Rust f32::acosh

Percentage Accurate: 52.7% → 98.2%

Time: 7.2s

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: 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 51.0%

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

2. Taylor expanded in x around inf 99.1%

   \[\leadsto \log \color{blue}{\left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right)} \]
3. Step-by-step derivation

   1. *-commutative 99.1%

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

   2. associate-*r/ 99.1%

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

   3. metadata-eval 99.1%

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

4. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 43.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.0%

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

   1. pow1/2 51.0%

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

   2. pow-to-exp 50.8%

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

   3. fma-neg 50.8%

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

   4. metadata-eval 50.8%

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

3. Applied egg-rr 50.8%

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

4. Taylor expanded in x around 0 -0.0%

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

6. Simplified 44.3%

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

7. Final simplification 44.3%

   \[\leadsto \log \left(x + -1\right) \]

Alternative 3: 96.9% 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 51.0%

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

2. Taylor expanded in x around inf 97.9%

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

3. Final simplification 97.9%

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

Alternative 4: 11.7% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* x 0.3333333333333333))

C

float code(float x) {
	return x * 0.3333333333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = x * 0.3333333333333333e0
end function

Julia

function code(x)
	return Float32(x * Float32(0.3333333333333333))
end

MATLAB

function tmp = code(x)
	tmp = x * single(0.3333333333333333);
end

Derivation

1. Initial program 51.0%

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

   1. add-cbrt-cube 35.3%

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

   2. pow1/3 35.0%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 34.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   4. pow3 34.9%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(x + \sqrt{x \cdot x - 1}\right)}^{3}\right)} \]

   5. log-pow 50.5%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   6. fma-neg 50.5%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right) \]

   7. metadata-eval 50.5%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right) \]

3. Applied egg-rr 50.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt{-1}\right) + 3 \cdot \frac{x}{\sqrt{-1}}\right)} \]

6. Simplified 11.4%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(-1 + \frac{-x}{-1}\right)} \]

7. Taylor expanded in x around inf 11.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot x} \]
8. Step-by-step derivation

   1. *-commutative 11.5%

      \[\leadsto \color{blue}{x \cdot 0.3333333333333333} \]

9. Simplified 11.5%

   \[\leadsto \color{blue}{x \cdot 0.3333333333333333} \]

10. Final simplification 11.5%

    \[\leadsto x \cdot 0.3333333333333333 \]

Alternative 5: 21.1% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (/ x x))

C

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

Fortran

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

Julia

function code(x)
	return Float32(x / x)
end

MATLAB

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

Derivation

1. Initial program 51.0%

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

   1. add-cbrt-cube 35.3%

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

   2. pow1/3 35.0%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 34.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   4. pow3 34.9%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(x + \sqrt{x \cdot x - 1}\right)}^{3}\right)} \]

   5. log-pow 50.5%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   6. fma-neg 50.5%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right) \]

   7. metadata-eval 50.5%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right) \]

3. Applied egg-rr 50.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt{-1}\right) + 3 \cdot \frac{x}{\sqrt{-1}}\right)} \]

6. Simplified 11.4%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(-1 + \frac{-x}{-1}\right)} \]

8. Applied egg-rr 19.7%

   \[\leadsto \color{blue}{\frac{\left(1 + x\right) \cdot 0.3333333333333333}{x + -1}} \]
9. Step-by-step derivation

   1. associate-/l* 19.4%

      \[\leadsto \color{blue}{\frac{1 + x}{\frac{x + -1}{0.3333333333333333}}} \]

   2. div-inv 19.4%

      \[\leadsto \color{blue}{\left(1 + x\right) \cdot \frac{1}{\frac{x + -1}{0.3333333333333333}}} \]

   3. +-commutative 19.4%

      \[\leadsto \color{blue}{\left(x + 1\right)} \cdot \frac{1}{\frac{x + -1}{0.3333333333333333}} \]

   4. div-inv 19.4%

      \[\leadsto \left(x + 1\right) \cdot \frac{1}{\color{blue}{\left(x + -1\right) \cdot \frac{1}{0.3333333333333333}}} \]

   5. metadata-eval 19.4%

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

10. Applied egg-rr 19.4%

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

12. Simplified 21.0%

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

13. Final simplification 21.0%

    \[\leadsto \frac{x}{x} \]

Alternative 6: 3.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -0.3333333333333333)

C

float code(float x) {
	return -0.3333333333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -0.3333333333333333e0
end function

Julia

function code(x)
	return Float32(-0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = single(-0.3333333333333333);
end

Derivation

1. Initial program 51.0%

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

   1. add-cbrt-cube 35.3%

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

   2. pow1/3 35.0%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 34.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   4. pow3 34.9%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(x + \sqrt{x \cdot x - 1}\right)}^{3}\right)} \]

   5. log-pow 50.5%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   6. fma-neg 50.5%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right) \]

   7. metadata-eval 50.5%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right) \]

3. Applied egg-rr 50.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt{-1}\right) + 3 \cdot \frac{x}{\sqrt{-1}}\right)} \]

6. Simplified 11.4%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(-1 + \frac{-x}{-1}\right)} \]

7. Taylor expanded in x around 0 3.1%

   \[\leadsto \color{blue}{-0.3333333333333333} \]

8. Final simplification 3.1%

   \[\leadsto -0.3333333333333333 \]

Developer target: 99.1% 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 2024024 
(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)))))