Rust f32::acosh

Percentage Accurate: 52.5% → 99.3%

Time: 6.1s

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: 52.5% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.6%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. difference-of-sqr-1 N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. lower-*.f32 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lower--.f32 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f32 99.6

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

4. Applied rewrites 99.6%

   \[\leadsto \log \left(x + \color{blue}{\sqrt{x - 1} \cdot \sqrt{1 + x}}\right) \]
5. 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 53.6%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 99.0%

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

Alternative 3: 97.7% 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 53.6%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. difference-of-sqr-1 N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. lower-*.f32 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lower--.f32 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f32 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around -inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. neg-mul-1 N/A

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

   5. mul-1-neg N/A

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

   6. remove-double-neg 97.5

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

7. Applied rewrites 97.5%

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

   1. lift-log.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. flip-+ N/A

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

   4. clear-num N/A

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

   5. log-rec N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

9. Applied rewrites 97.5%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f32 97.8

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

12. Applied rewrites 97.8%

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

Alternative 4: 97.0% accurate, 1.2× 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 53.6%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. difference-of-sqr-1 N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. lower-*.f32 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lower--.f32 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f32 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around -inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. neg-mul-1 N/A

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

   5. mul-1-neg N/A

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

   6. remove-double-neg 97.5

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

7. Applied rewrites 97.5%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
8. 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 2024308 
(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)))))