Rust f32::acosh

Percentage Accurate: 52.4% → 98.7%

Time: 10.5s

Alternatives: 4

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: 52.4% 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: 98.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 98.6%

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

   1. +-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 98.6%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

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

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

9. Applied egg-rr 98.6%

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

10. Taylor expanded in x around inf

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

    1. associate-*r/ N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\left(\frac{-1 \cdot \left(\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right)}{{x}^{3}}\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \left(\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right)\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

    3. distribute-rgt-in N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{8} \cdot -1 + \left(\frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{-1}{8} + \left(\frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{-1}{8}, \left(\left(\frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right)\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

    6. associate-*r/ N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{-1}{8}, \left(\frac{\frac{1}{16} \cdot 1}{{x}^{2}} \cdot -1\right)\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{-1}{8}, \left(\frac{\frac{1}{16}}{{x}^{2}} \cdot -1\right)\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

    8. associate-*l/ N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{-1}{8}, \left(\frac{\frac{1}{16} \cdot -1}{{x}^{2}}\right)\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{-1}{8}, \left(\frac{\frac{-1}{16}}{{x}^{2}}\right)\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

    10. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{/.f32}\left(\frac{-1}{16}, \left({x}^{2}\right)\right)\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{/.f32}\left(\frac{-1}{16}, \left(x \cdot x\right)\right)\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

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

        \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{/.f32}\left(\frac{-1}{16}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left({x}^{3}\right)\right), \mathsf{/.f32}\left(x, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

    13. cube-mult N/A

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

    14. unpow2 N/A

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

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

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

    16. unpow2 N/A

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

    17. *-lowering-*.f32 98.6%

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

12. Simplified 98.6%

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

Alternative 2: 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 56.9%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 98.4%

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

Alternative 3: 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 56.9%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 4: 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 56.9%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 96.4%

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

Developer Target 1: 99.2% 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 2024160 
(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)))))