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

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \sqrt{x}, 1 - x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1 - x\right) \]
6. Add Preprocessing

Alternative 2: 94.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+36} \lor \neg \left(y \leq 5.5 \cdot 10^{+19}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3e+36) (not (<= y 5.5e+19))) (+ 1.0 (* y (sqrt x))) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3e+36) || !(y <= 5.5e+19)) {
		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 <= (-3d+36)) .or. (.not. (y <= 5.5d+19))) 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 <= -3e+36) || !(y <= 5.5e+19)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3e+36) or not (y <= 5.5e+19):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3e+36) || !(y <= 5.5e+19))
		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 <= -3e+36) || ~((y <= 5.5e+19)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3e+36], N[Not[LessEqual[y, 5.5e+19]], $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 < -3e36 or 5.5e19 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 90.6%

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

   if -3e36 < y < 5.5e19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+36} \lor \neg \left(y \leq 5.5 \cdot 10^{+19}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+59} \lor \neg \left(y \leq 6.5 \cdot 10^{+60}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.3e+59) (not (<= y 6.5e+60))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3e+59) || !(y <= 6.5e+60)) {
		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 <= (-1.3d+59)) .or. (.not. (y <= 6.5d+60))) 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 <= -1.3e+59) || !(y <= 6.5e+60)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.3e+59) or not (y <= 6.5e+60):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.3e+59) || !(y <= 6.5e+60))
		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 <= -1.3e+59) || ~((y <= 6.5e+60)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.3e+59], N[Not[LessEqual[y, 6.5e+60]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e59 or 6.49999999999999931e60 < y

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 99.4%

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

      4. associate-*l* 99.4%

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

      5. fma-define 99.4%

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

      6. pow1/2 99.4%

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

      7. sqrt-pow1 99.5%

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

      8. metadata-eval 99.5%

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

      9. pow1/2 99.5%

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

      10. sqrt-pow1 99.4%

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

      11. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around inf 87.8%

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

   if -1.3e59 < y < 6.49999999999999931e60

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 95.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+59} \lor \neg \left(y \leq 6.5 \cdot 10^{+60}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
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 99.3%

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

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 75.6%

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

   4. Taylor expanded in x around inf 74.9%

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

   5. Taylor expanded in y around 0 99.2%

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

      1. neg-mul-1 99.2%

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

      2. +-commutative 99.2%

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

      3. sub-neg 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.3%

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

Alternative 5: 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. Final simplification 99.9%

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

Alternative 6: 68.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+116}:\\ \;\;\;\;\frac{y \cdot x}{-y}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+130}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e+116)
   (/ (* y x) (- y))
   (if (<= y 1.05e+130) (- 1.0 x) (/ (* y x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e+116) {
		tmp = (y * x) / -y;
	} else if (y <= 1.05e+130) {
		tmp = 1.0 - x;
	} else {
		tmp = (y * x) / 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 <= (-6d+116)) then
        tmp = (y * x) / -y
    else if (y <= 1.05d+130) then
        tmp = 1.0d0 - x
    else
        tmp = (y * x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6e+116) {
		tmp = (y * x) / -y;
	} else if (y <= 1.05e+130) {
		tmp = 1.0 - x;
	} else {
		tmp = (y * x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6e+116:
		tmp = (y * x) / -y
	elif y <= 1.05e+130:
		tmp = 1.0 - x
	else:
		tmp = (y * x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e+116)
		tmp = Float64(Float64(y * x) / Float64(-y));
	elseif (y <= 1.05e+130)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(y * x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6e+116)
		tmp = (y * x) / -y;
	elseif (y <= 1.05e+130)
		tmp = 1.0 - x;
	else
		tmp = (y * x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e+116], N[(N[(y * x), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[y, 1.05e+130], N[(1.0 - x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.9999999999999997e116

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf 99.7%

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

   4. Taylor expanded in y around 0 3.9%

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

   5. Taylor expanded in x around inf 4.9%

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

      1. neg-mul-1 4.9%

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

      2. distribute-neg-frac 4.9%

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

   7. Simplified 4.9%

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

      1. distribute-frac-neg 4.9%

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

      2. distribute-frac-neg2 4.9%

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

      3. associate-*r/ 28.4%

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

   9. Applied egg-rr 28.4%

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

   if -5.9999999999999997e116 < y < 1.04999999999999995e130

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 87.9%

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

   if 1.04999999999999995e130 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 99.8%

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

   4. Taylor expanded in y around 0 1.9%

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

   5. Taylor expanded in x around inf 0.9%

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

      1. neg-mul-1 0.9%

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

      2. distribute-neg-frac 0.9%

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

   7. Simplified 0.9%

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

      1. associate-*r/ 0.7%

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

      2. *-un-lft-identity 0.7%

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

      3. *-commutative 0.7%

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

      4. rgt-mult-inverse 0.7%

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

      5. *-commutative 0.7%

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

      6. associate-*l* 0.8%

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

      7. div-inv 0.8%

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

      8. *-commutative 0.8%

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

      9. *-commutative 0.8%

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

      10. div-inv 0.8%

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

      11. associate-*l* 0.7%

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

      12. *-commutative 0.7%

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

      13. rgt-mult-inverse 0.7%

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

      14. *-commutative 0.7%

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

      15. *-un-lft-identity 0.7%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 31.3%

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

      18. sqr-neg 31.3%

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

      19. sqrt-unprod 31.4%

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

      20. add-sqr-sqrt 31.4%

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

   9. Applied egg-rr 31.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+116}:\\ \;\;\;\;\frac{y \cdot x}{-y}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+130}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+129}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 9.5e+129) (- 1.0 x) (/ (* y x) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.5e+129) {
		tmp = 1.0 - x;
	} else {
		tmp = (y * x) / 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+129) then
        tmp = 1.0d0 - x
    else
        tmp = (y * x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.5e+129) {
		tmp = 1.0 - x;
	} else {
		tmp = (y * x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.5e+129:
		tmp = 1.0 - x
	else:
		tmp = (y * x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.5e+129)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(y * x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.5e+129)
		tmp = 1.0 - x;
	else
		tmp = (y * x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.5e+129], N[(1.0 - x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5000000000000004e129

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 72.9%

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

   if 9.5000000000000004e129 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 99.8%

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

   4. Taylor expanded in y around 0 1.9%

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

   5. Taylor expanded in x around inf 0.9%

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

      1. neg-mul-1 0.9%

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

      2. distribute-neg-frac 0.9%

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

   7. Simplified 0.9%

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

      1. associate-*r/ 0.7%

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

      2. *-un-lft-identity 0.7%

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

      3. *-commutative 0.7%

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

      4. rgt-mult-inverse 0.7%

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

      5. *-commutative 0.7%

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

      6. associate-*l* 0.8%

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

      7. div-inv 0.8%

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

      8. *-commutative 0.8%

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

      9. *-commutative 0.8%

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

      10. div-inv 0.8%

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

      11. associate-*l* 0.7%

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

      12. *-commutative 0.7%

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

      13. rgt-mult-inverse 0.7%

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

      14. *-commutative 0.7%

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

      15. *-un-lft-identity 0.7%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 31.3%

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

      18. sqr-neg 31.3%

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

      19. sqrt-unprod 31.4%

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

      20. add-sqr-sqrt 31.4%

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

   9. Applied egg-rr 31.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+129}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.8% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 235:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 235.0) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 235.0) {
		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 <= 235.0d0) 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 <= 235.0) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 235.0:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 235.0)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 235.0)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 235.0], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 235

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 99.7%

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

   4. Taylor expanded in y around 0 64.6%

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

   5. Taylor expanded in x around 0 64.3%

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

   if 235 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 75.4%

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

   4. Taylor expanded in y around 0 38.7%

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

   5. Taylor expanded in x around inf 62.5%

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

      1. neg-mul-1 62.5%

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

   7. Simplified 62.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 235:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% 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. Taylor expanded in y around 0 64.0%

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

4. Final simplification 64.0%

   \[\leadsto 1 - x \]
5. Add Preprocessing

Alternative 10: 31.3% 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 y around inf 87.8%

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

4. Taylor expanded in y around 0 52.0%

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

5. Taylor expanded in x around 0 33.7%

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

6. Final simplification 33.7%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))