Rust f32::acosh

Percentage Accurate: 53.3% → 98.3%

Time: 4.8s

Alternatives: 4

Speedup: 1.2×

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, 1.0× 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 54.3%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

      \[\leadsto \log \left(x \cdot \color{blue}{\left(2 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \log \color{blue}{\left(x \cdot 2 + x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \log \left(\color{blue}{2 \cdot x} + x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(2, x, x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \log \left(\mathsf{fma}\left(2, x, x \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{{x}^{2}} \cdot \frac{1}{2}}\right)\right)\right)\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \log \left(\mathsf{fma}\left(2, x, x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)\right) \]

   7. metadata-eval N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. *-rgt-identity N/A

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

   11. unpow2 N/A

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

   12. associate-/r* N/A

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

   13. *-inverses N/A

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

   14. associate-*l/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 -0.0

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

5. Applied rewrites -0.0%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(2, x, \frac{-0.5}{x}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 98.0%

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

Alternative 2: 98.3% accurate, 1.0× 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 54.3%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

      \[\leadsto \log \left(x + x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \log \left(x + \color{blue}{\left(x \cdot 1 + x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

   3. *-rgt-identity N/A

      \[\leadsto \log \left(x + \left(\color{blue}{x} + x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) \]

   4. distribute-rgt-neg-out N/A

      \[\leadsto \log \left(x + \left(x + \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right)\right) \]

   5. unsub-neg N/A

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

   6. remove-double-neg N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. distribute-lft-neg-out N/A

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

   9. mul-1-neg N/A

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

   10. lower--.f32 N/A

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

   11. mul-1-neg N/A

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

   12. distribute-lft-neg-out N/A

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

   13. *-commutative N/A

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

   14. distribute-rgt-neg-in N/A

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

   15. metadata-eval N/A

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

   16. associate-*r* N/A

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

5. Applied rewrites 98.0%

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

Alternative 3: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} + \log 2 \]

   4. log-rec N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \log 2 \]

   5. remove-double-neg N/A

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

   6. lower-log.f32 N/A

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

   7. lower-log.f32 96.4

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

5. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\log x + \log 2} \]
6. Step-by-step derivation

7. Applied rewrites 97.3%

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

Alternative 4: 97.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f32 96.6

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

5. Applied rewrites 96.6%

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

Developer Target 1: 99.3% accurate, 0.9× 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 2024326 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :alt
  (! :herbie-platform default (log (+ x (* (sqrt (- x 1)) (sqrt (+ x 1))))))

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