Rust f32::acosh

Percentage Accurate: 53.3% → 98.3%

Time: 8.1s

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: 53.3% 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: 98.3% 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 51.0%

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

3. Taylor expanded in x around inf 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*r/ 97.4%

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

   3. metadata-eval 97.4%

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

5. Simplified 97.4%

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

6. Final simplification 97.4%

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

Alternative 2: 43.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log (+ x -1.0)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.0%

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

   1. difference-of-sqr-1 51.0%

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

   2. sub-neg 51.0%

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

   3. metadata-eval 51.0%

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

4. Applied egg-rr 51.0%

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

5. Taylor expanded in x around 0 -0.0%

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

7. Simplified 44.0%

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

8. Final simplification 44.0%

   \[\leadsto \log \left(x + -1\right) \]
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 51.0%

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

3. Taylor expanded in x around inf 96.5%

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

4. Final simplification 96.5%

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

Alternative 4: 3.0% 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 Float32(-log(x))
end

MATLAB

function tmp = code(x)
	tmp = -log(x);
end

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in x around inf 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*r/ 97.4%

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

   3. metadata-eval 97.4%

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

5. Simplified 97.4%

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

6. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log -0.5 + -1 \cdot \log x} \]

8. Simplified 3.0%

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

9. Final simplification 3.0%

   \[\leadsto -\log x \]
10. Add Preprocessing

Alternative 5: 2.1% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ -1 + \frac{x}{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (+ -1.0 (/ x -1.0)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.0%

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

   1. difference-of-sqr-1 51.0%

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

   2. sub-neg 51.0%

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

   3. metadata-eval 51.0%

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

4. Applied egg-rr 51.0%

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

5. Taylor expanded in x around 0 -0.0%

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

7. Simplified 2.1%

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

8. Final simplification 2.1%

   \[\leadsto -1 + \frac{x}{-1} \]
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 2024043 
(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)))))