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

Percentage Accurate: 99.9% → 99.9%

Time: 13.0s

Alternatives: 13

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 99.8%

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

Alternative 2: 94.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y + 1\\ \mathbf{if}\;y \leq -6.4 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+87}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* (sqrt x) y) 1.0)))
   (if (<= y -6.4e+34) t_0 (if (<= y 2.15e+87) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = (sqrt(x) * y) + 1.0;
	double tmp;
	if (y <= -6.4e+34) {
		tmp = t_0;
	} else if (y <= 2.15e+87) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	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 = (sqrt(x) * y) + 1.0d0
    if (y <= (-6.4d+34)) then
        tmp = t_0
    else if (y <= 2.15d+87) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sqrt(x) * y) + 1.0;
	double tmp;
	if (y <= -6.4e+34) {
		tmp = t_0;
	} else if (y <= 2.15e+87) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sqrt(x) * y) + 1.0
	tmp = 0
	if y <= -6.4e+34:
		tmp = t_0
	elif y <= 2.15e+87:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(x) * y) + 1.0)
	tmp = 0.0
	if (y <= -6.4e+34)
		tmp = t_0;
	elseif (y <= 2.15e+87)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sqrt(x) * y) + 1.0;
	tmp = 0.0;
	if (y <= -6.4e+34)
		tmp = t_0;
	elseif (y <= 2.15e+87)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -6.4e+34], t$95$0, If[LessEqual[y, 2.15e+87], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.3999999999999997e34 or 2.15e87 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if -6.3999999999999997e34 < y < 2.15e87

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y\\ \mathbf{if}\;y \leq -6 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+88}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) y)))
   (if (<= y -6e+79) t_0 (if (<= y 1.32e+88) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (y <= -6e+79) {
		tmp = t_0;
	} else if (y <= 1.32e+88) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	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 = sqrt(x) * y
    if (y <= (-6d+79)) then
        tmp = t_0
    else if (y <= 1.32d+88) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * y;
	double tmp;
	if (y <= -6e+79) {
		tmp = t_0;
	} else if (y <= 1.32e+88) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * y
	tmp = 0
	if y <= -6e+79:
		tmp = t_0
	elif y <= 1.32e+88:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * y)
	tmp = 0.0
	if (y <= -6e+79)
		tmp = t_0;
	elseif (y <= 1.32e+88)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * y;
	tmp = 0.0;
	if (y <= -6e+79)
		tmp = t_0;
	elseif (y <= 1.32e+88)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -6e+79], t$95$0, If[LessEqual[y, 1.32e+88], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.99999999999999948e79 or 1.3200000000000001e88 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if -5.99999999999999948e79 < y < 1.3200000000000001e88

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 195:\\ \;\;\;\;\frac{y}{{x}^{-0.5}} + 1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + \frac{y}{\sqrt{x}}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 195.0) (+ (/ y (pow x -0.5)) 1.0) (* (+ -1.0 (/ y (sqrt x))) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 195.0) {
		tmp = (y / pow(x, -0.5)) + 1.0;
	} else {
		tmp = (-1.0 + (y / sqrt(x))) * 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 <= 195.0d0) then
        tmp = (y / (x ** (-0.5d0))) + 1.0d0
    else
        tmp = ((-1.0d0) + (y / sqrt(x))) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 195.0) {
		tmp = (y / Math.pow(x, -0.5)) + 1.0;
	} else {
		tmp = (-1.0 + (y / Math.sqrt(x))) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 195.0:
		tmp = (y / math.pow(x, -0.5)) + 1.0
	else:
		tmp = (-1.0 + (y / math.sqrt(x))) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 195.0)
		tmp = Float64(Float64(y / (x ^ -0.5)) + 1.0);
	else
		tmp = Float64(Float64(-1.0 + Float64(y / sqrt(x))) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 195.0)
		tmp = (y / (x ^ -0.5)) + 1.0;
	else
		tmp = (-1.0 + (y / sqrt(x))) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 195.0], N[(N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(-1.0 + N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 195

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   if 195 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.0) (+ (/ y (pow x -0.5)) 1.0) (- (* (sqrt x) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.0) {
		tmp = (y / pow(x, -0.5)) + 1.0;
	} else {
		tmp = (sqrt(x) * y) - 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 = (y / (x ** (-0.5d0))) + 1.0d0
    else
        tmp = (sqrt(x) * y) - x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. Add Preprocessing

   3. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (x <= 1.0) {
		tmp = t_0 + 1.0;
	} 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 = sqrt(x) * y
    if (x <= 1.0d0) then
        tmp = t_0 + 1.0d0
    else
        tmp = t_0 - x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. Add Preprocessing

   3. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+116}:\\ \;\;\;\;\frac{1}{y} \cdot \left(0 - x \cdot y\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+88}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{1 - x \cdot x}} \cdot \left(\left(y \cdot \left(1 - x\right)\right) \cdot \left(1 + x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.15e+116)
   (* (/ 1.0 y) (- 0.0 (* x y)))
   (if (<= y 6e+88)
     (- 1.0 x)
     (* (/ 1.0 (/ y (- 1.0 (* x x)))) (* (* y (- 1.0 x)) (+ 1.0 (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.15e+116) {
		tmp = (1.0 / y) * (0.0 - (x * y));
	} else if (y <= 6e+88) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 / (y / (1.0 - (x * x)))) * ((y * (1.0 - x)) * (1.0 + (x * 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 <= (-2.15d+116)) then
        tmp = (1.0d0 / y) * (0.0d0 - (x * y))
    else if (y <= 6d+88) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 / (y / (1.0d0 - (x * x)))) * ((y * (1.0d0 - x)) * (1.0d0 + (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.15e+116) {
		tmp = (1.0 / y) * (0.0 - (x * y));
	} else if (y <= 6e+88) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 / (y / (1.0 - (x * x)))) * ((y * (1.0 - x)) * (1.0 + (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.15e+116:
		tmp = (1.0 / y) * (0.0 - (x * y))
	elif y <= 6e+88:
		tmp = 1.0 - x
	else:
		tmp = (1.0 / (y / (1.0 - (x * x)))) * ((y * (1.0 - x)) * (1.0 + (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.15e+116)
		tmp = Float64(Float64(1.0 / y) * Float64(0.0 - Float64(x * y)));
	elseif (y <= 6e+88)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 / Float64(y / Float64(1.0 - Float64(x * x)))) * Float64(Float64(y * Float64(1.0 - x)) * Float64(1.0 + Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.15e+116)
		tmp = (1.0 / y) * (0.0 - (x * y));
	elseif (y <= 6e+88)
		tmp = 1.0 - x;
	else
		tmp = (1.0 / (y / (1.0 - (x * x)))) * ((y * (1.0 - x)) * (1.0 + (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.15e+116], N[(N[(1.0 / y), $MachinePrecision] * N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+88], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 / N[(y / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.15e116

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in x around inf 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if -2.15e116 < y < 6.00000000000000011e88

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 6.00000000000000011e88 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in x around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+116}:\\ \;\;\;\;\frac{1}{y} \cdot \left(0 - x \cdot y\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+88}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{1 - x \cdot x}} \cdot \left(y \cdot \left(1 - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.15e+116)
   (* (/ 1.0 y) (- 0.0 (* x y)))
   (if (<= y 6e+88)
     (- 1.0 x)
     (* (/ 1.0 (/ y (- 1.0 (* x x)))) (* y (- 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.15e+116) {
		tmp = (1.0 / y) * (0.0 - (x * y));
	} else if (y <= 6e+88) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 / (y / (1.0 - (x * x)))) * (y * (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 <= (-2.15d+116)) then
        tmp = (1.0d0 / y) * (0.0d0 - (x * y))
    else if (y <= 6d+88) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 / (y / (1.0d0 - (x * x)))) * (y * (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.15e+116) {
		tmp = (1.0 / y) * (0.0 - (x * y));
	} else if (y <= 6e+88) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 / (y / (1.0 - (x * x)))) * (y * (1.0 - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.15e+116:
		tmp = (1.0 / y) * (0.0 - (x * y))
	elif y <= 6e+88:
		tmp = 1.0 - x
	else:
		tmp = (1.0 / (y / (1.0 - (x * x)))) * (y * (1.0 - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.15e+116)
		tmp = Float64(Float64(1.0 / y) * Float64(0.0 - Float64(x * y)));
	elseif (y <= 6e+88)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 / Float64(y / Float64(1.0 - Float64(x * x)))) * Float64(y * Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.15e+116)
		tmp = (1.0 / y) * (0.0 - (x * y));
	elseif (y <= 6e+88)
		tmp = 1.0 - x;
	else
		tmp = (1.0 / (y / (1.0 - (x * x)))) * (y * (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.15e+116], N[(N[(1.0 / y), $MachinePrecision] * N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+88], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 / N[(y / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.15e116

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in x around inf 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if -2.15e116 < y < 6.00000000000000011e88

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 6.00000000000000011e88 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in x around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+116}:\\ \;\;\;\;\frac{1}{y} \cdot \left(0 - x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.15e+116) (* (/ 1.0 y) (- 0.0 (* x y))) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.15e+116) {
		tmp = (1.0 / y) * (0.0 - (x * y));
	} 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 <= (-2.15d+116)) then
        tmp = (1.0d0 / y) * (0.0d0 - (x * y))
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.15e+116) {
		tmp = (1.0 / y) * (0.0 - (x * y));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.15e+116:
		tmp = (1.0 / y) * (0.0 - (x * y))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.15e+116)
		tmp = Float64(Float64(1.0 / y) * Float64(0.0 - Float64(x * y)));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.15e+116)
		tmp = (1.0 / y) * (0.0 - (x * y));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.15e+116], N[(N[(1.0 / y), $MachinePrecision] * N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.15e116

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in x around inf 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if -2.15e116 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 65.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.02e+146) (/ (* (- 1.0 x) y) y) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.02e+146) {
		tmp = ((1.0 - x) * y) / y;
	} 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 <= (-1.02d+146)) then
        tmp = ((1.0d0 - x) * y) / y
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.02e+146) {
		tmp = ((1.0 - x) * y) / y;
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.02e+146:
		tmp = ((1.0 - x) * y) / y
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.02e+146)
		tmp = Float64(Float64(Float64(1.0 - x) * y) / y);
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.02e+146)
		tmp = ((1.0 - x) * y) / y;
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.02e+146], N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.01999999999999997e146

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if -1.01999999999999997e146 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 61.4% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 400000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 400000000.0) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 400000000.0) {
		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 <= 400000000.0d0) 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 <= 400000000.0) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 400000000.0:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 400000000.0)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 400000000.0)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 400000000.0], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 4e8

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   if 4e8 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   10. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.6% 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 99.8%

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

3. Taylor expanded in y around 0 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]
6. Add Preprocessing

Alternative 13: 31.5% 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 99.8%

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

3. Taylor expanded in y around 0 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in x around 0 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))