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

Percentage Accurate: 99.9% → 99.9%

Time: 5.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + (y * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + (y * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + (y * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+44} \lor \neg \left(y \leq 4.1 \cdot 10^{+41}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.8e+44) (not (<= y 4.1e+41)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.8e+44) || !(y <= 4.1e+41)) {
		tmp = 1.0 + (y * sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.8d+44)) .or. (.not. (y <= 4.1d+41))) then
        tmp = 1.0d0 + (y * sqrt(x))
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.8e+44) || !(y <= 4.1e+41)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.8e+44) or not (y <= 4.1e+41):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.8e+44) || !(y <= 4.1e+41))
		tmp = Float64(1.0 + Float64(y * sqrt(x)));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.8e+44) || ~((y <= 4.1e+41)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.8e+44], N[Not[LessEqual[y, 4.1e+41]], $MachinePrecision]], N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.8e44 or 4.1000000000000004e41 < y

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 99.3%

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

      4. associate-*l* 99.3%

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

      5. fma-def 99.3%

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

      6. pow1/2 99.3%

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

      7. sqrt-pow1 99.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right) \]

      8. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right) \]

      9. pow1/2 99.4%

         \[\leadsto \mathsf{fma}\left({x}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}} \cdot y, 1 - x\right) \]

      10. sqrt-pow1 99.3%

          \[\leadsto \mathsf{fma}\left({x}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}} \cdot y, 1 - x\right) \]

      11. metadata-eval 99.3%

          \[\leadsto \mathsf{fma}\left({x}^{0.25}, {x}^{\color{blue}{0.25}} \cdot y, 1 - x\right) \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25} \cdot y, 1 - x\right)} \]

   4. Taylor expanded in x around 0 97.6%

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

   if -6.8e44 < y < 4.1000000000000004e41

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.1%

      \[\leadsto \color{blue}{1 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+44} \lor \neg \left(y \leq 4.1 \cdot 10^{+41}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 3: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+57}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+44}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{{x}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+57)
   (* y (sqrt x))
   (if (<= y 1.25e+44) (- 1.0 x) (/ y (pow x -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+57) {
		tmp = y * sqrt(x);
	} else if (y <= 1.25e+44) {
		tmp = 1.0 - x;
	} else {
		tmp = y / pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.8d+57)) then
        tmp = y * sqrt(x)
    else if (y <= 1.25d+44) then
        tmp = 1.0d0 - x
    else
        tmp = y / (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+57) {
		tmp = y * Math.sqrt(x);
	} else if (y <= 1.25e+44) {
		tmp = 1.0 - x;
	} else {
		tmp = y / Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+57:
		tmp = y * math.sqrt(x)
	elif y <= 1.25e+44:
		tmp = 1.0 - x
	else:
		tmp = y / math.pow(x, -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+57)
		tmp = Float64(y * sqrt(x));
	elseif (y <= 1.25e+44)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y / (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.8e+57)
		tmp = y * sqrt(x);
	elseif (y <= 1.25e+44)
		tmp = 1.0 - x;
	else
		tmp = y / (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+57], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+44], N[(1.0 - x), $MachinePrecision], N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.80000000000000009e57

   1. Initial program 99.7%

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

   2. Taylor expanded in y around inf 89.0%

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

   if -4.80000000000000009e57 < y < 1.2499999999999999e44

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.5%

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

   if 1.2499999999999999e44 < y

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 99.3%

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

      4. associate-*l* 99.4%

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

      5. fma-def 99.4%

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

      6. pow1/2 99.4%

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

      7. sqrt-pow1 99.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right) \]

      8. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right) \]

      9. pow1/2 99.5%

         \[\leadsto \mathsf{fma}\left({x}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}} \cdot y, 1 - x\right) \]

      10. sqrt-pow1 99.3%

          \[\leadsto \mathsf{fma}\left({x}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}} \cdot y, 1 - x\right) \]

      11. metadata-eval 99.3%

          \[\leadsto \mathsf{fma}\left({x}^{0.25}, {x}^{\color{blue}{0.25}} \cdot y, 1 - x\right) \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25} \cdot y, 1 - x\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.3%

         \[\leadsto \color{blue}{{x}^{0.25} \cdot \left({x}^{0.25} \cdot y\right) + \left(1 - x\right)} \]

      2. associate-*r* 99.4%

         \[\leadsto \color{blue}{\left({x}^{0.25} \cdot {x}^{0.25}\right) \cdot y} + \left(1 - x\right) \]

      3. pow-prod-up 99.7%

         \[\leadsto \color{blue}{{x}^{\left(0.25 + 0.25\right)}} \cdot y + \left(1 - x\right) \]

      4. metadata-eval 99.7%

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

      5. pow1/2 99.7%

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

      6. *-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. flip-+ 49.5%

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

      9. swap-sqr 32.2%

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

      10. add-sqr-sqrt 32.3%

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

      11. unpow2 32.3%

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

      12. *-commutative 32.3%

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

      13. clear-num 32.2%

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

      14. clear-num 32.3%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around inf 87.8%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{y \cdot \sqrt{x}}}} \]
   7. Step-by-step derivation

      1. inv-pow 87.8%

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

      2. inv-pow 87.8%

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

      3. unpow-prod-down 87.5%

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

      4. inv-pow 87.5%

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

      5. inv-pow 87.5%

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

      6. metadata-eval 87.5%

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

      7. sqrt-div 87.5%

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

      8. unpow-prod-down 87.5%

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

      9. inv-pow 87.5%

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

      10. inv-pow 87.5%

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

      11. pow-flip 87.5%

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

      12. metadata-eval 87.5%

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

      13. pow1 87.5%

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

      14. inv-pow 87.5%

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

      15. sqrt-pow1 87.6%

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

      16. metadata-eval 87.6%

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

   8. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{y \cdot {\left({x}^{-0.5}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 87.6%

         \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{-0.5}}} \]

      2. associate-*r/ 87.9%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{{x}^{-0.5}}} \]

      3. *-rgt-identity 87.9%

         \[\leadsto \frac{\color{blue}{y}}{{x}^{-0.5}} \]

   10. Simplified 87.9%

       \[\leadsto \color{blue}{\frac{y}{{x}^{-0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+57}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+44}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{{x}^{-0.5}}\\ \end{array} \]

Alternative 4: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+57} \lor \neg \left(y \leq 1.15 \cdot 10^{+41}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.8e+57) (not (<= y 1.15e+41))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.8e+57) || !(y <= 1.15e+41)) {
		tmp = y * sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.8d+57)) .or. (.not. (y <= 1.15d+41))) then
        tmp = y * sqrt(x)
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.8e+57) || !(y <= 1.15e+41)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.8e+57) or not (y <= 1.15e+41):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.8e+57) || !(y <= 1.15e+41))
		tmp = Float64(y * sqrt(x));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.8e+57) || ~((y <= 1.15e+41)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.8e+57], N[Not[LessEqual[y, 1.15e+41]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.80000000000000009e57 or 1.1499999999999999e41 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in y around inf 88.5%

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

   if -4.80000000000000009e57 < y < 1.1499999999999999e41

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.5%

      \[\leadsto \color{blue}{1 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+57} \lor \neg \left(y \leq 1.15 \cdot 10^{+41}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 5: 62.2% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.0) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.0], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 64.8%

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

   if 1 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 60.9%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   3. Step-by-step derivation

      1. mul-1-neg 60.9%

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

   4. Simplified 60.9%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6: 63.0% accurate, 35.7× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- 1.0 x))

C

double code(double x, double y) {
	return 1.0 - x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - x
end function

Java

public static double code(double x, double y) {
	return 1.0 - x;
}

Python

def code(x, y):
	return 1.0 - x

Julia

function code(x, y)
	return Float64(1.0 - x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - x;
end

Wolfram

code[x_, y_] := N[(1.0 - x), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 64.0%

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

3. Final simplification 64.0%

   \[\leadsto 1 - x \]

Alternative 7: 32.0% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 31.9%

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

3. Final simplification 31.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))