Rust f32::acosh

Percentage Accurate: 53.5% → 99.5%

Time: 15.1s

Alternatives: 9

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: 53.5% 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: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ -\log \left(\frac{\frac{1}{t\_0}}{x + \frac{\frac{-0.5}{x}}{t\_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ 2.0 (/ (+ -0.125 (/ -0.0625 (* x x))) (* x (* x (* x x)))))))
   (- (log (/ (/ 1.0 t_0) (+ x (/ (/ -0.5 x) t_0)))))))

C

float code(float x) {
	float t_0 = 2.0f + ((-0.125f + (-0.0625f / (x * x))) / (x * (x * (x * x))));
	return -logf(((1.0f / t_0) / (x + ((-0.5f / x) / t_0))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    real(4) :: t_0
    t_0 = 2.0e0 + (((-0.125e0) + ((-0.0625e0) / (x * x))) / (x * (x * (x * x))))
    code = -log(((1.0e0 / t_0) / (x + (((-0.5e0) / x) / t_0))))
end function

Julia

function code(x)
	t_0 = Float32(Float32(2.0) + Float32(Float32(Float32(-0.125) + Float32(Float32(-0.0625) / Float32(x * x))) / Float32(x * Float32(x * Float32(x * x)))))
	return Float32(-log(Float32(Float32(Float32(1.0) / t_0) / Float32(x + Float32(Float32(Float32(-0.5) / x) / t_0)))))
end

MATLAB

function tmp = code(x)
	t_0 = single(2.0) + ((single(-0.125) + (single(-0.0625) / (x * x))) / (x * (x * (x * x))));
	tmp = -log(((single(1.0) / t_0) / (x + ((single(-0.5) / x) / t_0))));
end

Derivation

1. Initial program 55.0%

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

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

7. Applied egg-rr 97.7%

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

   1. clear-num N/A

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

   2. frac-add N/A

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

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

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

9. Applied egg-rr 98.4%

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

    1. clear-num N/A

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

    2. log-rec N/A

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

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

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

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

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

11. Applied egg-rr 99.2%

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

Alternative 2: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float x) {
	return logf(((-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((((-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(Float32(Float32(-0.5) / x) + 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))))))))
end

MATLAB

function tmp = code(x)
	tmp = log(((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 55.0%

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

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

Alternative 3: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.0%

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

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

6. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. unpow2 N/A

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

   13. cube-mult N/A

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

   14. times-frac N/A

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

   15. *-inverses N/A

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

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

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

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

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

8. Simplified 98.4%

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f32 98.4%

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

10. Applied egg-rr 98.4%

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

11. Final simplification 98.4%

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

Alternative 4: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \log \left(2 \cdot x + \frac{1}{x} \cdot \left(-0.5 + \frac{-0.125}{x \cdot x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log (+ (* 2.0 x) (* (/ 1.0 x) (+ -0.5 (/ -0.125 (* x x)))))))

C

float code(float x) {
	return logf(((2.0f * x) + ((1.0f / x) * (-0.5f + (-0.125f / (x * x))))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(((2.0e0 * x) + ((1.0e0 / x) * ((-0.5e0) + ((-0.125e0) / (x * x))))))
end function

Julia

function code(x)
	return log(Float32(Float32(Float32(2.0) * x) + Float32(Float32(Float32(1.0) / x) * Float32(Float32(-0.5) + Float32(Float32(-0.125) / Float32(x * x))))))
end

MATLAB

function tmp = code(x)
	tmp = log(((single(2.0) * x) + ((single(1.0) / x) * (single(-0.5) + (single(-0.125) / (x * x))))));
end

Derivation

1. Initial program 55.0%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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. mul-1-neg N/A

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

   3. distribute-neg-frac N/A

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

   4. associate-*l/ N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. associate-/l* N/A

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

   8. *-inverses N/A

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

   9. *-rgt-identity N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. /-lowering-/.f32 N/A

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

5. Simplified 98.2%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. sub-neg N/A

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

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

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. /-lowering-/.f32 N/A

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

   10. *-lowering-*.f32 98.2%

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

7. Applied egg-rr 98.2%

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

8. Final simplification 98.2%

   \[\leadsto \log \left(2 \cdot x + \frac{1}{x} \cdot \left(-0.5 + \frac{-0.125}{x \cdot x}\right)\right) \]
9. Add Preprocessing

Alternative 5: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.0%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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. mul-1-neg N/A

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

   3. distribute-neg-frac N/A

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

   4. associate-*l/ N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. associate-/l* N/A

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

   8. *-inverses N/A

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

   9. *-rgt-identity N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. /-lowering-/.f32 N/A

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

5. Simplified 98.2%

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

6. Final simplification 98.2%

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

Alternative 6: 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 55.0%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. cancel-sign-sub N/A

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

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

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

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

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

   8. metadata-eval N/A

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

   9. *-rgt-identity N/A

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

   10. *-commutative N/A

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

   11. associate-*r/ N/A

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

   12. unpow2 N/A

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

   13. times-frac N/A

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

   14. *-inverses N/A

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

   15. metadata-eval N/A

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

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

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

   17. neg-mul-1 N/A

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

5. Simplified 98.2%

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

6. Final simplification 98.2%

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

Alternative 7: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.0%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. metadata-eval N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. *-rgt-identity N/A

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

   11. unpow2 N/A

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

   12. associate-/r* N/A

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

   13. *-inverses N/A

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

   14. associate-*l/ N/A

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

   15. metadata-eval N/A

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

   16. /-lowering-/.f32 97.6%

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

5. Simplified 97.6%

   \[\leadsto \log \left(x + \color{blue}{\left(x + \frac{-0.5}{x}\right)}\right) \]
6. 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 55.0%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 95.8%

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

Alternative 9: 44.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(x)
	return log(x)
end

MATLAB

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

Derivation

1. Initial program 55.0%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 95.8%

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

   1. flip-+ N/A

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

   2. +-inverses N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{0}{x - x}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{0 + 0}{x - x}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\frac{-1}{2} \cdot 0 + 0}{x - x}\right)\right) \]

   5. +-inverses N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + 0}{x - x}\right)\right) \]

   6. +-inverses N/A

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

   7. distribute-lft-out-- N/A

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

   8. +-inverses N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + x \cdot 0}{x - x}\right)\right) \]

   9. +-inverses N/A

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

   10. +-inverses N/A

       \[\leadsto \mathsf{log.f32}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + x \cdot \left(x \cdot x - x \cdot x\right)}{0}\right)\right) \]

   11. +-inverses N/A

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

   12. distribute-lft-out-- N/A

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

   13. frac-add N/A

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

   14. flip-+ N/A

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

   15. associate-+r+ N/A

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

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

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

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

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

   18. /-lowering-/.f32 97.6%

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

7. Applied egg-rr 97.6%

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

8. Taylor expanded in x around 0

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

   1. /-lowering-/.f32 43.6%

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

10. Simplified 43.6%

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

11. Taylor expanded in x around inf

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

    1. mul-1-neg N/A

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

    2. log-rec N/A

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

    3. remove-double-neg N/A

       \[\leadsto \log x \]

    4. log-lowering-log.f32 43.8%

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

13. Simplified 43.8%

    \[\leadsto \color{blue}{\log x} \]
14. 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 2024191 
(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)))))