Rust f32::acosh

Percentage Accurate: 53.7% → 98.8%

Time: 8.3s

Alternatives: 7

Speedup: 2.0×

Specification

\[x \geq 1\]

Math

\[\begin{array}{l} \\ \cosh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (acosh x))

C

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

Julia

function code(x)
	return acosh(x)
end

MATLAB

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.125 + \frac{-0.0625}{x \cdot x}\\ t_1 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_2 := \frac{t\_0}{t\_1}\\ \log \left(x \cdot \frac{\frac{t\_2}{\frac{t\_1}{t\_0}} - 4}{t\_2 - 2} - \frac{0.5}{x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ -0.125 (/ -0.0625 (* x x))))
        (t_1 (* x (* x (* x x))))
        (t_2 (/ t_0 t_1)))
   (log (- (* x (/ (- (/ t_2 (/ t_1 t_0)) 4.0) (- t_2 2.0))) (/ 0.5 x)))))

C

float code(float x) {
	float t_0 = -0.125f + (-0.0625f / (x * x));
	float t_1 = x * (x * (x * x));
	float t_2 = t_0 / t_1;
	return logf(((x * (((t_2 / (t_1 / t_0)) - 4.0f) / (t_2 - 2.0f))) - (0.5f / x)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: t_2
    t_0 = (-0.125e0) + ((-0.0625e0) / (x * x))
    t_1 = x * (x * (x * x))
    t_2 = t_0 / t_1
    code = log(((x * (((t_2 / (t_1 / t_0)) - 4.0e0) / (t_2 - 2.0e0))) - (0.5e0 / x)))
end function

Julia

function code(x)
	t_0 = Float32(Float32(-0.125) + Float32(Float32(-0.0625) / Float32(x * x)))
	t_1 = Float32(x * Float32(x * Float32(x * x)))
	t_2 = Float32(t_0 / t_1)
	return log(Float32(Float32(x * Float32(Float32(Float32(t_2 / Float32(t_1 / t_0)) - Float32(4.0)) / Float32(t_2 - Float32(2.0)))) - Float32(Float32(0.5) / x)))
end

MATLAB

function tmp = code(x)
	t_0 = single(-0.125) + (single(-0.0625) / (x * x));
	t_1 = x * (x * (x * x));
	t_2 = t_0 / t_1;
	tmp = log(((x * (((t_2 / (t_1 / t_0)) - single(4.0)) / (t_2 - single(2.0)))) - (single(0.5) / x)));
end

Derivation

1. Initial program 56.8%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 99.1%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 - \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]
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. metadata-eval N/A

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

   6. distribute-frac-neg2 N/A

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

   7. fmm-undef N/A

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

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

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

7. Applied egg-rr 99.1%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

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

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

9. Applied egg-rr 99.1%

   \[\leadsto \log \left(x \cdot \color{blue}{\frac{\frac{\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}{-0.125 + \frac{-0.0625}{x \cdot x}}} - 4}{\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} - 2}} - \frac{0.5}{x}\right) \]
10. Add Preprocessing

Alternative 2: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 99.1%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 - \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]
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. metadata-eval N/A

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

   6. distribute-frac-neg2 N/A

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

   7. fmm-undef N/A

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

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

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

7. Applied egg-rr 99.1%

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

Alternative 3: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 99.1%

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

6. Taylor expanded in x around inf

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

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

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

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

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

   3. associate-*r/ N/A

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

   4. mul-1-neg N/A

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

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

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

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

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

   14. unpow2 N/A

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

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

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

   16. unpow2 N/A

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

   17. *-lowering-*.f32 98.9%

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

8. Simplified 98.9%

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

Alternative 4: 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 56.8%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 98.9%

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

Alternative 5: 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 56.8%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 98.9%

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

6. Taylor expanded in x around inf

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

   1. /-lowering-/.f32 98.5%

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

8. Simplified 98.5%

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

Alternative 6: 97.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 97.0%

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

   1. count-2 N/A

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

   2. *-commutative N/A

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

   3. log-prod N/A

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

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

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

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

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

   6. log-lowering-log.f32 96.6%

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

7. Applied egg-rr 96.6%

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

   1. sum-log N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. clear-num N/A

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

   5. log-rec N/A

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

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

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

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

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

   8. /-lowering-/.f32 97.3%

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

9. Applied egg-rr 97.3%

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

Alternative 7: 96.8% 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 56.8%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 97.0%

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

Developer Target 1: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Reproduce

herbie shell --seed 2024158 
(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)))))