Rust f32::acosh

Percentage Accurate: 53.2% → 98.8%

Time: 9.5s

Alternatives: 6

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: 53.2% 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: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{\frac{0.125 + \frac{0.001953125}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}{x}}{0.25 + \frac{\frac{0.015625}{x \cdot x} - 0.0625}{x \cdot x}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (-
  (log
   (/
    (/ (+ 0.125 (/ 0.001953125 (* (* x x) (* x (* x (* x x)))))) x)
    (+ 0.25 (/ (- (/ 0.015625 (* x x)) 0.0625) (* x x)))))))

C

float code(float x) {
	return -logf((((0.125f + (0.001953125f / ((x * x) * (x * (x * (x * x)))))) / x) / (0.25f + (((0.015625f / (x * x)) - 0.0625f) / (x * x)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -log((((0.125e0 + (0.001953125e0 / ((x * x) * (x * (x * (x * x)))))) / x) / (0.25e0 + (((0.015625e0 / (x * x)) - 0.0625e0) / (x * x)))))
end function

Julia

function code(x)
	return Float32(-log(Float32(Float32(Float32(Float32(0.125) + Float32(Float32(0.001953125) / Float32(Float32(x * x) * Float32(x * Float32(x * Float32(x * x)))))) / x) / Float32(Float32(0.25) + Float32(Float32(Float32(Float32(0.015625) / Float32(x * x)) - Float32(0.0625)) / Float32(x * x))))))
end

MATLAB

function tmp = code(x)
	tmp = -log((((single(0.125) + (single(0.001953125) / ((x * x) * (x * (x * (x * x)))))) / x) / (single(0.25) + (((single(0.015625) / (x * x)) - single(0.0625)) / (x * x)))));
end

Derivation

1. Initial program 54.7%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. log-rec N/A

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

   5. neg-lowering-neg.f32 N/A

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

   6. log-lowering-log.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

4. Applied egg-rr 54.7%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\color{blue}{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}}{x}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \color{blue}{\frac{\frac{1}{8} \cdot 1}{{x}^{2}}}}{x}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \frac{\color{blue}{\frac{1}{8}}}{{x}^{2}}}{x}\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \color{blue}{\frac{\frac{1}{8}}{{x}^{2}}}}{x}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \frac{\frac{1}{8}}{\color{blue}{x \cdot x}}}{x}\right)\right) \]

   7. *-lowering-*.f32 99.0

      \[\leadsto -\log \left(\frac{0.5 + \frac{0.125}{\color{blue}{x \cdot x}}}{x}\right) \]

7. Simplified 99.0%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5 + \frac{0.125}{x \cdot x}}{x}\right)} \]

9. Applied egg-rr 99.1%

   \[\leadsto -\log \color{blue}{\left(\frac{\frac{0.125 + \frac{0.001953125}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}{x}}{0.25 + \frac{\frac{0.015625}{x \cdot x} - 0.0625}{x \cdot x}}\right)} \]
10. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{\frac{0.125}{x}}{0.25 + \frac{\frac{0.015625}{x \cdot x} - 0.0625}{x \cdot x}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (-
  (log (/ (/ 0.125 x) (+ 0.25 (/ (- (/ 0.015625 (* x x)) 0.0625) (* x x)))))))

C

float code(float x) {
	return -logf(((0.125f / x) / (0.25f + (((0.015625f / (x * x)) - 0.0625f) / (x * x)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -log(((0.125e0 / x) / (0.25e0 + (((0.015625e0 / (x * x)) - 0.0625e0) / (x * x)))))
end function

Julia

function code(x)
	return Float32(-log(Float32(Float32(Float32(0.125) / x) / Float32(Float32(0.25) + Float32(Float32(Float32(Float32(0.015625) / Float32(x * x)) - Float32(0.0625)) / Float32(x * x))))))
end

MATLAB

function tmp = code(x)
	tmp = -log(((single(0.125) / x) / (single(0.25) + (((single(0.015625) / (x * x)) - single(0.0625)) / (x * x)))));
end

Derivation

1. Initial program 54.7%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. log-rec N/A

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

   5. neg-lowering-neg.f32 N/A

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

   6. log-lowering-log.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

4. Applied egg-rr 54.7%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\color{blue}{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}}{x}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \color{blue}{\frac{\frac{1}{8} \cdot 1}{{x}^{2}}}}{x}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \frac{\color{blue}{\frac{1}{8}}}{{x}^{2}}}{x}\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \color{blue}{\frac{\frac{1}{8}}{{x}^{2}}}}{x}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \frac{\frac{1}{8}}{\color{blue}{x \cdot x}}}{x}\right)\right) \]

   7. *-lowering-*.f32 99.0

      \[\leadsto -\log \left(\frac{0.5 + \frac{0.125}{\color{blue}{x \cdot x}}}{x}\right) \]

7. Simplified 99.0%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5 + \frac{0.125}{x \cdot x}}{x}\right)} \]

9. Applied egg-rr 99.1%

   \[\leadsto -\log \color{blue}{\left(\frac{\frac{0.125 + \frac{0.001953125}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}{x}}{0.25 + \frac{\frac{0.015625}{x \cdot x} - 0.0625}{x \cdot x}}\right)} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{neg}\left(\log \left(\frac{\color{blue}{\frac{\frac{1}{8}}{x}}}{\frac{1}{4} + \frac{\frac{\frac{1}{64}}{x \cdot x} - \frac{1}{16}}{x \cdot x}}\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f32 99.1

       \[\leadsto -\log \left(\frac{\color{blue}{\frac{0.125}{x}}}{0.25 + \frac{\frac{0.015625}{x \cdot x} - 0.0625}{x \cdot x}}\right) \]

12. Simplified 99.1%

    \[\leadsto -\log \left(\frac{\color{blue}{\frac{0.125}{x}}}{0.25 + \frac{\frac{0.015625}{x \cdot x} - 0.0625}{x \cdot x}}\right) \]
13. Add Preprocessing

Alternative 3: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{0.5 + \frac{0.125}{x \cdot x}}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (- (log (/ (+ 0.5 (/ 0.125 (* x x))) x))))

C

float code(float x) {
	return -logf(((0.5f + (0.125f / (x * x))) / x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -log(((0.5e0 + (0.125e0 / (x * x))) / x))
end function

Julia

function code(x)
	return Float32(-log(Float32(Float32(Float32(0.5) + Float32(Float32(0.125) / Float32(x * x))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = -log(((single(0.5) + (single(0.125) / (x * x))) / x));
end

Derivation

1. Initial program 54.7%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. log-rec N/A

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

   5. neg-lowering-neg.f32 N/A

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

   6. log-lowering-log.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

4. Applied egg-rr 54.7%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\color{blue}{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}}{x}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \color{blue}{\frac{\frac{1}{8} \cdot 1}{{x}^{2}}}}{x}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \frac{\color{blue}{\frac{1}{8}}}{{x}^{2}}}{x}\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \color{blue}{\frac{\frac{1}{8}}{{x}^{2}}}}{x}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{2} + \frac{\frac{1}{8}}{\color{blue}{x \cdot x}}}{x}\right)\right) \]

   7. *-lowering-*.f32 99.0

      \[\leadsto -\log \left(\frac{0.5 + \frac{0.125}{\color{blue}{x \cdot x}}}{x}\right) \]

7. Simplified 99.0%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5 + \frac{0.125}{x \cdot x}}{x}\right)} \]
8. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(x, 2, \frac{-0.5}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log (fma x 2.0 (/ -0.5 x))))

C

float code(float x) {
	return logf(fmaf(x, 2.0f, (-0.5f / x)));
}

Julia

function code(x)
	return log(fma(x, Float32(2.0), Float32(Float32(-0.5) / x)))
end

Derivation

1. Initial program 54.7%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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. accelerator-lowering-fma.f32 N/A

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

   4. *-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. associate-/l* N/A

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

   9. *-rgt-identity N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. *-inverses N/A

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

   13. associate-*l/ N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f32 98.8

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

5. Simplified 98.8%

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

Alternative 5: 97.5% 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 54.7%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. log-rec N/A

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

   5. neg-lowering-neg.f32 N/A

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

   6. log-lowering-log.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

4. Applied egg-rr 54.7%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f32 97.9

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

7. Simplified 97.9%

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

Alternative 6: 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 54.7%

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

3. Taylor expanded in x around inf

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

5. Simplified 97.6%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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 2024199 
(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)))))