Rust f32::acosh

Percentage Accurate: 53.6% → 97.5%

Time: 5.4s

Alternatives: 5

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: 53.6% 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: 97.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (* 2.0 (log (* (sqrt x) (sqrt 2.0)))))

C

float code(float x) {
	return 2.0f * logf((sqrtf(x) * sqrtf(2.0f)));
}

Fortran

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

Julia

function code(x)
	return Float32(Float32(2.0) * log(Float32(sqrt(x) * sqrt(Float32(2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = single(2.0) * log((sqrt(x) * sqrt(single(2.0))));
end

Derivation

1. Initial program 50.6%

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

   1. add-sqr-sqrt 50.6%

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

   2. pow2 50.6%

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

   3. log-pow 50.6%

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

   4. fma-neg 50.6%

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

   5. metadata-eval 50.6%

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

4. Applied egg-rr 50.6%

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

5. Taylor expanded in x around inf 96.7%

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

   1. pow1/2 96.7%

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

   2. count-2 96.7%

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

   3. unpow-prod-down 98.1%

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

   4. pow1/2 98.1%

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

7. Applied egg-rr 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

    \[\leadsto 2 \cdot \log \left(\sqrt{x} \cdot \sqrt{2}\right) \]
11. Add Preprocessing

Alternative 2: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 50.6%

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

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

Alternative 3: 96.7% 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 50.6%

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

3. Taylor expanded in x around inf 96.7%

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

4. Final simplification 96.7%

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

Alternative 4: 11.2% accurate, 29.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (* 2.0 (- -1.0 (/ x -1.0))))

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.0e0 * ((-1.0e0) - (x / (-1.0e0)))
end function

Julia

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

MATLAB

function tmp = code(x)
	tmp = single(2.0) * (single(-1.0) - (x / single(-1.0)));
end

Derivation

1. Initial program 50.6%

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

   1. add-sqr-sqrt 50.6%

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

   2. pow2 50.6%

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

   3. log-pow 50.6%

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

   4. fma-neg 50.6%

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

   5. metadata-eval 50.6%

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

4. Applied egg-rr 50.6%

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

5. Taylor expanded in x around 0 -0.0%

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

7. Simplified 10.9%

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

8. Final simplification 10.9%

   \[\leadsto 2 \cdot \left(-1 - \frac{x}{-1}\right) \]
9. Add Preprocessing

Alternative 5: 3.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -1.0)

C

float code(float x) {
	return -1.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -1.0e0
end function

Julia

function code(x)
	return Float32(-1.0)
end

MATLAB

function tmp = code(x)
	tmp = single(-1.0);
end

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in x around inf 96.7%

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

4. Taylor expanded in x around 0 97.2%

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

6. Simplified 40.2%

   \[\leadsto \color{blue}{\log x + -1} \]

7. Taylor expanded in x around 0 40.2%

   \[\leadsto \color{blue}{\log x - 1} \]

9. Simplified 3.1%

   \[\leadsto \color{blue}{-1} \]

10. Final simplification 3.1%

    \[\leadsto -1 \]
11. Add Preprocessing

Developer target: 99.1% 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 2024014 
(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)))))