Rust f32::acosh

Percentage Accurate: 53.4% → 98.8%

Time: 8.1s

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: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 49.9%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 98.4%

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

   1. +-commutative N/A

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

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

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 2: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 49.9%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 98.4%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. associate-+l+ N/A

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

   6. count-2 N/A

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

   7. distribute-rgt-in N/A

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

   8. fma-define N/A

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

7. Applied egg-rr 98.4%

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

Alternative 3: 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 49.9%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. cancel-sign-sub N/A

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

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

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

   5. mul-1-neg N/A

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

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

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

   7. remove-double-neg N/A

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

   8. associate-*r/ N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. *-inverses N/A

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

   12. metadata-eval N/A

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

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

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

   14. neg-mul-1 N/A

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

   15. remove-double-neg N/A

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

5. Simplified 98.0%

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

6. Final simplification 98.0%

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

Alternative 4: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 49.9%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 98.4%

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

6. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. rem-square-sqrt N/A

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

   6. unpow2 N/A

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

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

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

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

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

   9. mul-1-neg N/A

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

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

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

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

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

   12. mul-1-neg N/A

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

   13. distribute-lft-in N/A

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

   14. +-commutative N/A

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

   15. mul-1-neg N/A

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

   16. associate--l+ N/A

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

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

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

8. Simplified 97.4%

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

Alternative 5: 98.3% 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 49.9%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. cancel-sign-sub N/A

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

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

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

   5. mul-1-neg N/A

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

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

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

   7. remove-double-neg N/A

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

   8. associate-*r/ N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. *-inverses N/A

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

   12. metadata-eval N/A

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

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

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

   14. neg-mul-1 N/A

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

   15. remove-double-neg N/A

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

5. Simplified 98.0%

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

6. Taylor expanded in x around inf

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

   1. /-lowering-/.f32 97.4%

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

8. Simplified 97.4%

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

Alternative 6: 97.7% 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 49.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 96.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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 95.9%

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

7. Applied egg-rr 95.9%

   \[\leadsto \color{blue}{\log x + \log 2} \]
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 96.7%

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

9. Applied egg-rr 96.7%

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

Alternative 7: 97.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(x)
	return log(Float32(x + x))
end

MATLAB

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

Derivation

1. Initial program 49.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 96.0%

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

Developer Target 1: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Reproduce

herbie shell --seed 2024154 
(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)))))