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

Percentage Accurate: 99.9% → 99.9%

Time: 7.3s

Alternatives: 9

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: 95.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+63} \lor \neg \left(y \leq 6 \cdot 10^{+41}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.5e+63) (not (<= y 6e+41))) (+ 1.0 (* y (sqrt x))) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.5e+63) || !(y <= 6e+41)) {
		tmp = 1.0 + (y * sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3.5d+63)) .or. (.not. (y <= 6d+41))) then
        tmp = 1.0d0 + (y * sqrt(x))
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.5e+63) || !(y <= 6e+41)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.5e+63) or not (y <= 6e+41):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.5e+63) || !(y <= 6e+41))
		tmp = Float64(1.0 + Float64(y * sqrt(x)));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.5e+63) || ~((y <= 6e+41)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.5e+63], N[Not[LessEqual[y, 6e+41]], $MachinePrecision]], N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000029e63 or 5.9999999999999997e41 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 91.5%

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

   if -3.50000000000000029e63 < y < 5.9999999999999997e41

   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 100.0%

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

      2. pow2 100.0%

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

      3. pow1/2 100.0%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \left(1 - x\right) + y \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(1 - x\right) + y \cdot \color{blue}{{\left({x}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in y around 0 97.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+63} \lor \neg \left(y \leq 6 \cdot 10^{+41}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+84} \lor \neg \left(y \leq 6 \cdot 10^{+58}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.2e+84) (not (<= y 6e+58))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.2e+84) || !(y <= 6e+58)) {
		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 <= (-5.2d+84)) .or. (.not. (y <= 6d+58))) 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 <= -5.2e+84) || !(y <= 6e+58)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.2e+84) or not (y <= 6e+58):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.2e+84) || !(y <= 6e+58))
		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 <= -5.2e+84) || ~((y <= 6e+58)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.2e+84], N[Not[LessEqual[y, 6e+58]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2000000000000002e84 or 6.0000000000000005e58 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 83.2%

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

   4. Taylor expanded in y around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. unsub-neg 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in x around 0 88.3%

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

   if -5.2000000000000002e84 < y < 6.0000000000000005e58

   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 99.9%

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

      2. pow2 99.9%

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

      3. pow1/2 99.9%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \left(1 - x\right) + y \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(1 - x\right) + y \cdot \color{blue}{{\left({x}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in y around 0 96.3%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+84} \lor \neg \left(y \leq 6 \cdot 10^{+58}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
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 99.8%

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

   3. Taylor expanded in x around 0 97.4%

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

   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 99.4%

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

      1. neg-mul-1 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 98.4%

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

Alternative 5: 68.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+147}:\\ \;\;\;\;\frac{x \cdot y}{-y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+158}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e+147)
   (/ (* x y) (- y))
   (if (<= y 4.4e+158) (- 1.0 x) (/ (* x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+147) {
		tmp = (x * y) / -y;
	} else if (y <= 4.4e+158) {
		tmp = 1.0 - x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9.5d+147)) then
        tmp = (x * y) / -y
    else if (y <= 4.4d+158) then
        tmp = 1.0d0 - x
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+147) {
		tmp = (x * y) / -y;
	} else if (y <= 4.4e+158) {
		tmp = 1.0 - x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e+147:
		tmp = (x * y) / -y
	elif y <= 4.4e+158:
		tmp = 1.0 - x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e+147)
		tmp = Float64(Float64(x * y) / Float64(-y));
	elseif (y <= 4.4e+158)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.5e+147)
		tmp = (x * y) / -y;
	elseif (y <= 4.4e+158)
		tmp = 1.0 - x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e+147], N[(N[(x * y), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[y, 4.4e+158], N[(1.0 - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.4999999999999996e147

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 83.8%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around 0 4.3%

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

      1. mul-1-neg 4.3%

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

      2. distribute-frac-neg2 4.3%

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

   9. Simplified 4.3%

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

       1. associate-*r/ 20.9%

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

       2. frac-2neg 20.9%

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

       3. remove-double-neg 20.9%

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

   11. Applied egg-rr 20.9%

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

   if -9.4999999999999996e147 < y < 4.4000000000000002e158

   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 99.8%

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

      2. pow2 99.8%

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

      3. pow1/2 99.8%

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

      4. sqrt-pow1 99.8%

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

      5. metadata-eval 99.8%

         \[\leadsto \left(1 - x\right) + y \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left(1 - x\right) + y \cdot \color{blue}{{\left({x}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in y around 0 86.0%

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

   if 4.4000000000000002e158 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 76.1%

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

   4. Taylor expanded in y around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around 0 0.9%

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

      1. mul-1-neg 0.9%

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

      2. distribute-frac-neg2 0.9%

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

   9. Simplified 0.9%

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

       1. associate-*r/ 0.6%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 0.9%

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

       4. sqr-neg 0.9%

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

       5. sqrt-unprod 48.8%

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

       6. add-sqr-sqrt 48.8%

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

   11. Applied egg-rr 48.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+147}:\\ \;\;\;\;\frac{x \cdot y}{-y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+158}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+163}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.25e+163) (- 1.0 x) (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.25e+163) {
		tmp = 1.0 - x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.25d+163) then
        tmp = 1.0d0 - x
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.25e+163) {
		tmp = 1.0 - x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.25e+163:
		tmp = 1.0 - x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.25e+163)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.25e+163)
		tmp = 1.0 - x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.25e+163], N[(1.0 - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.25e163

   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 99.8%

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

      2. pow2 99.8%

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

      3. pow1/2 99.8%

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

      4. sqrt-pow1 99.8%

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

      5. metadata-eval 99.8%

         \[\leadsto \left(1 - x\right) + y \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left(1 - x\right) + y \cdot \color{blue}{{\left({x}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in y around 0 77.3%

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

   if 1.25e163 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 76.1%

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

   4. Taylor expanded in y around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around 0 0.9%

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

      1. mul-1-neg 0.9%

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

      2. distribute-frac-neg2 0.9%

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

   9. Simplified 0.9%

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

       1. associate-*r/ 0.6%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 0.9%

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

       4. sqr-neg 0.9%

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

       5. sqrt-unprod 48.8%

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

       6. add-sqr-sqrt 48.8%

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

   11. Applied egg-rr 48.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+163}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 300000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 300000.0) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 300000.0) {
		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 <= 300000.0d0) 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 <= 300000.0) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 300000.0:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 300000.0)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 300000.0)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 300000.0], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 3e5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 96.9%

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

   4. Taylor expanded in y around 0 64.4%

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

   if 3e5 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 99.8%

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

   4. Taylor expanded in y around 0 71.5%

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

      1. neg-mul-1 71.5%

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

   6. Simplified 71.5%

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

Alternative 8: 61.7% 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 99.7%

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

   2. pow2 99.7%

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

   3. pow1/2 99.7%

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

   4. sqrt-pow1 99.8%

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

   5. metadata-eval 99.8%

      \[\leadsto \left(1 - x\right) + y \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{2} \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(1 - x\right) + y \cdot \color{blue}{{\left({x}^{0.25}\right)}^{2}} \]

5. Taylor expanded in y around 0 69.0%

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

Alternative 9: 30.6% 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 62.2%

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

4. Taylor expanded in y around 0 32.2%

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

Reproduce

herbie shell --seed 2024146 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))