Rust f32::acosh

Percentage Accurate: 53.6% → 98.2%

Time: 7.1s

Alternatives: 7

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: 53.6% 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

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

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 56.1%

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

3. Taylor expanded in x around inf 98.8%

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

   1. associate-*r/ 98.8%

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

   2. metadata-eval 98.8%

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

5. Simplified 98.8%

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

   1. unpow2 98.8%

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

7. Applied egg-rr 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. *-commutative 98.8%

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

   3. log-prod 98.8%

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

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\log \left(x \cdot \mathsf{fma}\left(-0.5, {x}^{-2}, 2\right)\right) + 0} \]
10. Step-by-step derivation

    1. +-rgt-identity 98.8%

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

    2. fma-define 98.8%

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

    3. +-commutative 98.8%

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

    4. distribute-lft-out 98.8%

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

    5. fma-define 98.8%

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

    6. *-commutative 98.8%

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

    7. associate-*l* 98.8%

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

    8. metadata-eval 98.8%

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

    9. pow-plus 98.8%

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

    10. metadata-eval 98.8%

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

    11. unpow-1 98.8%

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

    12. distribute-lft-neg-in 98.8%

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

    13. associate-*r/ 98.8%

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

    14. metadata-eval 98.8%

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

    15. distribute-neg-frac 98.8%

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

    16. metadata-eval 98.8%

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

11. Simplified 98.8%

    \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(x, 2, \frac{-0.5}{x}\right)\right)} \]
12. Step-by-step derivation

    1. fma-undefine 98.8%

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

    2. add-exp-log 98.8%

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

    3. sum-log 97.9%

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

    4. add-sqr-sqrt 97.0%

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

    5. unpow2 97.0%

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

    6. +-commutative 97.0%

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

    7. +-commutative 97.0%

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

    8. +-commutative 97.0%

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

    9. unpow2 97.0%

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

    10. add-sqr-sqrt 97.9%

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

    11. sum-log 98.8%

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

    12. add-exp-log 98.8%

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

13. Applied egg-rr 98.8%

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

Alternative 2: 96.7% 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 56.1%

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

3. Taylor expanded in x around inf 97.5%

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

Alternative 3: 44.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + 2.125\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log (+ x 2.125)))

C

float code(float x) {
	return logf((x + 2.125f));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + 2.125e0))
end function

Julia

function code(x)
	return log(Float32(x + Float32(2.125)))
end

MATLAB

function tmp = code(x)
	tmp = log((x + single(2.125)));
end

Derivation

1. Initial program 56.1%

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

   1. add-exp-log 55.9%

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

   2. expm1-define 55.9%

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

   3. pow2 55.9%

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

   4. log-pow 55.9%

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

4. Applied egg-rr 55.9%

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

5. Taylor expanded in x around -inf 1.4%

   \[\leadsto \log \left(x + \color{blue}{-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)}\right) \]

7. Simplified 44.0%

   \[\leadsto \log \left(x + \color{blue}{2.125}\right) \]
8. Add Preprocessing

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 56.1%

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

   1. add-exp-log 55.9%

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

   2. expm1-define 55.9%

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

   3. pow2 55.9%

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

   4. log-pow 55.9%

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

4. Applied egg-rr 55.9%

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

5. Taylor expanded in x around -inf 1.4%

   \[\leadsto \log \left(x + \color{blue}{-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)}\right) \]

7. Simplified 44.0%

   \[\leadsto \log \left(x + \color{blue}{2.125}\right) \]

8. Taylor expanded in x around inf 43.9%

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

   1. log-rec 43.9%

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

   2. distribute-rgt-neg-in 43.9%

      \[\leadsto \color{blue}{--1 \cdot \log x} \]

   3. mul-1-neg 43.9%

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

   4. remove-double-neg 43.9%

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

10. Simplified 43.9%

    \[\leadsto \color{blue}{\log x} \]
11. Add Preprocessing

Alternative 5: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2.09375 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.09375)

C

float code(float x) {
	return 2.09375f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.09375e0
end function

Julia

function code(x)
	return Float32(2.09375)
end

MATLAB

function tmp = code(x)
	tmp = single(2.09375);
end

Derivation

1. Initial program 56.1%

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

   1. add-exp-log 55.9%

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

   2. expm1-define 55.9%

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

   3. pow2 55.9%

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

   4. log-pow 55.9%

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

4. Applied egg-rr 55.9%

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

5. Taylor expanded in x around inf 98.6%

   \[\leadsto \color{blue}{\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) + -1 \cdot \frac{0.25 + 0.09375 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)} \]

7. Simplified 22.4%

   \[\leadsto \color{blue}{2.09375} \]
8. Add Preprocessing

Alternative 6: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2.03125 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.03125)

C

float code(float x) {
	return 2.03125f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.03125e0
end function

Julia

function code(x)
	return Float32(2.03125)
end

MATLAB

function tmp = code(x)
	tmp = single(2.03125);
end

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in x around inf 98.8%

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

   1. associate-*r/ 98.8%

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

   2. metadata-eval 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in x around inf 98.3%

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

8. Simplified 22.4%

   \[\leadsto \color{blue}{2.03125} \]
9. Add Preprocessing

Alternative 7: 21.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.1458333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.1458333333333333)

C

float code(float x) {
	return 1.1458333333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.1458333333333333e0
end function

Julia

function code(x)
	return Float32(1.1458333333333333)
end

MATLAB

function tmp = code(x)
	tmp = single(1.1458333333333333);
end

Derivation

1. Initial program 56.1%

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

   1. add-exp-log 55.9%

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

   2. expm1-define 55.9%

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

   3. pow2 55.9%

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

   4. log-pow 55.9%

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

4. Applied egg-rr 55.9%

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

5. Taylor expanded in x around inf 98.8%

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

7. Simplified 21.3%

   \[\leadsto \color{blue}{1.1458333333333333} \]
8. Add Preprocessing

Developer Target 1: 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 2024147 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :alt
  (! :herbie-platform default (log (+ x (* (sqrt (- x 1)) (sqrt (+ x 1))))))

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