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

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 11

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: 93.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+82} \lor \neg \left(y \leq 8200\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.5e+82) (not (<= y 8200.0)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.5e+82) || !(y <= 8200.0)) {
		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 <= (-5.5d+82)) .or. (.not. (y <= 8200.0d0))) 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 <= -5.5e+82) || !(y <= 8200.0)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.5e+82) or not (y <= 8200.0):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.5e+82) || !(y <= 8200.0))
		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 <= -5.5e+82) || ~((y <= 8200.0)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.5e+82], N[Not[LessEqual[y, 8200.0]], $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 < -5.49999999999999997e82 or 8200 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 93.3%

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

   if -5.49999999999999997e82 < y < 8200

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 43.6%

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

      2. pow2 43.6%

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

   4. Applied egg-rr 43.6%

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

   5. Taylor expanded in y around 0 98.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+82} \lor \neg \left(y \leq 8200\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+83} \lor \neg \left(y \leq 3.2 \cdot 10^{+88}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.8e+83) (not (<= y 3.2e+88))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.8e+83) || !(y <= 3.2e+88)) {
		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 <= (-6.8d+83)) .or. (.not. (y <= 3.2d+88))) 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 <= -6.8e+83) || !(y <= 3.2e+88)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.8e+83) or not (y <= 3.2e+88):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.8e+83) || !(y <= 3.2e+88))
		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 <= -6.8e+83) || ~((y <= 3.2e+88)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.8e+83], N[Not[LessEqual[y, 3.2e+88]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.7999999999999996e83 or 3.1999999999999999e88 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 80.3%

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

   4. Taylor expanded in x around 0 97.3%

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

      1. neg-mul-1 97.3%

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

      2. +-commutative 97.3%

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

      3. fma-define 97.3%

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

      4. fmm-def 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in x around 0 95.2%

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

   if -6.7999999999999996e83 < y < 3.1999999999999999e88

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 49.4%

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

      2. pow2 49.4%

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

   4. Applied egg-rr 49.4%

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

   5. Taylor expanded in y around 0 94.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+83} \lor \neg \left(y \leq 3.2 \cdot 10^{+88}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, 1.0], N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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 99.9%

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

   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 99.6%

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

      1. *-commutative 99.6%

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

      2. sqrt-div 99.7%

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

      3. metadata-eval 99.7%

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

      4. un-div-inv 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto x \cdot \left(\color{blue}{\frac{y}{\sqrt{x}}} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

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

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 3.2 \cdot 10^{-16}:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x)))) (if (<= x 3.2e-16) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 3.2e-16) {
		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 <= 3.2d-16) 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 <= 3.2e-16) {
		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 <= 3.2e-16:
		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 <= 3.2e-16)
		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 <= 3.2e-16)
		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, 3.2e-16], N[(1.0 + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.20000000000000023e-16

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.9%

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

   if 3.20000000000000023e-16 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. neg-mul-1 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-define 99.6%

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

      4. fmm-def 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-16}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+193}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.2e+193)
   (/ (- 1.0 (* x x)) (+ 1.0 x))
   (if (<= y 3.2e+125) (- 1.0 x) (/ (+ 1.0 (* x x)) (+ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.2e+193) {
		tmp = (1.0 - (x * x)) / (1.0 + x);
	} else if (y <= 3.2e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 + (x * x)) / (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 <= (-8.2d+193)) then
        tmp = (1.0d0 - (x * x)) / (1.0d0 + x)
    else if (y <= 3.2d+125) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 + (x * x)) / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.2e+193) {
		tmp = (1.0 - (x * x)) / (1.0 + x);
	} else if (y <= 3.2e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 + (x * x)) / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.2e+193:
		tmp = (1.0 - (x * x)) / (1.0 + x)
	elif y <= 3.2e+125:
		tmp = 1.0 - x
	else:
		tmp = (1.0 + (x * x)) / (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.2e+193)
		tmp = Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x));
	elseif (y <= 3.2e+125)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 + Float64(x * x)) / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.2e+193)
		tmp = (1.0 - (x * x)) / (1.0 + x);
	elseif (y <= 3.2e+125)
		tmp = 1.0 - x;
	else
		tmp = (1.0 + (x * x)) / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.2e+193], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+125], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.1999999999999994e193

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in y around 0 4.0%

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

      1. sub-neg 4.0%

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

      2. flip-+ 35.3%

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

      3. metadata-eval 35.3%

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

   7. Applied egg-rr 35.3%

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

   if -8.1999999999999994e193 < y < 3.19999999999999983e125

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 45.7%

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

      2. pow2 45.7%

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

   4. Applied egg-rr 45.7%

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

   5. Taylor expanded in y around 0 83.4%

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

   if 3.19999999999999983e125 < y

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in y around 0 4.6%

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

      1. sub-neg 4.6%

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

      2. flip-+ 4.6%

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

      3. metadata-eval 4.6%

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

   7. Applied egg-rr 4.6%

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

      1. neg-sub0 4.6%

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

      2. sub-neg 4.6%

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

      3. pow1 4.6%

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

      4. metadata-eval 4.6%

         \[\leadsto \frac{1 - \left(0 + {\left(-x\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) \cdot \left(-x\right)}{1 - \left(-x\right)} \]

      5. sqrt-pow1 25.0%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{\sqrt{{\left(-x\right)}^{2}}}\right) \cdot \left(-x\right)}{1 - \left(-x\right)} \]

      6. pow2 25.0%

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

      7. sqr-neg 25.0%

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

      8. sqrt-prod 25.0%

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

      9. add-sqr-sqrt 25.0%

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

   9. Applied egg-rr 25.0%

      \[\leadsto \frac{1 - \color{blue}{\left(0 + x\right)} \cdot \left(-x\right)}{1 - \left(-x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+193}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+193}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.2e+193) (/ (- 1.0 (* x x)) (+ 1.0 x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.2e+193) {
		tmp = (1.0 - (x * x)) / (1.0 + 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 <= (-8.2d+193)) then
        tmp = (1.0d0 - (x * x)) / (1.0d0 + 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 <= -8.2e+193) {
		tmp = (1.0 - (x * x)) / (1.0 + x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.2e+193:
		tmp = (1.0 - (x * x)) / (1.0 + x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.2e+193)
		tmp = Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.2e+193)
		tmp = (1.0 - (x * x)) / (1.0 + x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.2e+193], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.1999999999999994e193

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in y around 0 4.0%

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

      1. sub-neg 4.0%

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

      2. flip-+ 35.3%

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

      3. metadata-eval 35.3%

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

   7. Applied egg-rr 35.3%

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

   if -8.1999999999999994e193 < y

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 55.4%

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

      2. pow2 55.4%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in y around 0 69.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+193}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+193}:\\ \;\;\;\;\frac{1 - x \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.2e+193) (/ (- 1.0 (* x x)) x) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.2e+193) {
		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 <= (-8.2d+193)) 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 <= -8.2e+193) {
		tmp = (1.0 - (x * x)) / x;
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.2e+193:
		tmp = (1.0 - (x * x)) / x
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.2e+193)
		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 <= -8.2e+193)
		tmp = (1.0 - (x * x)) / x;
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.2e+193], 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 < -8.1999999999999994e193

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in y around 0 4.0%

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

      1. sub-neg 4.0%

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

      2. flip-+ 35.3%

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

      3. metadata-eval 35.3%

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

   7. Applied egg-rr 35.3%

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

   8. Taylor expanded in x around inf 35.1%

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

   if -8.1999999999999994e193 < y

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 55.4%

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

      2. pow2 55.4%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in y around 0 69.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+193}:\\ \;\;\;\;\frac{1 - x \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.8% accurate, 15.3× 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 99.9%

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

   3. Taylor expanded in x around 0 99.9%

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

   4. Taylor expanded in y around 0 62.5%

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

   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 99.6%

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

   4. Taylor expanded in y around 0 63.3%

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

      1. neg-mul-1 63.3%

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

   6. Simplified 63.3%

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

Alternative 10: 62.9% 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-sqr-sqrt 50.2%

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

   2. pow2 50.2%

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

4. Applied egg-rr 50.2%

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

5. Taylor expanded in y around 0 63.1%

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

Alternative 11: 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 66.2%

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

4. Taylor expanded in y around 0 29.6%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))