Rust f32::acosh

Percentage Accurate: 52.7% → 99.0%

Time: 6.6s

Alternatives: 6

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: 52.7% 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: 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

Derivation

1. Initial program 52.4%

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

   1. difference-of-sqr-1 52.4%

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

   2. sqrt-prod 99.6%

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

   3. sub-neg 99.6%

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

   4. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 97.9% 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 52.4%

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

2. Taylor expanded in x around inf 97.5%

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

   1. associate-*r/ 97.5%

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

   2. metadata-eval 97.5%

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

4. Simplified 97.5%

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

5. Final simplification 97.5%

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

Alternative 3: 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 52.4%

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

   1. difference-of-sqr-1 52.4%

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

   2. sqrt-prod 99.6%

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

   3. sub-neg 99.6%

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

   4. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

   1. sqrt-unprod 52.4%

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

   2. metadata-eval 52.4%

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

   3. sub-neg 52.4%

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

   4. difference-of-sqr-1 52.4%

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

   5. fma-neg 52.4%

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

   6. metadata-eval 52.4%

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

   7. flip-+ 9.2%

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

   8. log-div 9.2%

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

   9. add-sqr-sqrt 9.1%

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

   10. fma-udef 9.1%

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

   11. associate--r+ 11.4%

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

   12. +-inverses 11.4%

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

   13. metadata-eval 11.4%

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

   14. metadata-eval 11.4%

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

5. Applied egg-rr 11.4%

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

6. Taylor expanded in x around inf 96.1%

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

7. Final simplification 96.1%

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

Alternative 4: 96.6% 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 52.4%

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

2. Taylor expanded in x around inf 95.7%

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

3. Final simplification 95.7%

   \[\leadsto \log \left(x + x\right) \]

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 52.4%

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

   1. add-cbrt-cube 39.3%

      \[\leadsto \log \left(x + \sqrt{x \cdot \color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}} - 1}\right) \]

   2. cbrt-prod 52.4%

      \[\leadsto \log \left(x + \sqrt{x \cdot \color{blue}{\left(\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x}\right)} - 1}\right) \]

   3. associate-*r* 52.4%

      \[\leadsto \log \left(x + \sqrt{\color{blue}{\left(x \cdot \sqrt[3]{x \cdot x}\right) \cdot \sqrt[3]{x}} - 1}\right) \]

   4. fma-neg 52.4%

      \[\leadsto \log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x \cdot \sqrt[3]{x \cdot x}, \sqrt[3]{x}, -1\right)}}\right) \]

   5. add-cube-cbrt 52.4%

      \[\leadsto \log \left(x + \sqrt{\mathsf{fma}\left(\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x \cdot x}, \sqrt[3]{x}, -1\right)}\right) \]

   6. pow3 52.4%

      \[\leadsto \log \left(x + \sqrt{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{3}} \cdot \sqrt[3]{x \cdot x}, \sqrt[3]{x}, -1\right)}\right) \]

   7. cbrt-prod 52.4%

      \[\leadsto \log \left(x + \sqrt{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{3} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}, \sqrt[3]{x}, -1\right)}\right) \]

   8. pow2 52.4%

      \[\leadsto \log \left(x + \sqrt{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt[3]{x}, -1\right)}\right) \]

   9. pow-prod-up 52.4%

      \[\leadsto \log \left(x + \sqrt{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(3 + 2\right)}}, \sqrt[3]{x}, -1\right)}\right) \]

   10. metadata-eval 52.4%

       \[\leadsto \log \left(x + \sqrt{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{\color{blue}{5}}, \sqrt[3]{x}, -1\right)}\right) \]

   11. metadata-eval 52.4%

       \[\leadsto \log \left(x + \sqrt{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{5}, \sqrt[3]{x}, \color{blue}{-1}\right)}\right) \]

3. Applied egg-rr 52.4%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{5}, \sqrt[3]{x}, -1\right)}}\right) \]
4. Step-by-step derivation

   1. fma-udef 52.4%

      \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt[3]{x}\right)}^{5} \cdot \sqrt[3]{x} + -1}}\right) \]

   2. +-commutative 52.4%

      \[\leadsto \log \left(x + \sqrt{\color{blue}{-1 + {\left(\sqrt[3]{x}\right)}^{5} \cdot \sqrt[3]{x}}}\right) \]

   3. pow-plus 52.4%

      \[\leadsto \log \left(x + \sqrt{-1 + \color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(5 + 1\right)}}}\right) \]

   4. metadata-eval 52.4%

      \[\leadsto \log \left(x + \sqrt{-1 + {\left(\sqrt[3]{x}\right)}^{\color{blue}{6}}}\right) \]

5. Simplified 52.4%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{-1 + {\left(\sqrt[3]{x}\right)}^{6}}}\right) \]

6. Taylor expanded in x around inf 43.9%

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

   1. mul-1-neg 43.9%

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

   2. log-rec 43.9%

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

   3. remove-double-neg 43.9%

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

8. Simplified 43.9%

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

9. Final simplification 43.9%

   \[\leadsto \log x \]

Alternative 6: 3.0% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -3.0)

C

float code(float x) {
	return -3.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -3.0e0
end function

Julia

function code(x)
	return Float32(-3.0)
end

MATLAB

function tmp = code(x)
	tmp = single(-3.0);
end

Derivation

1. Initial program 52.4%

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

   1. difference-of-sqr-1 52.4%

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

   2. sqrt-prod 99.6%

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

   3. sub-neg 99.6%

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

   4. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

4. Taylor expanded in x around 0 -0.0%

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

6. Simplified 2.1%

   \[\leadsto \color{blue}{-3 + \frac{x}{-3}} \]

7. Taylor expanded in x around 0 3.0%

   \[\leadsto \color{blue}{-3} \]

8. Final simplification 3.0%

   \[\leadsto -3 \]

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 2023334 
(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)))))