Rust f32::acosh

Percentage Accurate: 53.7% → 98.9%

Time: 9.4s

Alternatives: 8

Speedup: 2.0×

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: 53.7% 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: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\left(x + \left(x + \frac{-0.5}{x}\right)\right) + \frac{\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (+ (+ x (+ x (/ -0.5 x))) (/ (/ (+ -0.125 (/ -0.0625 (* x x))) (* x x)) x))))

C

float code(float x) {
	return logf(((x + (x + (-0.5f / x))) + (((-0.125f + (-0.0625f / (x * x))) / (x * x)) / x)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(((x + (x + ((-0.5e0) / x))) + ((((-0.125e0) + ((-0.0625e0) / (x * x))) / (x * x)) / x)))
end function

Julia

function code(x)
	return log(Float32(Float32(x + Float32(x + Float32(Float32(-0.5) / x))) + Float32(Float32(Float32(Float32(-0.125) + Float32(Float32(-0.0625) / Float32(x * x))) / Float32(x * x)) / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((x + (x + (single(-0.5) / x))) + (((single(-0.125) + (single(-0.0625) / (x * x))) / (x * x)) / x)));
end

Derivation

1. Initial program 46.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

5. Simplified 99.6%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(2 \cdot x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right) \]

   2. count-2 N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(\left(x + x\right) + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(x + \left(x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(x + \left(1 \cdot x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right)\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(x + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + x\right) + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) \cdot \left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) - \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) - x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)\right) \]

7. Applied egg-rr 5.0%

   \[\leadsto \log \color{blue}{\left(\frac{\left(x + \frac{-0.5}{x}\right) \cdot \left(x + \frac{-0.5}{x}\right) - \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)}{\left(x + \frac{-0.5}{x}\right) - x \cdot \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\left(x + \frac{\frac{-1}{2}}{x}\right) \cdot \left(x + \frac{\frac{-1}{2}}{x}\right) - \left(\left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot x\right) \cdot \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left(x + \frac{\frac{-1}{2}}{x}\right) - x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\left(x + \frac{\frac{-1}{2}}{x}\right) \cdot \left(x + \frac{\frac{-1}{2}}{x}\right) - \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left(x + \frac{\frac{-1}{2}}{x}\right) - x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)\right) \]

   3. flip-+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + \frac{\frac{-1}{2}}{x}\right) + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   4. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + \frac{\frac{-1}{2}}{x}\right) + \left(x \cdot 1 + x \cdot \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto \log \color{blue}{\left(\left(\left(x + \frac{-0.5}{x}\right) + x\right) + \frac{x}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{-0.125 + \frac{-0.0625}{x \cdot x}}}\right)} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \color{blue}{\left(-1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{3}}\right)}\right)\right) \]
11. Step-by-step derivation

    1. unpow3 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \left(-1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{\left(x \cdot x\right) \cdot x}\right)\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \left(-1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{2} \cdot x}\right)\right)\right) \]

    3. associate-/r* N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \left(-1 \cdot \frac{\frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{x}\right)\right)\right) \]

    4. associate-*r/ N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \left(\frac{-1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{x}\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right), x\right)\right)\right) \]

12. Simplified 99.6%

    \[\leadsto \log \left(\left(\left(x + \frac{-0.5}{x}\right) + x\right) + \color{blue}{\frac{\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot x}}{x}}\right) \]

13. Final simplification 99.6%

    \[\leadsto \log \left(\left(x + \left(x + \frac{-0.5}{x}\right)\right) + \frac{\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot x}}{x}\right) \]
14. Add Preprocessing

Alternative 2: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 46.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

5. Simplified 99.6%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(2 \cdot x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right) \]

   2. count-2 N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(\left(x + x\right) + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(x + \left(x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(x + \left(1 \cdot x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right)\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(x + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + x\right) + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) \cdot \left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) - \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) - x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)\right) \]

7. Applied egg-rr 5.0%

   \[\leadsto \log \color{blue}{\left(\frac{\left(x + \frac{-0.5}{x}\right) \cdot \left(x + \frac{-0.5}{x}\right) - \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)}{\left(x + \frac{-0.5}{x}\right) - x \cdot \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\left(x + \frac{\frac{-1}{2}}{x}\right) \cdot \left(x + \frac{\frac{-1}{2}}{x}\right) - \left(\left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot x\right) \cdot \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left(x + \frac{\frac{-1}{2}}{x}\right) - x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\left(x + \frac{\frac{-1}{2}}{x}\right) \cdot \left(x + \frac{\frac{-1}{2}}{x}\right) - \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left(x + \frac{\frac{-1}{2}}{x}\right) - x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)\right) \]

   3. flip-+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + \frac{\frac{-1}{2}}{x}\right) + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   4. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + \frac{\frac{-1}{2}}{x}\right) + \left(x \cdot 1 + x \cdot \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto \log \color{blue}{\left(\left(\left(x + \frac{-0.5}{x}\right) + x\right) + \frac{x}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{-0.125 + \frac{-0.0625}{x \cdot x}}}\right)} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \color{blue}{\left(\frac{\frac{-1}{8}}{{x}^{3}}\right)}\right)\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left({x}^{3}\right)\right)\right)\right) \]

    2. cube-mult N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left(x \cdot {x}^{2}\right)\right)\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \mathsf{/.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(x, \left({x}^{2}\right)\right)\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \mathsf{/.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(x, \left(x \cdot x\right)\right)\right)\right)\right) \]

    6. *-lowering-*.f32 99.4%

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right), \mathsf{/.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right) \]

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 3: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot 2 + \frac{-0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log (+ (* x 2.0) (/ (+ -0.5 (/ -0.125 (* x x))) x))))

C

float code(float x) {
	return logf(((x * 2.0f) + ((-0.5f + (-0.125f / (x * x))) / x)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(((x * 2.0e0) + (((-0.5e0) + ((-0.125e0) / (x * x))) / x)))
end function

Julia

function code(x)
	return log(Float32(Float32(x * Float32(2.0)) + Float32(Float32(Float32(-0.5) + Float32(Float32(-0.125) / Float32(x * x))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * single(2.0)) + ((single(-0.5) + (single(-0.125) / (x * x))) / x)));
end

Derivation

1. Initial program 46.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(2 + -1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)}\right) \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(2 \cdot x + \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot x\right)\right) \]

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

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(2 \cdot x\right), \left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(x \cdot 2\right), \left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(x \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(x \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right)\right) \]

   7. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(x \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{\mathsf{neg}\left({x}^{2}\right)}\right)\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{\mathsf{neg}\left({x}^{2}\right)}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{\mathsf{neg}\left(x \cdot x\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x \cdot \left(-1 \cdot x\right)}\right)\right)\right) \]

   12. times-frac N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(\frac{x}{x} \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{-1 \cdot x}\right)\right)\right) \]

   13. *-inverses N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{-1 \cdot x}\right)\right)\right) \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   15. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(1 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(x, 2\right), \left(\mathsf{neg}\left(1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

5. Simplified 99.4%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 + \frac{-0.5 + \frac{-0.125}{x \cdot x}}{x}\right)} \]
6. Add Preprocessing

Alternative 4: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot \left(2 + \frac{-0.5}{x \cdot x}\right)\right) \end{array} \]

FPCore

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

C

float code(float x) {
	return logf((x * (2.0f + (-0.5f / (x * x)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x * (2.0e0 + ((-0.5e0) / (x * x)))))
end function

Julia

function code(x)
	return log(Float32(x * Float32(Float32(2.0) + Float32(Float32(-0.5) / Float32(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = log((x * (single(2.0) + (single(-0.5) / (x * x)))));
end

Derivation

1. Initial program 46.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

5. Simplified 99.6%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(2 \cdot x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right) \]

   2. count-2 N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(\left(x + x\right) + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(x + \left(x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(x + \left(1 \cdot x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right)\right)\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(x + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + x\right) + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) \cdot \left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) - \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) - x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)\right) \]

7. Applied egg-rr 5.0%

   \[\leadsto \log \color{blue}{\left(\frac{\left(x + \frac{-0.5}{x}\right) \cdot \left(x + \frac{-0.5}{x}\right) - \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)}{\left(x + \frac{-0.5}{x}\right) - x \cdot \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\left(x + \frac{\frac{-1}{2}}{x}\right) \cdot \left(x + \frac{\frac{-1}{2}}{x}\right) - \left(\left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot x\right) \cdot \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left(x + \frac{\frac{-1}{2}}{x}\right) - x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\left(x + \frac{\frac{-1}{2}}{x}\right) \cdot \left(x + \frac{\frac{-1}{2}}{x}\right) - \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left(x + \frac{\frac{-1}{2}}{x}\right) - x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right)\right) \]

   3. flip-+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + \frac{\frac{-1}{2}}{x}\right) + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   4. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + \frac{\frac{-1}{2}}{x}\right) + \left(x \cdot 1 + x \cdot \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto \log \color{blue}{\left(\left(\left(x + \frac{-0.5}{x}\right) + x\right) + \frac{x}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{-0.125 + \frac{-0.0625}{x \cdot x}}}\right)} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(2 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(2 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(2 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

    4. associate-*r/ N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right) \]

    6. distribute-neg-frac N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{{x}^{2}}\right)\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right)\right)\right) \]

    8. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \left({x}^{2}\right)\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \left(x \cdot x\right)\right)\right)\right)\right) \]

    10. *-lowering-*.f32 99.1%

        \[\leadsto \mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right) \]

12. Simplified 99.1%

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

Alternative 5: 98.3% accurate, 1.9× 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 46.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \color{blue}{\left(x \cdot \left(1 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x \cdot 1 + x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   3. *-rgt-identity N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x + x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(x \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(x \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(x \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{{x}^{2}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(x \cdot \frac{\frac{-1}{2}}{{x}^{2}}\right)\right)\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(\frac{x \cdot \frac{-1}{2}}{{x}^{2}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(\frac{x \cdot \frac{-1}{2}}{x \cdot x}\right)\right)\right)\right) \]

   11. times-frac N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(\frac{x}{x} \cdot \frac{\frac{-1}{2}}{x}\right)\right)\right)\right) \]

   12. *-inverses N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(1 \cdot \frac{\frac{-1}{2}}{x}\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(1 \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right)\right)\right) \]

   14. distribute-neg-frac N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(1 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \mathsf{*.f32}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right)\right)\right) \]

   16. distribute-neg-frac N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \mathsf{*.f32}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \mathsf{*.f32}\left(1, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)\right)\right) \]

   18. /-lowering-/.f32 99.1%

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right)\right)\right)\right) \]

5. Simplified 99.1%

   \[\leadsto \log \left(x + \color{blue}{\left(x + 1 \cdot \frac{-0.5}{x}\right)}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right) + x\right)\right) \]

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

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(x + 1 \cdot \frac{\frac{-1}{2}}{x}\right), x\right)\right) \]

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

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \left(1 \cdot \frac{\frac{-1}{2}}{x}\right)\right), x\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right) \]

   5. /-lowering-/.f32 99.1%

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right)\right) \]

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 6: 97.5% accurate, 2.0× 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 46.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \color{blue}{x}\right)\right) \]
4. Step-by-step derivation

5. Simplified 98.4%

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

   1. count-2 N/A

      \[\leadsto \log \left(2 \cdot x\right) \]

   2. *-commutative N/A

      \[\leadsto \log \left(x \cdot 2\right) \]

   3. log-prod N/A

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

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

      \[\leadsto \mathsf{+.f32}\left(\log x, \color{blue}{\log 2}\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \log \color{blue}{2}\right) \]

   6. log-lowering-log.f32 97.7%

      \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{log.f32}\left(2\right)\right) \]

7. Applied egg-rr 97.7%

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

   1. sum-log N/A

      \[\leadsto \log \left(x \cdot 2\right) \]

   2. metadata-eval N/A

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

   3. div-inv N/A

      \[\leadsto \log \left(\frac{x}{\frac{1}{2}}\right) \]

   4. clear-num N/A

      \[\leadsto \log \left(\frac{1}{\frac{\frac{1}{2}}{x}}\right) \]

   5. log-rec N/A

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

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

      \[\leadsto \mathsf{neg.f32}\left(\log \left(\frac{\frac{1}{2}}{x}\right)\right) \]

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

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

   8. /-lowering-/.f32 98.7%

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

9. Applied egg-rr 98.7%

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

Alternative 7: 97.0% accurate, 2.0× 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 46.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \color{blue}{x}\right)\right) \]
4. Step-by-step derivation

5. Simplified 98.4%

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

Alternative 8: 44.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log x \end{array} \]

FPCore

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

C

float code(float x) {
	return logf(x);
}

Fortran

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

Julia

function code(x)
	return log(x)
end

MATLAB

function tmp = code(x)
	tmp = log(x);
end

Derivation

1. Initial program 46.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \color{blue}{\left(x \cdot \left(1 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x \cdot 1 + x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   3. *-rgt-identity N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x + x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(x \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(x \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(x \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{{x}^{2}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(x \cdot \frac{\frac{-1}{2}}{{x}^{2}}\right)\right)\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(\frac{x \cdot \frac{-1}{2}}{{x}^{2}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(\frac{x \cdot \frac{-1}{2}}{x \cdot x}\right)\right)\right)\right) \]

   11. times-frac N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(\frac{x}{x} \cdot \frac{\frac{-1}{2}}{x}\right)\right)\right)\right) \]

   12. *-inverses N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(1 \cdot \frac{\frac{-1}{2}}{x}\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(1 \cdot \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right)\right)\right) \]

   14. distribute-neg-frac N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \left(1 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \mathsf{*.f32}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right)\right)\right) \]

   16. distribute-neg-frac N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \mathsf{*.f32}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \mathsf{*.f32}\left(1, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)\right)\right) \]

   18. /-lowering-/.f32 99.1%

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{+.f32}\left(x, \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right)\right)\right)\right) \]

5. Simplified 99.1%

   \[\leadsto \log \left(x + \color{blue}{\left(x + 1 \cdot \frac{-0.5}{x}\right)}\right) \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f32 44.9%

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right)\right) \]

8. Simplified 44.9%

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

9. Taylor expanded in x around inf

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

    1. mul-1-neg N/A

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

    2. log-rec N/A

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

    3. remove-double-neg N/A

       \[\leadsto \log x \]

    4. log-lowering-log.f32 44.9%

       \[\leadsto \mathsf{log.f32}\left(x\right) \]

11. Simplified 44.9%

    \[\leadsto \color{blue}{\log x} \]
12. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× 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 2024138 
(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)))))