Rust f32::acosh

Percentage Accurate: 54.1% → 98.2%

Time: 5.6s

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: 54.1% 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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

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

2. Taylor expanded in x around inf 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*r/ 96.9%

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

   3. metadata-eval 96.9%

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

4. Simplified 96.9%

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

5. Final simplification 96.9%

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

Alternative 2: 98.2% 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 52.8%

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

2. Taylor expanded in x around inf 96.9%

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

   1. associate-*r/ 96.9%

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

   2. metadata-eval 96.9%

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

4. Simplified 96.9%

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

5. Final simplification 96.9%

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

Alternative 3: 44.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + 1.0625\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log (+ x 1.0625)))

C

float code(float x) {
	return logf((x + 1.0625f));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + 1.0625e0))
end function

Julia

function code(x)
	return log(Float32(x + Float32(1.0625)))
end

MATLAB

function tmp = code(x)
	tmp = log((x + single(1.0625)));
end

Derivation

1. Initial program 52.8%

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

   1. difference-of-sqr-1 52.8%

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

   2. sub-neg 52.8%

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

   3. metadata-eval 52.8%

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

3. Applied egg-rr 52.8%

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

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto \log \left(x + \color{blue}{\left(-0.125 \cdot \frac{{x}^{4}}{{\left(\sqrt{-1}\right)}^{3}} + \left(\sqrt{-1} + \left(0.5 \cdot \frac{{x}^{2}}{\sqrt{-1}} + 0.0625 \cdot \frac{{x}^{6}}{{\left(\sqrt{-1}\right)}^{5}}\right)\right)\right)}\right) \]

6. Simplified 44.2%

   \[\leadsto \log \left(x + \color{blue}{1.0625}\right) \]

7. Final simplification 44.2%

   \[\leadsto \log \left(x + 1.0625\right) \]

Alternative 4: 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 52.8%

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

2. Taylor expanded in x around inf 95.2%

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

3. Final simplification 95.2%

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

Alternative 5: 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 52.8%

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

   1. difference-of-sqr-1 52.8%

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

   2. sub-neg 52.8%

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

   3. metadata-eval 52.8%

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

3. Applied egg-rr 52.8%

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

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto \log \left(x + \color{blue}{\left(-0.125 \cdot \frac{{x}^{4}}{{\left(\sqrt{-1}\right)}^{3}} + \left(\sqrt{-1} + \left(0.5 \cdot \frac{{x}^{2}}{\sqrt{-1}} + 0.0625 \cdot \frac{{x}^{6}}{{\left(\sqrt{-1}\right)}^{5}}\right)\right)\right)}\right) \]

6. Simplified 44.2%

   \[\leadsto \log \left(x + \color{blue}{1.0625}\right) \]

7. Taylor expanded in x around inf 44.1%

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

   1. mul-1-neg 44.1%

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

   2. log-rec 44.1%

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

   3. remove-double-neg 44.1%

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

9. Simplified 44.1%

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

10. Final simplification 44.1%

    \[\leadsto \log x \]

Alternative 6: 20.8% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 0.8333333333333334)

C

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

Fortran

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

Julia

function code(x)
	return Float32(0.8333333333333334)
end

MATLAB

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

Derivation

1. Initial program 52.8%

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

   1. difference-of-sqr-1 52.8%

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

   2. sub-neg 52.8%

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

   3. metadata-eval 52.8%

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

3. Applied egg-rr 52.8%

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

4. Taylor expanded in x around 0 -0.0%

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

6. Simplified 20.8%

   \[\leadsto \color{blue}{0.8333333333333334} \]

7. Final simplification 20.8%

   \[\leadsto 0.8333333333333334 \]

Alternative 7: 21.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.1666666666666667 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.1666666666666667)

C

float code(float x) {
	return 1.1666666666666667f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.1666666666666667e0
end function

Julia

function code(x)
	return Float32(1.1666666666666667)
end

MATLAB

function tmp = code(x)
	tmp = single(1.1666666666666667);
end

Derivation

1. Initial program 52.8%

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

   1. difference-of-sqr-1 52.8%

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

   2. sub-neg 52.8%

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

   3. metadata-eval 52.8%

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

3. Applied egg-rr 52.8%

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

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

   \[\leadsto \color{blue}{0.001388888888888889 \cdot \frac{45 \cdot \frac{1}{{\left({\left(\sqrt{-1}\right)}^{2} - 1\right)}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}} + \left(45 \cdot \frac{1}{\left({\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot {\left(\sqrt{-1}\right)}^{4}} + 30 \cdot \frac{1}{{\left({\left(\sqrt{-1}\right)}^{2} - 1\right)}^{3}}\right)}{{x}^{6}} + \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \left(\log \left({\left(\sqrt{-1}\right)}^{2} - 1\right) + \left(-0.041666666666666664 \cdot \frac{3 \cdot \frac{1}{{\left({\left(\sqrt{-1}\right)}^{2} - 1\right)}^{2}} + 3 \cdot \frac{1}{\left({\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}{{x}^{4}} + 0.5 \cdot \frac{1}{\left({\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot {x}^{2}}\right)\right)\right)} \]

6. Simplified 21.4%

   \[\leadsto \color{blue}{1.1666666666666667} \]

7. Final simplification 21.4%

   \[\leadsto 1.1666666666666667 \]

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 2023182 
(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)))))