Rust f32::acosh

Percentage Accurate: 52.4% → 99.3%

Time: 4.1s

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: 52.4% 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: 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

Derivation

1. Initial program 51.4%

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

   1. pow1/2 51.4%

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

   2. difference-of-sqr-1 51.4%

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

   3. unpow-prod-down 99.3%

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

   4. sub-neg 99.3%

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

   5. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. unpow1/2 99.3%

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

   2. unpow1/2 99.3%

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

   3. +-commutative 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 2: 98.3% 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 51.4%

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

3. Taylor expanded in x around inf 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*r/ 97.8%

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

   3. metadata-eval 97.8%

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

5. Simplified 97.8%

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

6. Final simplification 97.8%

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

Alternative 3: 97.1% 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 96.4%

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

4. Final simplification 96.4%

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

Alternative 4: 22.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.9583333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.9583333333333333)

C

float code(float x) {
	return 1.9583333333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.9583333333333333e0
end function

Julia

function code(x)
	return Float32(1.9583333333333333)
end

MATLAB

function tmp = code(x)
	tmp = single(1.9583333333333333);
end

Derivation

1. Initial program 51.4%

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

   1. pow1/2 51.4%

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

   2. difference-of-sqr-1 51.4%

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

   3. unpow-prod-down 99.3%

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

   4. sub-neg 99.3%

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

   5. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. unpow1/2 99.3%

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

   2. unpow1/2 99.3%

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

   3. +-commutative 99.3%

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

6. Simplified 99.3%

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

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

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

9. Simplified 22.1%

   \[\leadsto \color{blue}{1.9583333333333333} \]

10. Final simplification 22.1%

    \[\leadsto 1.9583333333333333 \]
11. 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 2024033 
(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)))))