Rust f32::acosh

Percentage Accurate: 51.9% → 99.0%

Time: 7.4s

Alternatives: 8

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: 51.9% 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: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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. Final simplification 99.6%

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

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.25}{x \cdot x} - \log \left(\frac{0.5}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (- (/ (- 0.25) (* x x)) (log (/ 0.5 x))))

C

float code(float x) {
	return (-0.25f / (x * x)) - logf((0.5f / x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (-0.25e0 / (x * x)) - log((0.5e0 / x))
end function

Julia

function code(x)
	return Float32(Float32(Float32(-Float32(0.25)) / Float32(x * x)) - log(Float32(Float32(0.5) / x)))
end

MATLAB

function tmp = code(x)
	tmp = (-single(0.25) / (x * x)) - log((single(0.5) / x));
end

Derivation

1. Initial program 58.3%

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

3. Taylor expanded in x around inf

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   5. log-rec N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   6. remove-double-neg N/A

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

   7. lower-log.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 97.5

       \[\leadsto \left(\log x + \log 2\right) - \frac{0.25}{\color{blue}{x \cdot x}} \]

5. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\left(\log x + \log 2\right) - \frac{0.25}{x \cdot x}} \]
6. Step-by-step derivation

7. Applied rewrites 98.3%

   \[\leadsto \left(-\log \left(\frac{0.5}{x}\right)\right) - \frac{\color{blue}{0.25}}{x \cdot x} \]

8. Final simplification 98.3%

   \[\leadsto \frac{-0.25}{x \cdot x} - \log \left(\frac{0.5}{x}\right) \]
9. Add Preprocessing

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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. *-commutative N/A

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

   11. lower--.f32 N/A

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

   12. *-commutative N/A

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

   13. mul-1-neg N/A

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

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

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

   15. *-commutative N/A

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

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

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

   17. metadata-eval N/A

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

   18. 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. Final simplification 98.0%

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

Alternative 4: 96.8% 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 58.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity 96.4

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

7. Applied rewrites 96.4%

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

8. Final simplification 96.4%

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

Alternative 5: 5.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.25}{x \cdot x}\\ \frac{1}{\frac{0 \cdot t\_0 - \frac{0.25}{x}}{\frac{0.25}{x} \cdot t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 0.25 (* x x))))
   (/ 1.0 (/ (- (* 0.0 t_0) (/ 0.25 x)) (* (/ 0.25 x) t_0)))))

C

float code(float x) {
	float t_0 = 0.25f / (x * x);
	return 1.0f / (((0.0f * t_0) - (0.25f / x)) / ((0.25f / x) * t_0));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    real(4) :: t_0
    t_0 = 0.25e0 / (x * x)
    code = 1.0e0 / (((0.0e0 * t_0) - (0.25e0 / x)) / ((0.25e0 / x) * t_0))
end function

Julia

function code(x)
	t_0 = Float32(Float32(0.25) / Float32(x * x))
	return Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.0) * t_0) - Float32(Float32(0.25) / x)) / Float32(Float32(Float32(0.25) / x) * t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = single(0.25) / (x * x);
	tmp = single(1.0) / (((single(0.0) * t_0) - (single(0.25) / x)) / ((single(0.25) / x) * t_0));
end

Derivation

1. Initial program 58.3%

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

3. Taylor expanded in x around inf

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   5. log-rec N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   6. remove-double-neg N/A

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

   7. lower-log.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 97.5

       \[\leadsto \left(\log x + \log 2\right) - \frac{0.25}{\color{blue}{x \cdot x}} \]

5. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\left(\log x + \log 2\right) - \frac{0.25}{x \cdot x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-1}{4}}{\color{blue}{{x}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 5.2%

   \[\leadsto \frac{-0.25}{\color{blue}{x \cdot x}} \]
9. Step-by-step derivation

10. Applied rewrites 5.2%

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

12. Applied rewrites 5.2%

    \[\leadsto \frac{1}{\frac{0 \cdot \frac{0.25}{x \cdot x} - \frac{0.25}{x} \cdot 1}{\frac{0.25}{x} \cdot \color{blue}{\frac{0.25}{x \cdot x}}}} \]

13. Final simplification 5.2%

    \[\leadsto \frac{1}{\frac{0 \cdot \frac{0.25}{x \cdot x} - \frac{0.25}{x}}{\frac{0.25}{x} \cdot \frac{0.25}{x \cdot x}}} \]
14. Add Preprocessing

Alternative 6: 5.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\frac{\frac{-0.25}{x}}{x}}} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (/ 1.0 (/ 1.0 (/ (/ -0.25 x) x))))

C

float code(float x) {
	return 1.0f / (1.0f / ((-0.25f / x) / x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.0e0 / (1.0e0 / (((-0.25e0) / x) / x))
end function

Julia

function code(x)
	return Float32(Float32(1.0) / Float32(Float32(1.0) / Float32(Float32(Float32(-0.25) / x) / x)))
end

MATLAB

function tmp = code(x)
	tmp = single(1.0) / (single(1.0) / ((single(-0.25) / x) / x));
end

Derivation

1. Initial program 58.3%

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

3. Taylor expanded in x around inf

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   5. log-rec N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   6. remove-double-neg N/A

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

   7. lower-log.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 97.5

       \[\leadsto \left(\log x + \log 2\right) - \frac{0.25}{\color{blue}{x \cdot x}} \]

5. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\left(\log x + \log 2\right) - \frac{0.25}{x \cdot x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-1}{4}}{\color{blue}{{x}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 5.2%

   \[\leadsto \frac{-0.25}{\color{blue}{x \cdot x}} \]
9. Step-by-step derivation

10. Applied rewrites 5.2%

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

12. Applied rewrites 5.2%

    \[\leadsto \frac{1}{\frac{1}{\frac{\frac{-0.25}{x}}{\color{blue}{x}}}} \]
13. Add Preprocessing

Alternative 7: 5.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-4 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (/ 1.0 (* -4.0 (* x x))))

C

float code(float x) {
	return 1.0f / (-4.0f * (x * x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.0e0 / ((-4.0e0) * (x * x))
end function

Julia

function code(x)
	return Float32(Float32(1.0) / Float32(Float32(-4.0) * Float32(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = single(1.0) / (single(-4.0) * (x * x));
end

Derivation

1. Initial program 58.3%

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

3. Taylor expanded in x around inf

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   5. log-rec N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   6. remove-double-neg N/A

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

   7. lower-log.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 97.5

       \[\leadsto \left(\log x + \log 2\right) - \frac{0.25}{\color{blue}{x \cdot x}} \]

5. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\left(\log x + \log 2\right) - \frac{0.25}{x \cdot x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-1}{4}}{\color{blue}{{x}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 5.2%

   \[\leadsto \frac{-0.25}{\color{blue}{x \cdot x}} \]
9. Step-by-step derivation

10. Applied rewrites 5.2%

    \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{-4}} \]

11. Final simplification 5.2%

    \[\leadsto \frac{1}{-4 \cdot \left(x \cdot x\right)} \]
12. Add Preprocessing

Alternative 8: 5.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.25}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (/ -0.25 (* x x)))

C

float code(float x) {
	return -0.25f / (x * x);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (-0.25e0) / (x * x)
end function

Julia

function code(x)
	return Float32(Float32(-0.25) / Float32(x * x))
end

MATLAB

function tmp = code(x)
	tmp = single(-0.25) / (x * x);
end

Derivation

1. Initial program 58.3%

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

3. Taylor expanded in x around inf

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   5. log-rec N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   6. remove-double-neg N/A

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

   7. lower-log.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 97.5

       \[\leadsto \left(\log x + \log 2\right) - \frac{0.25}{\color{blue}{x \cdot x}} \]

5. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\left(\log x + \log 2\right) - \frac{0.25}{x \cdot x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-1}{4}}{\color{blue}{{x}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 5.2%

   \[\leadsto \frac{-0.25}{\color{blue}{x \cdot x}} \]
9. Add Preprocessing

Developer Target 1: 99.0% 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 2024249 
(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)))))