Rust f32::acosh

Percentage Accurate: 52.7% → 98.3%

Time: 4.6s

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: 52.7% 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: 98.3% 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 56.9%

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

2. Taylor expanded in x around inf 98.1%

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

   1. *-commutative 98.1%

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

   2. associate-*r/ 98.1%

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

   3. metadata-eval 98.1%

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

4. Simplified 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 97.1% 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 56.9%

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

2. Taylor expanded in x around inf 97.2%

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

3. Final simplification 97.2%

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

Alternative 3: 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 56.9%

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

   1. pow1/2 56.9%

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

   2. add-cube-cbrt 56.9%

      \[\leadsto \log \left(x + {\color{blue}{\left(\left(\sqrt[3]{x \cdot x - 1} \cdot \sqrt[3]{x \cdot x - 1}\right) \cdot \sqrt[3]{x \cdot x - 1}\right)}}^{0.5}\right) \]

   3. pow3 56.9%

      \[\leadsto \log \left(x + {\color{blue}{\left({\left(\sqrt[3]{x \cdot x - 1}\right)}^{3}\right)}}^{0.5}\right) \]

   4. pow-pow 56.9%

      \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{x \cdot x - 1}\right)}^{\left(3 \cdot 0.5\right)}}\right) \]

   5. fma-neg 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   6. metadata-eval 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   7. metadata-eval 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{1.5}}\right) \]

3. Applied egg-rr 56.9%

   \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}}\right) \]

4. Taylor expanded in x around inf 44.1%

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

5. Final simplification 44.1%

   \[\leadsto \log x \]

Alternative 4: 8.8% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. pow1/2 56.9%

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

   2. add-cube-cbrt 56.9%

      \[\leadsto \log \left(x + {\color{blue}{\left(\left(\sqrt[3]{x \cdot x - 1} \cdot \sqrt[3]{x \cdot x - 1}\right) \cdot \sqrt[3]{x \cdot x - 1}\right)}}^{0.5}\right) \]

   3. pow3 56.9%

      \[\leadsto \log \left(x + {\color{blue}{\left({\left(\sqrt[3]{x \cdot x - 1}\right)}^{3}\right)}}^{0.5}\right) \]

   4. pow-pow 56.9%

      \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{x \cdot x - 1}\right)}^{\left(3 \cdot 0.5\right)}}\right) \]

   5. fma-neg 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   6. metadata-eval 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   7. metadata-eval 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{1.5}}\right) \]

3. Applied egg-rr 56.9%

   \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}}\right) \]

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{-1}\right)}^{1.5}\right) + \left(0.5 \cdot \left(\left(1 + -1 \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot {x}^{2}\right) + \sqrt{-1} \cdot x\right)} \]

6. Simplified 1.7%

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

   1. +-commutative 1.7%

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

   2. add-sqr-sqrt -0.0%

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

   3. difference-of-sqr--1 -0.0%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod -0.0%

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

   6. mul-1-neg -0.0%

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

   7. mul-1-neg -0.0%

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

   8. sqr-neg -0.0%

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

   9. sqrt-unprod -0.0%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-prod -0.0%

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

   12. add-sqr-sqrt -0.0%

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

8. Applied egg-rr 8.7%

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

9. Taylor expanded in x around inf 8.8%

   \[\leadsto \color{blue}{-1 \cdot x + {x}^{2}} \]
10. Step-by-step derivation

    1. unpow2 8.8%

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

    2. distribute-rgt-in 8.8%

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

11. Simplified 8.8%

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

12. Final simplification 8.8%

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

Alternative 5: 8.8% accurate, 41.4× speedup

Math

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

FPCore

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

C

float code(float x) {
	return (x * x) - x;
}

Fortran

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * x) - x;
end

Derivation

1. Initial program 56.9%

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

   1. pow1/2 56.9%

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

   2. add-cube-cbrt 56.9%

      \[\leadsto \log \left(x + {\color{blue}{\left(\left(\sqrt[3]{x \cdot x - 1} \cdot \sqrt[3]{x \cdot x - 1}\right) \cdot \sqrt[3]{x \cdot x - 1}\right)}}^{0.5}\right) \]

   3. pow3 56.9%

      \[\leadsto \log \left(x + {\color{blue}{\left({\left(\sqrt[3]{x \cdot x - 1}\right)}^{3}\right)}}^{0.5}\right) \]

   4. pow-pow 56.9%

      \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{x \cdot x - 1}\right)}^{\left(3 \cdot 0.5\right)}}\right) \]

   5. fma-neg 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   6. metadata-eval 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   7. metadata-eval 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{1.5}}\right) \]

3. Applied egg-rr 56.9%

   \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}}\right) \]

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{-1}\right)}^{1.5}\right) + \left(0.5 \cdot \left(\left(1 + -1 \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot {x}^{2}\right) + \sqrt{-1} \cdot x\right)} \]

6. Simplified 1.7%

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

   1. +-commutative 1.7%

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

   2. add-sqr-sqrt -0.0%

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

   3. difference-of-sqr--1 -0.0%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod -0.0%

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

   6. mul-1-neg -0.0%

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

   7. mul-1-neg -0.0%

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

   8. sqr-neg -0.0%

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

   9. sqrt-unprod -0.0%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-prod -0.0%

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

   12. add-sqr-sqrt -0.0%

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

8. Applied egg-rr 8.7%

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

9. Taylor expanded in x around inf 8.8%

   \[\leadsto \color{blue}{-1 \cdot x + {x}^{2}} \]
10. Step-by-step derivation

    1. unpow2 8.8%

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

    2. +-commutative 8.8%

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

    3. mul-1-neg 8.8%

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

    4. unsub-neg 8.8%

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

11. Simplified 8.8%

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

12. Final simplification 8.8%

    \[\leadsto x \cdot x - x \]

Alternative 6: 8.7% accurate, 69.0× speedup

Math

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

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 56.9%

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

   1. pow1/2 56.9%

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

   2. add-cube-cbrt 56.9%

      \[\leadsto \log \left(x + {\color{blue}{\left(\left(\sqrt[3]{x \cdot x - 1} \cdot \sqrt[3]{x \cdot x - 1}\right) \cdot \sqrt[3]{x \cdot x - 1}\right)}}^{0.5}\right) \]

   3. pow3 56.9%

      \[\leadsto \log \left(x + {\color{blue}{\left({\left(\sqrt[3]{x \cdot x - 1}\right)}^{3}\right)}}^{0.5}\right) \]

   4. pow-pow 56.9%

      \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{x \cdot x - 1}\right)}^{\left(3 \cdot 0.5\right)}}\right) \]

   5. fma-neg 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   6. metadata-eval 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   7. metadata-eval 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{1.5}}\right) \]

3. Applied egg-rr 56.9%

   \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}}\right) \]

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{-1}\right)}^{1.5}\right) + \left(0.5 \cdot \left(\left(1 + -1 \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot {x}^{2}\right) + \sqrt{-1} \cdot x\right)} \]

6. Simplified 1.7%

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

   1. +-commutative 1.7%

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

   2. add-sqr-sqrt -0.0%

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

   3. difference-of-sqr--1 -0.0%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod -0.0%

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

   6. mul-1-neg -0.0%

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

   7. mul-1-neg -0.0%

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

   8. sqr-neg -0.0%

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

   9. sqrt-unprod -0.0%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-prod -0.0%

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

   12. add-sqr-sqrt -0.0%

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

8. Applied egg-rr 8.7%

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

9. Taylor expanded in x around inf 8.8%

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

    1. unpow2 8.8%

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

11. Simplified 8.8%

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

12. Final simplification 8.8%

    \[\leadsto x \cdot x \]

Alternative 7: 3.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. pow1/2 56.9%

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

   2. add-cube-cbrt 56.9%

      \[\leadsto \log \left(x + {\color{blue}{\left(\left(\sqrt[3]{x \cdot x - 1} \cdot \sqrt[3]{x \cdot x - 1}\right) \cdot \sqrt[3]{x \cdot x - 1}\right)}}^{0.5}\right) \]

   3. pow3 56.9%

      \[\leadsto \log \left(x + {\color{blue}{\left({\left(\sqrt[3]{x \cdot x - 1}\right)}^{3}\right)}}^{0.5}\right) \]

   4. pow-pow 56.9%

      \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{x \cdot x - 1}\right)}^{\left(3 \cdot 0.5\right)}}\right) \]

   5. fma-neg 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   6. metadata-eval 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)}^{\left(3 \cdot 0.5\right)}\right) \]

   7. metadata-eval 56.9%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{1.5}}\right) \]

3. Applied egg-rr 56.9%

   \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}}\right) \]

4. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{-1}\right)}^{1.5}\right) + \left(0.5 \cdot \left(\left(1 + -1 \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot {x}^{2}\right) + \sqrt{-1} \cdot x\right)} \]

6. Simplified 1.7%

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

7. Taylor expanded in x around 0 3.1%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 3.1%

   \[\leadsto -1 \]

Developer target: 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 2023199 
(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)))))