Rust f32::acosh

Percentage Accurate: 53.0% → 98.9%

Time: 9.2s

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: 53.0% 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, 1.6× speedup

Math

\[\begin{array}{l} \\ \log \left(x + x \cdot \left(\left(1 - \frac{0.125 + \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) + \frac{-0.5}{x \cdot x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (+
   x
   (*
    x
    (+
     (- 1.0 (/ (+ 0.125 (/ 0.0625 (* x x))) (* x (* x (* x x)))))
     (/ -0.5 (* x x)))))))

C

float code(float x) {
	return logf((x + (x * ((1.0f - ((0.125f + (0.0625f / (x * x))) / (x * (x * (x * x))))) + (-0.5f / (x * x))))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + (x * ((1.0e0 - ((0.125e0 + (0.0625e0 / (x * x))) / (x * (x * (x * x))))) + ((-0.5e0) / (x * x))))))
end function

Julia

function code(x)
	return log(Float32(x + Float32(x * Float32(Float32(Float32(1.0) - Float32(Float32(Float32(0.125) + Float32(Float32(0.0625) / Float32(x * x))) / Float32(x * Float32(x * Float32(x * x))))) + Float32(Float32(-0.5) / Float32(x * x))))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + (x * ((single(1.0) - ((single(0.125) + (single(0.0625) / (x * x))) / (x * (x * (x * x))))) + (single(-0.5) / (x * x))))));
end

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around inf 99.4%

   \[\leadsto \log \left(x + \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - 0.5 \cdot \frac{1}{{x}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. sub-neg 99.4%

      \[\leadsto \log \left(x + x \cdot \color{blue}{\left(\left(1 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

   2. mul-1-neg 99.4%

      \[\leadsto \log \left(x + x \cdot \left(\left(1 + \color{blue}{\left(-\frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   3. unsub-neg 99.4%

      \[\leadsto \log \left(x + x \cdot \left(\color{blue}{\left(1 - \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   4. unpow2 99.4%

      \[\leadsto \log \left(x + x \cdot \left(\left(1 - \frac{0.125 + 0.0625 \cdot \frac{1}{\color{blue}{x \cdot x}}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ 99.4%

      \[\leadsto \log \left(x + x \cdot \left(\left(1 - \frac{0.125 + \color{blue}{\frac{0.0625 \cdot 1}{x \cdot x}}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval 99.4%

      \[\leadsto \log \left(x + x \cdot \left(\left(1 - \frac{0.125 + \frac{\color{blue}{0.0625}}{x \cdot x}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   7. metadata-eval 99.4%

      \[\leadsto \log \left(x + x \cdot \left(\left(1 - \frac{0.125 + \frac{0.0625}{x \cdot x}}{{x}^{\color{blue}{\left(3 + 1\right)}}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   8. pow-plus 99.4%

      \[\leadsto \log \left(x + x \cdot \left(\left(1 - \frac{0.125 + \frac{0.0625}{x \cdot x}}{\color{blue}{{x}^{3} \cdot x}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   9. cube-unmult 99.4%

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

   10. *-commutative 99.4%

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

   11. unpow2 99.4%

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

   12. associate-*r/ 99.4%

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

   13. metadata-eval 99.4%

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

5. Simplified 99.4%

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

Alternative 2: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around inf 99.4%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)} \]
4. Step-by-step derivation

   1. sub-neg 99.4%

      \[\leadsto \log \left(x \cdot \color{blue}{\left(\left(2 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

   2. mul-1-neg 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 + \color{blue}{\left(-\frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   3. unsub-neg 99.4%

      \[\leadsto \log \left(x \cdot \left(\color{blue}{\left(2 - \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   4. unpow2 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 - \frac{0.125 + 0.0625 \cdot \frac{1}{\color{blue}{x \cdot x}}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 - \frac{0.125 + \color{blue}{\frac{0.0625 \cdot 1}{x \cdot x}}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 - \frac{0.125 + \frac{\color{blue}{0.0625}}{x \cdot x}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   7. metadata-eval 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 - \frac{0.125 + \frac{0.0625}{x \cdot x}}{{x}^{\color{blue}{\left(3 + 1\right)}}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   8. pow-plus 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 - \frac{0.125 + \frac{0.0625}{x \cdot x}}{\color{blue}{{x}^{3} \cdot x}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   9. cube-unmult 99.4%

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

   10. *-commutative 99.4%

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

   11. unpow2 99.4%

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

   12. associate-*r/ 99.4%

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

   13. metadata-eval 99.4%

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

5. Simplified 99.4%

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

   1. associate-/r* 99.4%

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

7. Applied egg-rr 99.4%

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

Alternative 3: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around inf 99.4%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)} \]
4. Step-by-step derivation

   1. sub-neg 99.4%

      \[\leadsto \log \left(x \cdot \color{blue}{\left(\left(2 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

   2. mul-1-neg 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 + \color{blue}{\left(-\frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   3. unsub-neg 99.4%

      \[\leadsto \log \left(x \cdot \left(\color{blue}{\left(2 - \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   4. unpow2 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 - \frac{0.125 + 0.0625 \cdot \frac{1}{\color{blue}{x \cdot x}}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 - \frac{0.125 + \color{blue}{\frac{0.0625 \cdot 1}{x \cdot x}}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 - \frac{0.125 + \frac{\color{blue}{0.0625}}{x \cdot x}}{{x}^{4}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   7. metadata-eval 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 - \frac{0.125 + \frac{0.0625}{x \cdot x}}{{x}^{\color{blue}{\left(3 + 1\right)}}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   8. pow-plus 99.4%

      \[\leadsto \log \left(x \cdot \left(\left(2 - \frac{0.125 + \frac{0.0625}{x \cdot x}}{\color{blue}{{x}^{3} \cdot x}}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

   9. cube-unmult 99.4%

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

   10. *-commutative 99.4%

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

   11. unpow2 99.4%

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

   12. associate-*r/ 99.4%

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

   13. metadata-eval 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \left(\left(1 + x \cdot \left(1 - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right)\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (+ x (+ (+ 1.0 (* x (- 1.0 (/ (+ 0.5 (/ 0.125 (* x x))) (* x x))))) -1.0))))

C

float code(float x) {
	return logf((x + ((1.0f + (x * (1.0f - ((0.5f + (0.125f / (x * x))) / (x * x))))) + -1.0f)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + ((1.0e0 + (x * (1.0e0 - ((0.5e0 + (0.125e0 / (x * x))) / (x * x))))) + (-1.0e0))))
end function

Julia

function code(x)
	return log(Float32(x + Float32(Float32(Float32(1.0) + Float32(x * Float32(Float32(1.0) - Float32(Float32(Float32(0.5) + Float32(Float32(0.125) / Float32(x * x))) / Float32(x * x))))) + Float32(-1.0))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + ((single(1.0) + (x * (single(1.0) - ((single(0.5) + (single(0.125) / (x * x))) / (x * x))))) + single(-1.0))));
end

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around inf 99.2%

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

   1. mul-1-neg 99.2%

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

   2. unsub-neg 99.2%

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

   3. unpow2 99.2%

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

   4. associate-*r/ 99.2%

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

   5. metadata-eval 99.2%

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

   6. unpow2 99.2%

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

5. Simplified 99.2%

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

   1. expm1-log1p-u 98.7%

      \[\leadsto \log \left(x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(1 - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right)\right)\right)}\right) \]

   2. expm1-undefine 98.7%

      \[\leadsto \log \left(x + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(1 - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right)\right)} - 1\right)}\right) \]

7. Applied egg-rr 98.7%

   \[\leadsto \log \left(x + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(1 - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right)\right)} - 1\right)}\right) \]
8. Step-by-step derivation

   1. sub-neg 98.7%

      \[\leadsto \log \left(x + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(1 - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right)\right)} + \left(-1\right)\right)}\right) \]

   2. log1p-undefine 98.7%

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

   3. rem-exp-log 99.2%

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

   4. metadata-eval 99.2%

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

9. Simplified 99.2%

   \[\leadsto \log \left(x + \color{blue}{\left(\left(1 + x \cdot \left(1 - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right)\right) + -1\right)}\right) \]
10. Add Preprocessing

Alternative 5: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \log \left(x + x \cdot \left(1 - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log (+ x (* x (- 1.0 (/ (+ 0.5 (/ 0.125 (* x x))) (* x x)))))))

C

float code(float x) {
	return logf((x + (x * (1.0f - ((0.5f + (0.125f / (x * x))) / (x * x))))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + (x * (1.0e0 - ((0.5e0 + (0.125e0 / (x * x))) / (x * x))))))
end function

Julia

function code(x)
	return log(Float32(x + Float32(x * Float32(Float32(1.0) - Float32(Float32(Float32(0.5) + Float32(Float32(0.125) / Float32(x * x))) / Float32(x * x))))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + (x * (single(1.0) - ((single(0.5) + (single(0.125) / (x * x))) / (x * x))))));
end

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around inf 99.2%

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

   1. mul-1-neg 99.2%

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

   2. unsub-neg 99.2%

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

   3. unpow2 99.2%

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

   4. associate-*r/ 99.2%

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

   5. metadata-eval 99.2%

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

   6. unpow2 99.2%

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

5. Simplified 99.2%

   \[\leadsto \log \left(x + \color{blue}{x \cdot \left(1 - \frac{0.5 + \frac{0.125}{x \cdot x}}{x \cdot x}\right)}\right) \]
6. Add Preprocessing

Alternative 6: 98.7% 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 54.3%

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

3. Taylor expanded in x around inf 99.2%

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

   1. mul-1-neg 99.2%

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

   2. unsub-neg 99.2%

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

   3. unpow2 99.2%

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

   4. associate-*r/ 99.2%

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

   5. metadata-eval 99.2%

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

   6. unpow2 99.2%

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

5. Simplified 99.2%

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

Alternative 7: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around inf 98.7%

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

   1. sub-neg 98.7%

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

   2. unpow2 98.7%

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

   3. associate-*r/ 98.7%

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

   4. metadata-eval 98.7%

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

   5. distribute-neg-frac 98.7%

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

   6. metadata-eval 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 53.0%

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

   1. unpow2 53.0%

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

8. Simplified 53.0%

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

   1. div-sub 53.0%

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

   2. pow2 53.0%

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

   3. pow1 53.0%

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

   4. pow-div 98.7%

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

   5. metadata-eval 98.7%

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

   6. pow1 98.7%

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

10. Applied egg-rr 98.7%

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

    1. associate-+r- 98.7%

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

    2. sub-neg 98.7%

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

    3. count-2 98.7%

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

    4. *-commutative 98.7%

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

    5. distribute-neg-frac 98.7%

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

    6. metadata-eval 98.7%

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

12. Simplified 98.7%

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

13. Final simplification 98.7%

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

Alternative 8: 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 54.3%

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

3. Taylor expanded in x around inf 96.5%

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

Developer target: 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 2024107 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :alt
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

  (log (+ x (sqrt (- (* x x) 1.0)))))