Rust f32::acosh

Percentage Accurate: 52.5% → 98.8%

Time: 9.5s

Alternatives: 10

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: 52.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\\ \log \left(t\_0 \cdot \frac{x - \frac{0.5}{t\_0}}{x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0
         (* x (+ 2.0 (/ (+ -0.125 (/ -0.0625 (* x x))) (* x (* x (* x x))))))))
   (log (* t_0 (/ (- x (/ 0.5 t_0)) x)))))

C

float code(float x) {
	float t_0 = x * (2.0f + ((-0.125f + (-0.0625f / (x * x))) / (x * (x * (x * x)))));
	return logf((t_0 * ((x - (0.5f / t_0)) / x)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    real(4) :: t_0
    t_0 = x * (2.0e0 + (((-0.125e0) + ((-0.0625e0) / (x * x))) / (x * (x * (x * x)))))
    code = log((t_0 * ((x - (0.5e0 / t_0)) / x)))
end function

Julia

function code(x)
	t_0 = Float32(x * Float32(Float32(2.0) + Float32(Float32(Float32(-0.125) + Float32(Float32(-0.0625) / Float32(x * x))) / Float32(x * Float32(x * Float32(x * x))))))
	return log(Float32(t_0 * Float32(Float32(x - Float32(Float32(0.5) / t_0)) / x)))
end

MATLAB

function tmp = code(x)
	t_0 = x * (single(2.0) + ((single(-0.125) + (single(-0.0625) / (x * x))) / (x * (x * (x * x)))));
	tmp = log((t_0 * ((x - (single(0.5) / t_0)) / x)));
end

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 98.7%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 - \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

7. Applied egg-rr 54.7%

   \[\leadsto \log \left(1 \cdot \frac{-0.5}{x} + \color{blue}{\frac{1}{\frac{\frac{-0.5}{x} - x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{\left(x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \left(\frac{-0.5}{x} - x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}}}\right) \]

9. Applied egg-rr 98.6%

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

    1. *-lft-identity N/A

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

    2. associate-*l/ N/A

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

    3. neg-mul-1 N/A

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

    4. associate-/r/ N/A

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

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

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

11. Applied egg-rr 98.7%

    \[\leadsto \log \color{blue}{\left(\frac{\frac{0.5}{x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)} - x}{-x} \cdot \left(x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)} \]

12. Final simplification 98.7%

    \[\leadsto \log \left(\left(x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \frac{x - \frac{0.5}{x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}}{x}\right) \]
13. Add Preprocessing

Alternative 2: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot \left(\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(2 + \frac{-0.5}{x \cdot x}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (*
   x
   (+
    (/ (+ -0.125 (/ -0.0625 (* x x))) (* x (* x (* x x))))
    (+ 2.0 (/ -0.5 (* x x)))))))

C

float code(float x) {
	return logf((x * (((-0.125f + (-0.0625f / (x * x))) / (x * (x * (x * x)))) + (2.0f + (-0.5f / (x * x))))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x * ((((-0.125e0) + ((-0.0625e0) / (x * x))) / (x * (x * (x * x)))) + (2.0e0 + ((-0.5e0) / (x * x))))))
end function

Julia

function code(x)
	return log(Float32(x * Float32(Float32(Float32(Float32(-0.125) + Float32(Float32(-0.0625) / Float32(x * x))) / Float32(x * Float32(x * Float32(x * x)))) + Float32(Float32(2.0) + Float32(Float32(-0.5) / Float32(x * x))))))
end

MATLAB

function tmp = code(x)
	tmp = log((x * (((single(-0.125) + (single(-0.0625) / (x * x))) / (x * (x * (x * x)))) + (single(2.0) + (single(-0.5) / (x * x))))));
end

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 98.7%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 - \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

6. Taylor expanded in x around inf

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

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

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

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

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

   6. sub-neg N/A

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

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

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f32 N/A

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

   13. unpow2 N/A

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

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

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

8. Simplified 98.7%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(\left(2 + \frac{-0.5}{x \cdot x}\right) + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

9. Final simplification 98.7%

   \[\leadsto \log \left(x \cdot \left(\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(2 + \frac{-0.5}{x \cdot x}\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) - \frac{0.5}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (-
   (* x (+ 2.0 (/ (+ -0.125 (/ -0.0625 (* x x))) (* x (* x (* x x))))))
   (/ 0.5 x))))

C

float code(float x) {
	return logf(((x * (2.0f + ((-0.125f + (-0.0625f / (x * x))) / (x * (x * (x * x)))))) - (0.5f / x)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(((x * (2.0e0 + (((-0.125e0) + ((-0.0625e0) / (x * x))) / (x * (x * (x * x)))))) - (0.5e0 / x)))
end function

Julia

function code(x)
	return log(Float32(Float32(x * Float32(Float32(2.0) + Float32(Float32(Float32(-0.125) + Float32(Float32(-0.0625) / Float32(x * x))) / Float32(x * Float32(x * Float32(x * x)))))) - Float32(Float32(0.5) / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * (single(2.0) + ((single(-0.125) + (single(-0.0625) / (x * x))) / (x * (x * (x * x)))))) - (single(0.5) / x)));
end

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 98.7%

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

   1. +-commutative N/A

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

   2. fma-define N/A

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

   3. *-lft-identity N/A

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg2 N/A

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

   6. fmm-undef N/A

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

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

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

7. Applied egg-rr 98.7%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) - \frac{0.5}{x}\right)} \]
8. Add Preprocessing

Alternative 4: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 98.7%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 - \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

6. Taylor expanded in x around inf

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

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

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

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

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

   6. sub-neg N/A

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

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

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f32 N/A

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

   13. unpow2 N/A

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

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

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

8. Simplified 98.7%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(\left(2 + \frac{-0.5}{x \cdot x}\right) + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

9. Taylor expanded in x around inf

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

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

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

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

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

    3. associate-*r/ N/A

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

    4. unpow2 N/A

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

    5. associate-/r* N/A

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

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

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

11. Simplified 98.5%

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

Alternative 5: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 98.7%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 - \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

6. Taylor expanded in x around inf

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

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

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

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

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

   3. associate-*r/ N/A

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

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

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

   5. distribute-lft-in N/A

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

   6. metadata-eval N/A

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

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

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f32 N/A

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

   13. unpow2 N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 98.5%

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

8. Simplified 98.5%

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

Alternative 6: 98.6% 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 55.4%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 98.5%

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

Alternative 7: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. cancel-sign-sub N/A

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

   4. distribute-lft-neg-in N/A

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

   5. mul-1-neg N/A

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

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

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

   7. remove-double-neg N/A

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

   8. associate-*r/ N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. *-inverses N/A

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

   12. metadata-eval N/A

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

   13. distribute-lft-neg-in N/A

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

   14. neg-mul-1 N/A

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

   15. remove-double-neg N/A

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

5. Simplified 98.4%

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

6. Final simplification 98.4%

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

Alternative 8: 98.2% 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 55.4%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 98.7%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 - \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(2 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
7. 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 98.0%

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

8. Simplified 98.0%

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

Alternative 9: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. associate-*r/ N/A

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

   12. unpow2 N/A

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

   13. times-frac N/A

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

   14. *-inverses N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

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

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. /-lowering-/.f32 98.0%

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

5. Simplified 98.0%

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

   1. *-lft-identity N/A

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

   2. /-lowering-/.f32 98.0%

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

7. Applied egg-rr 98.0%

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

Alternative 10: 96.9% 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 55.4%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 96.8%

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

Developer Target 1: 99.3% 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 2024155 
(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)))))