Rust f32::acosh

Percentage Accurate: 52.9% → 98.1%

Time: 5.0s

Alternatives: 4

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: 52.9% 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: 98.1% 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 49.9%

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

2. Taylor expanded in x around inf 97.7%

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

   1. associate-*r/ 97.7%

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

   2. metadata-eval 97.7%

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

4. Simplified 97.7%

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

5. Final simplification 97.7%

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

Alternative 2: 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 95.9%

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

3. Final simplification 95.9%

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

Alternative 3: 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 49.9%

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

   1. add-cube-cbrt 49.9%

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

   2. pow3 49.9%

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

   3. pow1/3 49.7%

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

   4. pow1/2 49.7%

      \[\leadsto \log \left(x + {\left({\color{blue}{\left({\left(x \cdot x - 1\right)}^{0.5}\right)}}^{0.3333333333333333}\right)}^{3}\right) \]

   5. pow-pow 49.7%

      \[\leadsto \log \left(x + {\color{blue}{\left({\left(x \cdot x - 1\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}}^{3}\right) \]

   6. fma-neg 49.7%

      \[\leadsto \log \left(x + {\left({\color{blue}{\left(\mathsf{fma}\left(x, x, -1\right)\right)}}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}^{3}\right) \]

   7. metadata-eval 49.7%

      \[\leadsto \log \left(x + {\left({\left(\mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}^{3}\right) \]

   8. metadata-eval 49.7%

      \[\leadsto \log \left(x + {\left({\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3}\right) \]

3. Applied egg-rr 49.7%

   \[\leadsto \log \left(x + \color{blue}{{\left({\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{0.16666666666666666}\right)}^{3}}\right) \]

4. Taylor expanded in x around -inf 3.0%

   \[\leadsto \log \color{blue}{\left(0.16666666666666666 \cdot \left({1}^{0.3333333333333333} \cdot \frac{1}{x}\right) + \left(-1 \cdot x + \left(0.3333333333333333 \cdot \frac{1}{x} + x\right)\right)\right)} \]

6. Simplified 44.5%

   \[\leadsto \log \color{blue}{\left(0.041666666666666664 + \left(1.3333333333333333 + x\right)\right)} \]

7. Taylor expanded in x around inf 44.2%

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

   1. mul-1-neg 44.2%

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

   2. log-rec 44.2%

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

   3. remove-double-neg 44.2%

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

9. Simplified 44.2%

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

10. Final simplification 44.2%

    \[\leadsto \log x \]

Alternative 4: 21.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.3333333333333333)

C

float code(float x) {
	return 1.3333333333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.3333333333333333e0
end function

Julia

function code(x)
	return Float32(1.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = single(1.3333333333333333);
end

Derivation

1. Initial program 49.9%

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

   1. add-cube-cbrt 49.9%

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

   2. pow3 49.9%

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

   3. pow1/3 49.7%

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

   4. pow1/2 49.7%

      \[\leadsto \log \left(x + {\left({\color{blue}{\left({\left(x \cdot x - 1\right)}^{0.5}\right)}}^{0.3333333333333333}\right)}^{3}\right) \]

   5. pow-pow 49.7%

      \[\leadsto \log \left(x + {\color{blue}{\left({\left(x \cdot x - 1\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}}^{3}\right) \]

   6. fma-neg 49.7%

      \[\leadsto \log \left(x + {\left({\color{blue}{\left(\mathsf{fma}\left(x, x, -1\right)\right)}}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}^{3}\right) \]

   7. metadata-eval 49.7%

      \[\leadsto \log \left(x + {\left({\left(\mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}^{3}\right) \]

   8. metadata-eval 49.7%

      \[\leadsto \log \left(x + {\left({\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3}\right) \]

3. Applied egg-rr 49.7%

   \[\leadsto \log \left(x + \color{blue}{{\left({\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{0.16666666666666666}\right)}^{3}}\right) \]

4. Taylor expanded in x around inf 14.5%

   \[\leadsto \color{blue}{0.5 + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 0.16666666666666666 \cdot \frac{2 \cdot {x}^{3} + 6 \cdot \left(-0.16666666666666666 \cdot \left({1}^{0.3333333333333333} \cdot x\right) - 0.3333333333333333 \cdot x\right)}{{x}^{3}}\right)} \]

6. Simplified 21.6%

   \[\leadsto \color{blue}{1.3333333333333333} \]

7. Final simplification 21.6%

   \[\leadsto 1.3333333333333333 \]

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 2023200 
(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)))))