Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D

Percentage Accurate: 100.0% → 100.0%

Time: 12.8s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))

C

double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + ((0.99229d0 + (x * 0.04481d0)) * x)))
end function

Java

public static double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

Python

def code(x):
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)))

Julia

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(Float64(0.99229 + Float64(x * 0.04481)) * x))))
end

MATLAB

function tmp = code(x)
	tmp = x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
end

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))

C

double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + ((0.99229d0 + (x * 0.04481d0)) * x)))
end function

Java

public static double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

Python

def code(x):
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)))

Julia

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(Float64(0.99229 + Float64(x * 0.04481)) * x))))
end

MATLAB

function tmp = code(x)
	tmp = x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
end

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ x - \frac{2.30753 + 3 \cdot \left(\left(x \cdot 0.09020333333333333 + 1\right) + -1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  x
  (/
   (+ 2.30753 (* 3.0 (+ (+ (* x 0.09020333333333333) 1.0) -1.0)))
   (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))))

C

double code(double x) {
	return x - ((2.30753 + (3.0 * (((x * 0.09020333333333333) + 1.0) + -1.0))) / (1.0 + (x * (0.99229 + (x * 0.04481)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (3.0d0 * (((x * 0.09020333333333333d0) + 1.0d0) + (-1.0d0)))) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0)))))
end function

Java

public static double code(double x) {
	return x - ((2.30753 + (3.0 * (((x * 0.09020333333333333) + 1.0) + -1.0))) / (1.0 + (x * (0.99229 + (x * 0.04481)))));
}

Python

def code(x):
	return x - ((2.30753 + (3.0 * (((x * 0.09020333333333333) + 1.0) + -1.0))) / (1.0 + (x * (0.99229 + (x * 0.04481)))))

Julia

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(3.0 * Float64(Float64(Float64(x * 0.09020333333333333) + 1.0) + -1.0))) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))))
end

MATLAB

function tmp = code(x)
	tmp = x - ((2.30753 + (3.0 * (((x * 0.09020333333333333) + 1.0) + -1.0))) / (1.0 + (x * (0.99229 + (x * 0.04481)))));
end

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(3.0 * N[(N[(N[(x * 0.09020333333333333), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. log1p-expm1-u 48.5%

      \[\leadsto x - \frac{2.30753 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 0.27061\right)\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   2. log1p-udef 48.6%

      \[\leadsto x - \frac{2.30753 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(x \cdot 0.27061\right)\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

4. Applied egg-rr 48.6%

   \[\leadsto x - \frac{2.30753 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(x \cdot 0.27061\right)\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
5. Step-by-step derivation

   1. add-cube-cbrt 48.5%

      \[\leadsto x - \frac{2.30753 + \log \color{blue}{\left(\left(\sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)} \cdot \sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)}\right) \cdot \sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)}\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   2. log-prod 48.6%

      \[\leadsto x - \frac{2.30753 + \color{blue}{\left(\log \left(\sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)} \cdot \sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)}\right) + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)}\right)\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   3. pow2 48.6%

      \[\leadsto x - \frac{2.30753 + \left(\log \color{blue}{\left({\left(\sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)}\right)}^{2}\right)} + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)}\right)\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   4. add-exp-log 48.6%

      \[\leadsto x - \frac{2.30753 + \left(\log \left({\left(\sqrt[3]{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(x \cdot 0.27061\right)\right)}}}\right)}^{2}\right) + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)}\right)\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   5. log1p-def 48.6%

      \[\leadsto x - \frac{2.30753 + \left(\log \left({\left(\sqrt[3]{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 0.27061\right)\right)}}}\right)}^{2}\right) + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)}\right)\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   6. log1p-expm1-u 48.6%

      \[\leadsto x - \frac{2.30753 + \left(\log \left({\left(\sqrt[3]{e^{\color{blue}{x \cdot 0.27061}}}\right)}^{2}\right) + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)}\right)\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   7. exp-prod 48.6%

      \[\leadsto x - \frac{2.30753 + \left(\log \left({\left(\sqrt[3]{\color{blue}{{\left(e^{x}\right)}^{0.27061}}}\right)}^{2}\right) + \log \left(\sqrt[3]{1 + \mathsf{expm1}\left(x \cdot 0.27061\right)}\right)\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   8. add-exp-log 48.6%

      \[\leadsto x - \frac{2.30753 + \left(\log \left({\left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(x \cdot 0.27061\right)\right)}}}\right)\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   9. log1p-def 48.6%

      \[\leadsto x - \frac{2.30753 + \left(\log \left({\left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 0.27061\right)\right)}}}\right)\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   10. log1p-expm1-u 48.6%

       \[\leadsto x - \frac{2.30753 + \left(\log \left({\left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\color{blue}{x \cdot 0.27061}}}\right)\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   11. exp-prod 48.6%

       \[\leadsto x - \frac{2.30753 + \left(\log \left({\left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{{\left(e^{x}\right)}^{0.27061}}}\right)\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

6. Applied egg-rr 48.6%

   \[\leadsto x - \frac{2.30753 + \color{blue}{\left(\log \left({\left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)}^{2}\right) + \log \left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
7. Step-by-step derivation

   1. log-pow 48.5%

      \[\leadsto x - \frac{2.30753 + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)} + \log \left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   2. distribute-lft1-in 48.5%

      \[\leadsto x - \frac{2.30753 + \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   3. metadata-eval 48.5%

      \[\leadsto x - \frac{2.30753 + \color{blue}{3} \cdot \log \left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

8. Simplified 48.5%

   \[\leadsto x - \frac{2.30753 + \color{blue}{3 \cdot \log \left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
9. Step-by-step derivation

   1. expm1-log1p-u 48.5%

      \[\leadsto x - \frac{2.30753 + 3 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)\right)\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   2. expm1-udef 48.5%

      \[\leadsto x - \frac{2.30753 + 3 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\sqrt[3]{{\left(e^{x}\right)}^{0.27061}}\right)\right)} - 1\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   3. pow1/3 48.5%

      \[\leadsto x - \frac{2.30753 + 3 \cdot \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left({\left({\left(e^{x}\right)}^{0.27061}\right)}^{0.3333333333333333}\right)}\right)} - 1\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   4. pow-pow 48.5%

      \[\leadsto x - \frac{2.30753 + 3 \cdot \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left({\left(e^{x}\right)}^{\left(0.27061 \cdot 0.3333333333333333\right)}\right)}\right)} - 1\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   5. log-pow 48.5%

      \[\leadsto x - \frac{2.30753 + 3 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(0.27061 \cdot 0.3333333333333333\right) \cdot \log \left(e^{x}\right)}\right)} - 1\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   6. metadata-eval 48.5%

      \[\leadsto x - \frac{2.30753 + 3 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{0.09020333333333333} \cdot \log \left(e^{x}\right)\right)} - 1\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

   7. add-log-exp 73.0%

      \[\leadsto x - \frac{2.30753 + 3 \cdot \left(e^{\mathsf{log1p}\left(0.09020333333333333 \cdot \color{blue}{x}\right)} - 1\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

10. Applied egg-rr 73.0%

    \[\leadsto x - \frac{2.30753 + 3 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.09020333333333333 \cdot x\right)} - 1\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
11. Step-by-step derivation

    1. log1p-udef 73.0%

       \[\leadsto x - \frac{2.30753 + 3 \cdot \left(e^{\color{blue}{\log \left(1 + 0.09020333333333333 \cdot x\right)}} - 1\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

    2. rem-exp-log 100.0%

       \[\leadsto x - \frac{2.30753 + 3 \cdot \left(\color{blue}{\left(1 + 0.09020333333333333 \cdot x\right)} - 1\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

    3. +-commutative 100.0%

       \[\leadsto x - \frac{2.30753 + 3 \cdot \left(\color{blue}{\left(0.09020333333333333 \cdot x + 1\right)} - 1\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

12. Applied egg-rr 100.0%

    \[\leadsto x - \frac{2.30753 + 3 \cdot \left(\color{blue}{\left(0.09020333333333333 \cdot x + 1\right)} - 1\right)}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

13. Final simplification 100.0%

    \[\leadsto x - \frac{2.30753 + 3 \cdot \left(\left(x \cdot 0.09020333333333333 + 1\right) + -1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \]
14. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))))

C

double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0)))))
end function

Java

public static double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481)))));
}

Python

def code(x):
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481)))))

Julia

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))))
end

MATLAB

function tmp = code(x)
	tmp = x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481)))));
end

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \]
4. Add Preprocessing

Alternative 3: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x 0.99229)))))

C

double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * 0.99229d0)))
end function

Java

public static double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229)));
}

Python

def code(x):
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229)))

Julia

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * 0.99229))))
end

MATLAB

function tmp = code(x)
	tmp = x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229)));
end

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * 0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.7%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} \]
4. Step-by-step derivation

   1. *-commutative 98.7%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} \]

5. Simplified 98.7%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} \]

6. Final simplification 98.7%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} \]
7. Add Preprocessing

Alternative 4: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -3.6) x (if (<= x 1.15) -2.30753 x)))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = x;
	} else if (x <= 1.15) {
		tmp = -2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.6d0)) then
        tmp = x
    else if (x <= 1.15d0) then
        tmp = -2.30753d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = x;
	} else if (x <= 1.15) {
		tmp = -2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = x
	elif x <= 1.15:
		tmp = -2.30753
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = x;
	elseif (x <= 1.15)
		tmp = -2.30753;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.6)
		tmp = x;
	elseif (x <= 1.15)
		tmp = -2.30753;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], x, If[LessEqual[x, 1.15], -2.30753, x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009 or 1.1499999999999999 < x

   1. Initial program 100.0%

      \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.7%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} \]
   4. Step-by-step derivation

      1. *-commutative 98.7%

         \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} \]

   5. Simplified 98.7%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} \]

   6. Taylor expanded in x around inf 99.7%

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

   if -3.60000000000000009 < x < 1.1499999999999999

   1. Initial program 100.0%

      \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.6%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} \]
   4. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} \]

   5. Simplified 98.6%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} \]

   6. Taylor expanded in x around 0 97.9%

      \[\leadsto \color{blue}{-2.30753} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ x - 2.30753 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x 2.30753))

C

double code(double x) {
	return x - 2.30753;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - 2.30753d0
end function

Java

public static double code(double x) {
	return x - 2.30753;
}

Python

def code(x):
	return x - 2.30753

Julia

function code(x)
	return Float64(x - 2.30753)
end

MATLAB

function tmp = code(x)
	tmp = x - 2.30753;
end

Wolfram

code[x_] := N[(x - 2.30753), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.2%

   \[\leadsto x - \color{blue}{2.30753} \]

4. Final simplification 98.2%

   \[\leadsto x - 2.30753 \]
5. Add Preprocessing

Alternative 6: 50.0% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ -2.30753 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.30753)

C

double code(double x) {
	return -2.30753;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.30753d0
end function

Java

public static double code(double x) {
	return -2.30753;
}

Python

def code(x):
	return -2.30753

Julia

function code(x)
	return -2.30753
end

MATLAB

function tmp = code(x)
	tmp = -2.30753;
end

Wolfram

code[x_] := -2.30753

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.7%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} \]
4. Step-by-step derivation

   1. *-commutative 98.7%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} \]

5. Simplified 98.7%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} \]

6. Taylor expanded in x around 0 49.1%

   \[\leadsto \color{blue}{-2.30753} \]

7. Final simplification 49.1%

   \[\leadsto -2.30753 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))