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

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 10

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{+79} \lor \neg \left(y \leq 4.5 \cdot 10^{+45}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.95e+79) (not (<= y 4.5e+45)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.95e+79) || !(y <= 4.5e+45)) {
		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 <= (-2.95d+79)) .or. (.not. (y <= 4.5d+45))) 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 <= -2.95e+79) || !(y <= 4.5e+45)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.95e+79) or not (y <= 4.5e+45):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.95e+79) || !(y <= 4.5e+45))
		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 <= -2.95e+79) || ~((y <= 4.5e+45)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.95e+79], N[Not[LessEqual[y, 4.5e+45]], $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 < -2.95e79 or 4.4999999999999998e45 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 91.1%

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

   if -2.95e79 < y < 4.4999999999999998e45

   1. Initial program 100.0%

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

      1. add-cbrt-cube 95.4%

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

      2. pow3 95.4%

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in y around 0 98.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{+79} \lor \neg \left(y \leq 4.5 \cdot 10^{+45}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+108} \lor \neg \left(y \leq 5.8 \cdot 10^{+90}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.8e+108) (not (<= y 5.8e+90))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.8e+108) || !(y <= 5.8e+90)) {
		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+108)) .or. (.not. (y <= 5.8d+90))) 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+108) || !(y <= 5.8e+90)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.8e+108) or not (y <= 5.8e+90):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.8e+108) || !(y <= 5.8e+90))
		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+108) || ~((y <= 5.8e+90)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.8e+108], N[Not[LessEqual[y, 5.8e+90]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.80000000000000037e108 or 5.8000000000000003e90 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 89.5%

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

      1. associate--l+ 89.5%

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

      2. distribute-rgt-in 89.5%

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

   5. Simplified 99.5%

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

      1. pow1 99.5%

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

      2. *-commutative 99.5%

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

      3. sqrt-div 99.5%

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

      4. metadata-eval 99.5%

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

      5. un-div-inv 99.6%

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

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(y \cdot \frac{x}{\sqrt{x}}\right)}^{1}} + \left(1 - x\right) \]
   8. Step-by-step derivation

      1. unpow1 99.6%

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

      2. associate-*r/ 78.0%

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

   9. Simplified 78.0%

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

   10. Taylor expanded in y around inf 90.3%

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

   if -4.80000000000000037e108 < y < 5.8000000000000003e90

   1. Initial program 100.0%

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

      1. add-cbrt-cube 93.2%

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

      2. pow3 93.2%

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

   4. Applied egg-rr 93.2%

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

   5. Taylor expanded in y around 0 91.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+108} \lor \neg \left(y \leq 5.8 \cdot 10^{+90}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x)))) (if (<= x 1.0) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(x)
    if (x <= 1.0d0) then
        tmp = 1.0d0 + t_0
    else
        tmp = t_0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * Math.sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + t_0
	else:
		tmp = t_0 - x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(x))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 + t_0);
	else
		tmp = Float64(t_0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * sqrt(x);
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 + t_0;
	else
		tmp = t_0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(1.0 + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 98.8%

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

   if 1 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 99.3%

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

      1. neg-mul-1 99.3%

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\left(-x\right)} + y \cdot \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot x\\ \mathbf{if}\;y \leq -1.08 \cdot 10^{+162}:\\ \;\;\;\;\frac{t\_0}{1 + x}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+121}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x x))))
   (if (<= y -1.08e+162)
     (/ t_0 (+ 1.0 x))
     (if (<= y 2.6e+121) (- 1.0 x) (/ t_0 (- 1.0 x))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= -1.08e+162) {
		tmp = t_0 / (1.0 + x);
	} else if (y <= 2.6e+121) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0 / (1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x * x)
    if (y <= (-1.08d+162)) then
        tmp = t_0 / (1.0d0 + x)
    else if (y <= 2.6d+121) then
        tmp = 1.0d0 - x
    else
        tmp = t_0 / (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= -1.08e+162) {
		tmp = t_0 / (1.0 + x);
	} else if (y <= 2.6e+121) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0 / (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x * x)
	tmp = 0
	if y <= -1.08e+162:
		tmp = t_0 / (1.0 + x)
	elif y <= 2.6e+121:
		tmp = 1.0 - x
	else:
		tmp = t_0 / (1.0 - x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x * x))
	tmp = 0.0
	if (y <= -1.08e+162)
		tmp = Float64(t_0 / Float64(1.0 + x));
	elseif (y <= 2.6e+121)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(t_0 / Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x * x);
	tmp = 0.0;
	if (y <= -1.08e+162)
		tmp = t_0 / (1.0 + x);
	elseif (y <= 2.6e+121)
		tmp = 1.0 - x;
	else
		tmp = t_0 / (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.08e+162], N[(t$95$0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+121], N[(1.0 - x), $MachinePrecision], N[(t$95$0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0799999999999999e162

   1. Initial program 99.7%

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

      1. add-cbrt-cube 45.3%

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

      2. pow3 45.3%

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

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in y around 0 3.5%

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

      1. sub-neg 3.5%

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

      2. flip-+ 26.8%

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

      3. metadata-eval 26.8%

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

   7. Applied egg-rr 26.8%

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

   if -1.0799999999999999e162 < y < 2.5999999999999999e121

   1. Initial program 99.9%

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

      1. add-cbrt-cube 88.3%

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

      2. pow3 88.3%

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

   4. Applied egg-rr 88.3%

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

   5. Taylor expanded in y around 0 85.0%

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

   if 2.5999999999999999e121 < y

   1. Initial program 99.6%

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

      1. add-cbrt-cube 32.4%

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

      2. pow3 32.4%

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

   4. Applied egg-rr 32.4%

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

   5. Taylor expanded in y around 0 2.3%

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

      1. sub-neg 2.3%

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

      2. flip-+ 2.3%

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

      3. metadata-eval 2.3%

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

   7. Applied egg-rr 2.3%

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

      1. neg-sub0 2.3%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{\left(0 - x\right)}} \]

      2. sub-neg 2.3%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{\left(0 + \left(-x\right)\right)}} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \left(0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)} \]

      4. sqrt-unprod 3.5%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \left(0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)} \]

      5. sqr-neg 3.5%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \left(0 + \sqrt{\color{blue}{x \cdot x}}\right)} \]

      6. sqrt-unprod 14.3%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \left(0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)} \]

      7. add-sqr-sqrt 14.3%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \left(0 + \color{blue}{x}\right)} \]

   9. Applied egg-rr 14.3%

      \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{\left(0 + x\right)}} \]
   10. Step-by-step derivation

       1. +-lft-identity 14.3%

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

   11. Simplified 14.3%

       \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{+162}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+121}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot x\\ \mathbf{if}\;y \leq -4.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x x))))
   (if (<= y -4.9e+162)
     (/ t_0 x)
     (if (<= y 7.8e+121) (- 1.0 x) (/ t_0 (- 1.0 x))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= -4.9e+162) {
		tmp = t_0 / x;
	} else if (y <= 7.8e+121) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0 / (1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x * x)
    if (y <= (-4.9d+162)) then
        tmp = t_0 / x
    else if (y <= 7.8d+121) then
        tmp = 1.0d0 - x
    else
        tmp = t_0 / (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= -4.9e+162) {
		tmp = t_0 / x;
	} else if (y <= 7.8e+121) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0 / (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x * x)
	tmp = 0
	if y <= -4.9e+162:
		tmp = t_0 / x
	elif y <= 7.8e+121:
		tmp = 1.0 - x
	else:
		tmp = t_0 / (1.0 - x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x * x))
	tmp = 0.0
	if (y <= -4.9e+162)
		tmp = Float64(t_0 / x);
	elseif (y <= 7.8e+121)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(t_0 / Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x * x);
	tmp = 0.0;
	if (y <= -4.9e+162)
		tmp = t_0 / x;
	elseif (y <= 7.8e+121)
		tmp = 1.0 - x;
	else
		tmp = t_0 / (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.9e+162], N[(t$95$0 / x), $MachinePrecision], If[LessEqual[y, 7.8e+121], N[(1.0 - x), $MachinePrecision], N[(t$95$0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.90000000000000033e162

   1. Initial program 99.7%

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

      1. add-cbrt-cube 45.3%

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

      2. pow3 45.3%

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

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in y around 0 3.5%

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

      1. sub-neg 3.5%

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

      2. flip-+ 26.8%

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

      3. metadata-eval 26.8%

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

   7. Applied egg-rr 26.8%

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

   8. Taylor expanded in x around inf 26.6%

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

   if -4.90000000000000033e162 < y < 7.79999999999999967e121

   1. Initial program 99.9%

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

      1. add-cbrt-cube 88.3%

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

      2. pow3 88.3%

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

   4. Applied egg-rr 88.3%

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

   5. Taylor expanded in y around 0 85.0%

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

   if 7.79999999999999967e121 < y

   1. Initial program 99.6%

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

      1. add-cbrt-cube 32.4%

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

      2. pow3 32.4%

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

   4. Applied egg-rr 32.4%

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

   5. Taylor expanded in y around 0 2.3%

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

      1. sub-neg 2.3%

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

      2. flip-+ 2.3%

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

      3. metadata-eval 2.3%

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

   7. Applied egg-rr 2.3%

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

      1. neg-sub0 2.3%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{\left(0 - x\right)}} \]

      2. sub-neg 2.3%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{\left(0 + \left(-x\right)\right)}} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \left(0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)} \]

      4. sqrt-unprod 3.5%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \left(0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)} \]

      5. sqr-neg 3.5%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \left(0 + \sqrt{\color{blue}{x \cdot x}}\right)} \]

      6. sqrt-unprod 14.3%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \left(0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)} \]

      7. add-sqr-sqrt 14.3%

         \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \left(0 + \color{blue}{x}\right)} \]

   9. Applied egg-rr 14.3%

      \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{\left(0 + x\right)}} \]
   10. Step-by-step derivation

       1. +-lft-identity 14.3%

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

   11. Simplified 14.3%

       \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{1 - x \cdot x}{x}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+162}:\\ \;\;\;\;\frac{1 - x \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+162) (/ (- 1.0 (* x x)) x) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+162) {
		tmp = (1.0 - (x * x)) / 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 <= (-1.2d+162)) then
        tmp = (1.0d0 - (x * x)) / 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 <= -1.2e+162) {
		tmp = (1.0 - (x * x)) / x;
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e+162:
		tmp = (1.0 - (x * x)) / x
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+162)
		tmp = Float64(Float64(1.0 - Float64(x * x)) / x);
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e+162)
		tmp = (1.0 - (x * x)) / x;
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e+162], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.20000000000000005e162

   1. Initial program 99.7%

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

      1. add-cbrt-cube 45.3%

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

      2. pow3 45.3%

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

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in y around 0 3.5%

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

      1. sub-neg 3.5%

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

      2. flip-+ 26.8%

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

      3. metadata-eval 26.8%

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

   7. Applied egg-rr 26.8%

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

   8. Taylor expanded in x around inf 26.6%

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

   if -1.20000000000000005e162 < y

   1. Initial program 99.9%

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

      1. add-cbrt-cube 77.9%

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

      2. pow3 77.9%

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

   4. Applied egg-rr 77.9%

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

   5. Taylor expanded in y around 0 69.7%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+162}:\\ \;\;\;\;\frac{1 - x \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.0% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 75000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 75000.0) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 75000.0) {
		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 <= 75000.0d0) 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 <= 75000.0) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 75000.0:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 75000.0)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 75000.0)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 75000.0], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 75000

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 98.8%

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

   4. Taylor expanded in y around 0 66.9%

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

   if 75000 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 99.2%

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

   4. Taylor expanded in y around 0 57.3%

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

      1. neg-mul-1 57.3%

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

   6. Simplified 57.3%

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

Alternative 9: 62.2% 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. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 74.9%

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

   2. pow3 74.9%

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

4. Applied egg-rr 74.9%

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

5. Taylor expanded in y around 0 63.5%

   \[\leadsto \color{blue}{1 - x} \]
6. Add Preprocessing

Alternative 10: 31.4% 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. Add Preprocessing

3. Taylor expanded in x around 0 73.4%

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

4. Taylor expanded in y around 0 38.2%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))