Rust f32::acosh

Percentage Accurate: 52.6% → 98.0%

Time: 5.5s

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: 52.6% 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: 98.0% 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 48.1%

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

3. Taylor expanded in x around inf 98.8%

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

   1. associate-*r/ 98.8%

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

   2. metadata-eval 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 2: 97.6% 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 48.1%

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

   1. expm1-log1p-u 47.2%

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

   2. fma-neg 47.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right)\right) \]

   3. metadata-eval 47.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right)\right) \]

4. Applied egg-rr 47.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 48.1%

      \[\leadsto \color{blue}{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)} \]

   2. flip-+ 4.6%

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \sqrt{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, -1\right)}}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right)} \]

   3. log-div 4.6%

      \[\leadsto \color{blue}{\log \left(x \cdot x - \sqrt{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)} \]

   4. add-sqr-sqrt 4.5%

      \[\leadsto \log \left(x \cdot x - \color{blue}{\mathsf{fma}\left(x, x, -1\right)}\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   5. fma-udef 4.5%

      \[\leadsto \log \left(x \cdot x - \color{blue}{\left(x \cdot x + -1\right)}\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   6. associate--r+ 7.0%

      \[\leadsto \log \color{blue}{\left(\left(x \cdot x - x \cdot x\right) - -1\right)} - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   7. +-inverses 7.0%

      \[\leadsto \log \left(\color{blue}{0} - -1\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   8. metadata-eval 7.0%

      \[\leadsto \log \color{blue}{1} - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   9. metadata-eval 7.0%

      \[\leadsto \color{blue}{0} - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

6. Applied egg-rr 7.0%

   \[\leadsto \color{blue}{0 - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)} \]

7. Taylor expanded in x around inf 98.3%

   \[\leadsto 0 - \log \color{blue}{\left(\frac{0.5}{x}\right)} \]

8. Final simplification 98.3%

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

Alternative 3: 96.7% 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 48.1%

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

3. Taylor expanded in x around inf 97.9%

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

4. Final simplification 97.9%

   \[\leadsto \log \left(x + x\right) \]
5. Add Preprocessing

Alternative 4: 24.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(2.0520833333333335\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (expm1 2.0520833333333335))

C

float code(float x) {
	return expm1f(2.0520833333333335f);
}

Julia

function code(x)
	return expm1(Float32(2.0520833333333335))
end

Derivation

1. Initial program 48.1%

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

   1. expm1-log1p-u 47.2%

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

   2. fma-neg 47.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right)\right) \]

   3. metadata-eval 47.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right)\right) \]

4. Applied egg-rr 47.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)\right)} \]

5. Taylor expanded in x around -inf -0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\log -0.5 + \log \left(\frac{-1}{x}\right)\right)\right) + \left(0.041666666666666664 \cdot \frac{2.25 \cdot \frac{1}{1 + \left(\log -0.5 + \log \left(\frac{-1}{x}\right)\right)} - 0.75 \cdot \frac{1}{{\left(1 + \left(\log -0.5 + \log \left(\frac{-1}{x}\right)\right)\right)}^{2}}}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2} \cdot \left(1 + \left(\log -0.5 + \log \left(\frac{-1}{x}\right)\right)\right)}\right)}\right) \]

7. Simplified 24.4%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{2.0520833333333335}\right) \]

8. Final simplification 24.4%

   \[\leadsto \mathsf{expm1}\left(2.0520833333333335\right) \]
9. Add Preprocessing

Alternative 5: 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 48.1%

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

3. Taylor expanded in x around inf 98.8%

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

   1. associate-*r/ 98.8%

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

   2. metadata-eval 98.8%

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

5. Simplified 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. fma-def 98.8%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(1, x, x - \frac{0.5}{x}\right)\right)} \]

   3. clear-num 98.8%

      \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \color{blue}{\frac{1}{\frac{x}{0.5}}}\right)\right) \]

   4. div-inv 98.8%

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

   5. metadata-eval 98.8%

      \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \frac{1}{x \cdot \color{blue}{2}}\right)\right) \]

   6. *-commutative 98.8%

      \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \frac{1}{\color{blue}{2 \cdot x}}\right)\right) \]

   7. count-2 98.8%

      \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \frac{1}{\color{blue}{x + x}}\right)\right) \]

   8. flip-+ -0.0%

      \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \frac{1}{\color{blue}{\frac{x \cdot x - x \cdot x}{x - x}}}\right)\right) \]

   9. +-inverses -0.0%

      \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \frac{1}{\frac{\color{blue}{0}}{x - x}}\right)\right) \]

   10. +-inverses -0.0%

       \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \frac{1}{\frac{\color{blue}{x - x}}{x - x}}\right)\right) \]

   11. +-inverses -0.0%

       \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \frac{1}{\frac{x - x}{\color{blue}{0}}}\right)\right) \]

   12. +-inverses -0.0%

       \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \frac{1}{\frac{x - x}{\color{blue}{x \cdot x - x \cdot x}}}\right)\right) \]

   13. clear-num -0.0%

       \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \color{blue}{\frac{x \cdot x - x \cdot x}{x - x}}\right)\right) \]

   14. +-inverses -0.0%

       \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \frac{\color{blue}{0}}{x - x}\right)\right) \]

   15. +-inverses -0.0%

       \[\leadsto \log \left(\mathsf{fma}\left(1, x, x - \frac{0}{\color{blue}{0}}\right)\right) \]

7. Applied egg-rr -0.0%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(1, x, x - \frac{0}{0}\right)\right)} \]

9. Simplified 44.1%

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

10. Taylor expanded in x around inf 44.6%

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

11. Final simplification 44.6%

    \[\leadsto \log x \]
12. Add Preprocessing

Alternative 6: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2.09375 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.09375)

C

float code(float x) {
	return 2.09375f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.09375e0
end function

Julia

function code(x)
	return Float32(2.09375)
end

MATLAB

function tmp = code(x)
	tmp = single(2.09375);
end

Derivation

1. Initial program 48.1%

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

   1. expm1-log1p-u 47.2%

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

   2. fma-neg 47.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right)\right) \]

   3. metadata-eval 47.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right)\right) \]

4. Applied egg-rr 47.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)\right)} \]

5. Taylor expanded in x around -inf -0.0%

   \[\leadsto \color{blue}{\log -0.5 + \left(\log \left(\frac{-1}{x}\right) + \left(0.09375 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

7. Simplified 22.0%

   \[\leadsto \color{blue}{2.09375} \]

8. Final simplification 22.0%

   \[\leadsto 2.09375 \]
9. Add Preprocessing

Alternative 7: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2.1458333333333335 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.1458333333333335)

C

float code(float x) {
	return 2.1458333333333335f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.1458333333333335e0
end function

Julia

function code(x)
	return Float32(2.1458333333333335)
end

MATLAB

function tmp = code(x)
	tmp = single(2.1458333333333335);
end

Derivation

1. Initial program 48.1%

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

   1. expm1-log1p-u 47.2%

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

   2. fma-neg 47.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right)\right) \]

   3. metadata-eval 47.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right)\right) \]

4. Applied egg-rr 47.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)\right)} \]

5. Taylor expanded in x around -inf -0.0%

   \[\leadsto \color{blue}{\log -0.5 + \left(\log \left(\frac{-1}{x}\right) + \left(0.052083333333333336 \cdot \frac{1}{{x}^{6}} + \left(0.09375 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

7. Simplified 22.0%

   \[\leadsto \color{blue}{2.1458333333333335} \]

8. Final simplification 22.0%

   \[\leadsto 2.1458333333333335 \]
9. Add Preprocessing

Developer target: 99.0% 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 2024013 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

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