Rust f32::acosh

Percentage Accurate: 53.9% → 97.0%

Time: 5.6s

Alternatives: 8

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: 53.9% 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: 97.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(2 \cdot x + -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log1p (+ (* 2.0 x) -1.0)))

C

float code(float x) {
	return log1pf(((2.0f * x) + -1.0f));
}

Julia

function code(x)
	return log1p(Float32(Float32(Float32(2.0) * x) + Float32(-1.0)))
end

Derivation

1. Initial program 55.0%

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

3. Taylor expanded in x around inf 97.5%

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

   1. log1p-expm1-u 97.5%

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

   2. expm1-undefine 97.5%

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

   3. add-exp-log 97.5%

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

   4. count-2 97.5%

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

   5. *-commutative 97.5%

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

   6. fma-neg 97.5%

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

   7. metadata-eval 97.5%

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

5. Applied egg-rr 97.5%

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

6. Taylor expanded in x around 0 97.5%

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

7. Final simplification 97.5%

   \[\leadsto \mathsf{log1p}\left(2 \cdot x + -1\right) \]
8. Add Preprocessing

Alternative 2: 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 55.0%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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: 44.1% 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 55.0%

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

3. Taylor expanded in x around inf 97.5%

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

4. Taylor expanded in x around 0 96.8%

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

6. Simplified 44.3%

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

Alternative 4: 11.2% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.9083333333333333 - \frac{2}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* x (- 0.9083333333333333 (/ 2.0 x))))

C

float code(float x) {
	return x * (0.9083333333333333f - (2.0f / x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x * (0.9083333333333333e0 - (2.0e0 / x))
end function

Julia

function code(x)
	return Float32(x * Float32(Float32(0.9083333333333333) - Float32(Float32(2.0) / x)))
end

MATLAB

function tmp = code(x)
	tmp = x * (single(0.9083333333333333) - (single(2.0) / x));
end

Derivation

1. Initial program 55.0%

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

   1. pow2 55.0%

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

   2. add-sqr-sqrt 55.0%

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

   3. pow2 55.0%

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

   4. pow-pow 55.0%

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

   5. metadata-eval 55.0%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt{x}\right)}^{\color{blue}{4}} - 1}\right) \]

4. Applied egg-rr 55.0%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{4}} - 1}\right) \]

5. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{-1}\right) + x \cdot \left({x}^{2} \cdot \left(0.075 \cdot \frac{{x}^{2}}{{\left(\sqrt{-1}\right)}^{5}} - 0.16666666666666666 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{3}}\right) + \frac{1}{\sqrt{-1}}\right)} \]

7. Simplified 11.3%

   \[\leadsto \color{blue}{-2 + x \cdot 0.9083333333333333} \]

8. Taylor expanded in x around inf 11.3%

   \[\leadsto \color{blue}{x \cdot \left(0.9083333333333333 - 2 \cdot \frac{1}{x}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 11.3%

      \[\leadsto x \cdot \left(0.9083333333333333 - \color{blue}{\frac{2 \cdot 1}{x}}\right) \]

   2. metadata-eval 11.3%

      \[\leadsto x \cdot \left(0.9083333333333333 - \frac{\color{blue}{2}}{x}\right) \]

10. Simplified 11.3%

    \[\leadsto \color{blue}{x \cdot \left(0.9083333333333333 - \frac{2}{x}\right)} \]
11. Add Preprocessing

Alternative 5: 11.2% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ -2 + x \cdot 0.9083333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 (+ -2.0 (* x 0.9083333333333333)))

C

float code(float x) {
	return -2.0f + (x * 0.9083333333333333f);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (-2.0e0) + (x * 0.9083333333333333e0)
end function

Julia

function code(x)
	return Float32(Float32(-2.0) + Float32(x * Float32(0.9083333333333333)))
end

MATLAB

function tmp = code(x)
	tmp = single(-2.0) + (x * single(0.9083333333333333));
end

Derivation

1. Initial program 55.0%

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

   1. pow2 55.0%

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

   2. add-sqr-sqrt 55.0%

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

   3. pow2 55.0%

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

   4. pow-pow 55.0%

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

   5. metadata-eval 55.0%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt{x}\right)}^{\color{blue}{4}} - 1}\right) \]

4. Applied egg-rr 55.0%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{4}} - 1}\right) \]

5. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{-1}\right) + x \cdot \left({x}^{2} \cdot \left(0.075 \cdot \frac{{x}^{2}}{{\left(\sqrt{-1}\right)}^{5}} - 0.16666666666666666 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{3}}\right) + \frac{1}{\sqrt{-1}}\right)} \]

7. Simplified 11.3%

   \[\leadsto \color{blue}{-2 + x \cdot 0.9083333333333333} \]
8. Add Preprocessing

Alternative 6: 11.2% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ -2 + x \cdot 0.8333333333333334 \end{array} \]

FPCore

(FPCore (x) :precision binary32 (+ -2.0 (* x 0.8333333333333334)))

C

float code(float x) {
	return -2.0f + (x * 0.8333333333333334f);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (-2.0e0) + (x * 0.8333333333333334e0)
end function

Julia

function code(x)
	return Float32(Float32(-2.0) + Float32(x * Float32(0.8333333333333334)))
end

MATLAB

function tmp = code(x)
	tmp = single(-2.0) + (x * single(0.8333333333333334));
end

Derivation

1. Initial program 55.0%

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

   1. pow2 55.0%

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

   2. add-sqr-sqrt 55.0%

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

   3. pow2 55.0%

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

   4. pow-pow 55.0%

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

   5. metadata-eval 55.0%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt{x}\right)}^{\color{blue}{4}} - 1}\right) \]

4. Applied egg-rr 55.0%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{4}} - 1}\right) \]

5. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{-1}\right) + x \cdot \left(-0.16666666666666666 \cdot \frac{{x}^{2}}{{\left(\sqrt{-1}\right)}^{3}} + \frac{1}{\sqrt{-1}}\right)} \]

7. Simplified 11.3%

   \[\leadsto \color{blue}{-2 + x \cdot 0.8333333333333334} \]
8. Add Preprocessing

Alternative 7: 11.4% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.8333333333333334 \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* x 0.8333333333333334))

C

float code(float x) {
	return x * 0.8333333333333334f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x * 0.8333333333333334e0
end function

Julia

function code(x)
	return Float32(x * Float32(0.8333333333333334))
end

MATLAB

function tmp = code(x)
	tmp = x * single(0.8333333333333334);
end

Derivation

1. Initial program 55.0%

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

   1. pow2 55.0%

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

   2. add-sqr-sqrt 55.0%

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

   3. pow2 55.0%

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

   4. pow-pow 55.0%

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

   5. metadata-eval 55.0%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt{x}\right)}^{\color{blue}{4}} - 1}\right) \]

4. Applied egg-rr 55.0%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{4}} - 1}\right) \]

5. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{-1}\right) + x \cdot \left(-0.16666666666666666 \cdot \frac{{x}^{2}}{{\left(\sqrt{-1}\right)}^{3}} + \frac{1}{\sqrt{-1}}\right)} \]

7. Simplified 11.3%

   \[\leadsto \color{blue}{-2 + x \cdot 0.8333333333333334} \]

8. Taylor expanded in x around inf 11.3%

   \[\leadsto \color{blue}{x \cdot \left(0.8333333333333334 - 2 \cdot \frac{1}{x}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 11.3%

      \[\leadsto x \cdot \left(0.8333333333333334 - \color{blue}{\frac{2 \cdot 1}{x}}\right) \]

   2. metadata-eval 11.3%

      \[\leadsto x \cdot \left(0.8333333333333334 - \frac{\color{blue}{2}}{x}\right) \]

10. Simplified 11.3%

    \[\leadsto \color{blue}{x \cdot \left(0.8333333333333334 - \frac{2}{x}\right)} \]

11. Taylor expanded in x around inf 11.3%

    \[\leadsto x \cdot \color{blue}{0.8333333333333334} \]
12. Add Preprocessing

Alternative 8: 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 55.0%

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

   1. pow2 55.0%

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

   2. add-sqr-sqrt 55.0%

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

   3. pow2 55.0%

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

   4. pow-pow 55.0%

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

   5. metadata-eval 55.0%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt{x}\right)}^{\color{blue}{4}} - 1}\right) \]

4. Applied egg-rr 55.0%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{4}} - 1}\right) \]

5. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{-1}\right) + x \cdot \left(-0.16666666666666666 \cdot \frac{{x}^{2}}{{\left(\sqrt{-1}\right)}^{3}} + \frac{1}{\sqrt{-1}}\right)} \]

7. Simplified 11.3%

   \[\leadsto \color{blue}{-2 + x \cdot 0.8333333333333334} \]

8. Taylor expanded in x around 0 3.1%

   \[\leadsto \color{blue}{-2} \]
9. 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 2024146 
(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)))))