Rust f32::acosh

Percentage Accurate: 53.4% → 98.3%

Time: 5.4s

Alternatives: 5

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: 98.3% 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 98.9%

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

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 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. Step-by-step derivation

   1. add-sqr-sqrt 49.3%

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

   2. pow2 49.3%

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

   3. fma-neg 49.3%

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

   4. metadata-eval 49.3%

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

4. Applied egg-rr 49.3%

   \[\leadsto \color{blue}{{\left(\sqrt{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}\right)}^{2}} \]
5. Step-by-step derivation

   1. unpow2 49.3%

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

   2. add-sqr-sqrt 49.9%

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

   3. flip-+ 7.0%

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

   4. log-div 7.1%

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

   5. add-sqr-sqrt 6.7%

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

   6. fma-udef 6.7%

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

   7. associate--r+ 9.1%

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

   8. +-inverses 9.1%

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

   9. metadata-eval 9.1%

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

   10. metadata-eval 9.1%

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

6. Applied egg-rr 9.1%

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

7. Taylor expanded in x around inf 98.0%

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

8. Final simplification 98.0%

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

Alternative 3: 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 97.6%

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

4. Final simplification 97.6%

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

Alternative 4: 21.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.1068793402777777 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.1068793402777777)

C

float code(float x) {
	return 1.1068793402777777f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.1068793402777777e0
end function

Julia

function code(x)
	return Float32(1.1068793402777777)
end

MATLAB

function tmp = code(x)
	tmp = single(1.1068793402777777);
end

Derivation

1. Initial program 49.9%

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

   1. add-sqr-sqrt 49.3%

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

   2. pow2 49.3%

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

   3. fma-neg 49.3%

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

   4. metadata-eval 49.3%

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

4. Applied egg-rr 49.3%

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

5. Taylor expanded in x around inf 97.6%

   \[\leadsto {\color{blue}{\left(\sqrt{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)} + \left(-0.5 \cdot \left(\frac{0.052083333333333336 + 0.125 \cdot \frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{6}} \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right) + \left(-0.5 \cdot \left(\frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{{x}^{4}} \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right) + -0.125 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)\right)\right)\right)}}^{2} \]

7. Simplified 21.1%

   \[\leadsto {\color{blue}{1.0520833333333333}}^{2} \]
8. Step-by-step derivation

   1. metadata-eval 21.1%

      \[\leadsto \color{blue}{1.1068793402777777} \]

9. Applied egg-rr 21.1%

   \[\leadsto \color{blue}{1.1068793402777777} \]

10. Final simplification 21.1%

    \[\leadsto 1.1068793402777777 \]
11. Add Preprocessing

Alternative 5: 23.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 4.348958333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 4.348958333333333)

C

float code(float x) {
	return 4.348958333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 4.348958333333333e0
end function

Julia

function code(x)
	return Float32(4.348958333333333)
end

MATLAB

function tmp = code(x)
	tmp = single(4.348958333333333);
end

Derivation

1. Initial program 49.9%

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

   1. add-sqr-sqrt 49.3%

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

   2. pow2 49.3%

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

   3. fma-neg 49.3%

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

   4. metadata-eval 49.3%

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

4. Applied egg-rr 49.3%

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

5. Taylor expanded in x around inf 98.7%

   \[\leadsto \color{blue}{\left(\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1 \cdot \frac{0.052083333333333336 + 0.125 \cdot \frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}{{x}^{6}} + \left(-1 \cdot \frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{{x}^{4}} + \left(0.015625 \cdot \frac{1}{{x}^{4} \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right)} + 0.125 \cdot \frac{0.09375 + {\left(-0.125 \cdot \sqrt{\frac{1}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}}\right)}^{2}}{{x}^{6} \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right)}\right)\right)\right)\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]

7. Simplified 23.8%

   \[\leadsto \color{blue}{4.348958333333333} \]

8. Final simplification 23.8%

   \[\leadsto 4.348958333333333 \]
9. Add Preprocessing

Developer target: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Reproduce

herbie shell --seed 2024031 
(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)))))