Rust f32::acosh

Percentage Accurate: 52.9% → 98.1%

Time: 7.1s

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: 52.9% 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: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\log 2 + \log x\right) - \frac{0.25}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (- (+ (log 2.0) (log x)) (/ 0.25 (* x x))))

C

float code(float x) {
	return (logf(2.0f) + logf(x)) - (0.25f / (x * x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (log(2.0e0) + log(x)) - (0.25e0 / (x * x))
end function

Julia

function code(x)
	return Float32(Float32(log(Float32(2.0)) + log(x)) - Float32(Float32(0.25) / Float32(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (log(single(2.0)) + log(x)) - (single(0.25) / (x * x));
end

Derivation

1. Initial program 51.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 51.4

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 28.6

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

3. Applied rewrites 29.1%

   \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}}} \]
6. 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 97.9

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

7. Applied rewrites 97.9%

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

8. Final simplification 97.9%

   \[\leadsto \left(\log 2 + \log x\right) - \frac{0.25}{x \cdot x} \]
9. Add Preprocessing

Alternative 2: 98.2% 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 51.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 51.4

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 28.4

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

3. Applied rewrites 28.3%

   \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)} \]
6. 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

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

7. Applied rewrites -0.0%

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

9. Applied rewrites 97.8%

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

Alternative 3: 98.2% 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 51.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 51.4

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 28.4

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

3. Applied rewrites 28.8%

   \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. unsub-neg N/A

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

   6. remove-double-neg N/A

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

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

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

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

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

   9. mul-1-neg N/A

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

   10. *-commutative N/A

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

   11. lower--.f32 N/A

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

   12. *-commutative N/A

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

   13. mul-1-neg N/A

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

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

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

   15. *-commutative N/A

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

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

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

   17. metadata-eval N/A

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

7. Applied rewrites 97.8%

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

Alternative 4: 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 51.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 51.4

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 28.6

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

3. Applied rewrites 29.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)} \]
6. 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 96.6

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

7. Applied rewrites 96.6%

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

9. Applied rewrites 97.3%

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

Alternative 5: 96.9% 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 51.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 51.4

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 28.4

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

3. Applied rewrites 28.5%

   \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

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

7. Applied rewrites 96.3%

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

Alternative 6: 5.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-4}{\frac{1}{x}} \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 51.4

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 28.5

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

3. Applied rewrites 28.5%

   \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}}} \]
6. 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 97.9

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

7. Applied rewrites 97.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.3%

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

12. Applied rewrites 5.3%

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

14. Applied rewrites 5.3%

    \[\leadsto \frac{1}{\frac{-4}{\frac{1}{x}} \cdot x} \]
15. 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 51.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 51.4

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 28.5

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

3. Applied rewrites 29.1%

   \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}}} \]
6. 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 97.9

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

7. Applied rewrites 97.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.3%

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

12. Applied rewrites 5.3%

    \[\leadsto \frac{1}{-4 \cdot \color{blue}{\left(x \cdot x\right)}} \]
13. 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 51.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 51.4

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 28.6

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

3. Applied rewrites 28.2%

   \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}}} \]
6. 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 97.9

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

7. Applied rewrites 97.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.3%

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

Developer Target 1: 99.1% 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 2024272 
(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)))))