Rust f32::acosh

Percentage Accurate: 52.0% → 97.1%

Time: 5.5s

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: 52.0% 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: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \log \left(\sqrt[3]{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (+ (log (* 2.0 (pow (cbrt x) 2.0))) (log (cbrt x))))

C

float code(float x) {
	return logf((2.0f * powf(cbrtf(x), 2.0f))) + logf(cbrtf(x));
}

Julia

function code(x)
	return Float32(log(Float32(Float32(2.0) * (cbrt(x) ^ Float32(2.0)))) + log(cbrt(x)))
end

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in x around inf 97.4%

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

   1. count-2 97.4%

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

   2. add-cube-cbrt 97.3%

      \[\leadsto \log \left(2 \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}\right) \]

   3. associate-*r* 97.3%

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}\right)} \]

   4. log-prod 97.6%

      \[\leadsto \color{blue}{\log \left(2 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \log \left(\sqrt[3]{x}\right)} \]

   5. pow2 97.6%

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

5. Applied egg-rr 97.6%

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

6. Final simplification 97.6%

   \[\leadsto \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \log \left(\sqrt[3]{x}\right) \]
7. Add Preprocessing

Alternative 2: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \log x \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (+ (log (* 2.0 (pow (cbrt x) 2.0))) (* (log x) 0.3333333333333333)))

C

float code(float x) {
	return logf((2.0f * powf(cbrtf(x), 2.0f))) + (logf(x) * 0.3333333333333333f);
}

Julia

function code(x)
	return Float32(log(Float32(Float32(2.0) * (cbrt(x) ^ Float32(2.0)))) + Float32(log(x) * Float32(0.3333333333333333)))
end

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in x around inf 97.4%

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

   1. count-2 97.4%

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

   2. add-cube-cbrt 97.3%

      \[\leadsto \log \left(2 \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}\right) \]

   3. associate-*r* 97.3%

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}\right)} \]

   4. log-prod 97.6%

      \[\leadsto \color{blue}{\log \left(2 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \log \left(\sqrt[3]{x}\right)} \]

   5. pow2 97.6%

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

5. Applied egg-rr 97.6%

   \[\leadsto \color{blue}{\log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \log \left(\sqrt[3]{x}\right)} \]
6. Step-by-step derivation

   1. log1p-expm1-u 97.6%

      \[\leadsto \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt[3]{x}\right)\right)\right)} \]

   2. expm1-undefine 97.6%

      \[\leadsto \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \mathsf{log1p}\left(\color{blue}{e^{\log \left(\sqrt[3]{x}\right)} - 1}\right) \]

   3. add-exp-log 97.6%

      \[\leadsto \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \mathsf{log1p}\left(\color{blue}{\sqrt[3]{x}} - 1\right) \]

   4. pow1/3 97.5%

      \[\leadsto \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \mathsf{log1p}\left(\color{blue}{{x}^{0.3333333333333333}} - 1\right) \]

   5. pow-to-exp 97.4%

      \[\leadsto \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \mathsf{log1p}\left(\color{blue}{e^{\log x \cdot 0.3333333333333333}} - 1\right) \]

   6. expm1-define 97.4%

      \[\leadsto \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log x \cdot 0.3333333333333333\right)}\right) \]

   7. log1p-expm1-u 97.4%

      \[\leadsto \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \color{blue}{\log x \cdot 0.3333333333333333} \]

7. Applied egg-rr 97.4%

   \[\leadsto \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \color{blue}{\log x \cdot 0.3333333333333333} \]

8. Final simplification 97.4%

   \[\leadsto \log \left(2 \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \log x \cdot 0.3333333333333333 \]
9. Add Preprocessing

Alternative 3: 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 51.4%

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

3. Taylor expanded in x around inf 97.4%

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

4. Final simplification 97.4%

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

Alternative 4: 6.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary32 0.0)

C

float code(float x) {
	return 0.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.0e0
end function

Julia

function code(x)
	return Float32(0.0)
end

MATLAB

function tmp = code(x)
	tmp = single(0.0);
end

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in x around inf 97.4%

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

   1. flip-+ -0.0%

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

   2. difference-of-squares -0.0%

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

   3. +-inverses -0.0%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \color{blue}{0}}{x - x}\right) \]

   4. +-inverses -0.0%

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

   5. +-inverses -0.0%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \left(x \cdot x - x \cdot x\right)}{\color{blue}{0}}\right) \]

   6. +-inverses -0.0%

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

   7. associate-*r/ -0.0%

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

   8. +-inverses -0.0%

      \[\leadsto \log \left(\left(x + x\right) \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{0}}\right) \]

   9. +-inverses -0.0%

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

   10. flip-+ 16.9%

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

   11. sum-log 28.0%

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

5. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\log \left(\frac{0}{0}\right) + \log \left(\frac{0}{0}\right)} \]

7. Simplified 6.1%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 6.1%

   \[\leadsto 0 \]
9. Add Preprocessing

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 2024084 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

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

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