   \[\leadsto \log 2 + \log x \]
7. Add Preprocessing

Alternative 2: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = log((x + (x - (single(0.5) / 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%

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

   1. associate-*r/ 97.2%

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

   2. metadata-eval 97.2%

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

5. Simplified 97.2%

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

6. Final simplification 97.2%

   \[\leadsto \log \left(x + \left(x - \frac{0.5}{x}\right)\right) \]
7. 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 48.8%

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

3. Taylor expanded in x around inf 96.3%

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

4. Final simplification 96.3%

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

Alternative 4: 44.2% 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 48.8%

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

3. Taylor expanded in x around inf 97.2%

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

   1. associate-*r/ 97.2%

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

   2. metadata-eval 97.2%

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

5. Simplified 97.2%

   \[\leadsto \log \left(x + \color{blue}{\left(x - \frac{0.5}{x}\right)}\right) \]
6. Step-by-step derivation

   1. *-un-lft-identity 97.2%

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

   2. fma-def 97.2%

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

   3. clear-num 97.2%

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

   4. div-inv 97.2%

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

   5. metadata-eval 97.2%

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

   6. *-commutative 97.2%

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

   7. count-2 97.2%

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

   8. flip-+ -0.0%

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

   9. +-inverses -0.0%

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

   10. +-inverses -0.0%

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

   11. +-inverses -0.0%

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

   12. +-inverses -0.0%

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

   13. clear-num -0.0%

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

   14. +-inverses -0.0%

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

   15. +-inverses -0.0%

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

7. Applied egg-rr -0.0%

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

9. Simplified 44.1%

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

10. Taylor expanded in x around inf 44.5%

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

    1. mul-1-neg 44.5%

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

    2. log-rec 44.5%

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

    3. remove-double-neg 44.5%

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

12. Simplified 44.5%

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

13. Final simplification 44.5%

    \[\leadsto \log x \]
14. Add Preprocessing

Alternative 5: 3.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -0.03125 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -0.03125)

C

float code(float x) {
	return -0.03125f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -0.03125e0
end function

Julia

function code(x)
	return Float32(-0.03125)
end

MATLAB

function tmp = code(x)
	tmp = single(-0.03125);
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%

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

   1. associate-*r/ 97.2%

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

   2. metadata-eval 97.2%

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

5. Simplified 97.2%

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

6. Taylor expanded in x around inf 98.5%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - \left(0.03125 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

8. Simplified 3.2%

   \[\leadsto \color{blue}{-0.03125} \]

9. Final simplification 3.2%

   \[\leadsto -0.03125 \]
10. Add Preprocessing

Alternative 6: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2.1458333333333335 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.1458333333333335)

C

float code(float x) {
	return 2.1458333333333335f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.1458333333333335e0
end function

Julia

function code(x)
	return Float32(2.1458333333333335)
end

MATLAB

function tmp = code(x)
	tmp = single(2.1458333333333335);
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. add-cbrt-cube 34.9%

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

   2. pow1/3 34.6%

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

   3. log-pow 34.7%

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

   4. pow3 34.7%

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

   5. log-pow 48.4%

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

   6. fma-neg 48.4%

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

   7. metadata-eval 48.4%

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

4. Applied egg-rr 48.4%

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

5. Taylor expanded in x around -inf -0.0%

   \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(\log -0.5 + \left(\log \left(\frac{-1}{x}\right) + \left(0.052083333333333336 \cdot \frac{1}{{x}^{6}} + \left(0.09375 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}\right) \]

7. Simplified 22.1%

   \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \color{blue}{2.1458333333333335}\right) \]

8. Final simplification 22.1%

   \[\leadsto 2.1458333333333335 \]
9. Add Preprocessing

Developer target: 99.0% 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 2024017 
(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)))))