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

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

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. Add Preprocessing

Alternative 2: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+20} \lor \neg \left(y \leq 5 \cdot 10^{+47}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.55e+20) (not (<= y 5e+47)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.55e+20) || !(y <= 5e+47)) {
		tmp = 1.0 + (y * sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.55d+20)) .or. (.not. (y <= 5d+47))) then
        tmp = 1.0d0 + (y * sqrt(x))
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.55e+20) || !(y <= 5e+47)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.55e+20) or not (y <= 5e+47):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.55e+20) || !(y <= 5e+47))
		tmp = Float64(1.0 + Float64(y * sqrt(x)));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.55e+20) || ~((y <= 5e+47)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.55e+20], N[Not[LessEqual[y, 5e+47]], $MachinePrecision]], N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.55e20 or 5.00000000000000022e47 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 93.0%

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

   if -2.55e20 < y < 5.00000000000000022e47

   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 56.3%

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

      2. sqrt-unprod 98.6%

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

      3. *-commutative 98.6%

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

      4. *-commutative 98.6%

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

      5. swap-sqr 98.6%

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

      6. add-sqr-sqrt 98.6%

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

      7. pow2 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in y around 0 98.3%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+20} \lor \neg \left(y \leq 5 \cdot 10^{+47}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+81} \lor \neg \left(y \leq 1.1 \cdot 10^{+91}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.9e+81) (not (<= y 1.1e+91))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e+81) || !(y <= 1.1e+91)) {
		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 <= (-2.9d+81)) .or. (.not. (y <= 1.1d+91))) 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 <= -2.9e+81) || !(y <= 1.1e+91)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.9e+81) or not (y <= 1.1e+91):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.9e+81) || !(y <= 1.1e+91))
		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 <= -2.9e+81) || ~((y <= 1.1e+91)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.9e+81], N[Not[LessEqual[y, 1.1e+91]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e81 or 1.1e91 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 76.3%

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

      1. associate--l+ 76.3%

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

      2. distribute-rgt-in 76.3%

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

   5. Simplified 99.5%

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

      1. sqrt-div 99.4%

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

      2. metadata-eval 99.4%

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

      3. un-div-inv 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. clear-num 99.4%

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

      2. clear-num 99.4%

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

      3. pow1 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-div 99.5%

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

      6. metadata-eval 99.5%

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

      7. pow1/2 99.5%

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

      8. associate-*l/ 99.5%

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

      9. *-un-lft-identity 99.5%

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

      10. pow1/2 99.5%

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

      11. pow-flip 99.5%

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

      12. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in y around inf 96.8%

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

   if -2.9e81 < y < 1.1e91

   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 54.6%

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

      2. sqrt-unprod 92.7%

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

      3. *-commutative 92.7%

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

      4. *-commutative 92.7%

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

      5. swap-sqr 92.7%

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

      6. add-sqr-sqrt 92.7%

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

      7. pow2 92.7%

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

   4. Applied egg-rr 92.7%

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

   5. Taylor expanded in y around 0 93.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+81} \lor \neg \left(y \leq 1.1 \cdot 10^{+91}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 0.26:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x)))) (if (<= x 0.26) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 0.26) {
		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 <= 0.26d0) 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 <= 0.26) {
		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 <= 0.26:
		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 <= 0.26)
		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 <= 0.26)
		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, 0.26], N[(1.0 + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.26000000000000001

   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 0.26000000000000001 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 99.0%

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

      1. neg-mul-1 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in y around 0 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.26:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
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. Add Preprocessing

Alternative 6: 68.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+115}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{y}\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+227}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.85e+115)
   (/ (* y (- x)) y)
   (if (<= y 2.65e+227) (- 1.0 x) (/ (* y x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.85e+115) {
		tmp = (y * -x) / y;
	} else if (y <= 2.65e+227) {
		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 <= (-1.85d+115)) then
        tmp = (y * -x) / y
    else if (y <= 2.65d+227) 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 <= -1.85e+115) {
		tmp = (y * -x) / y;
	} else if (y <= 2.65e+227) {
		tmp = 1.0 - x;
	} else {
		tmp = (y * x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.85e+115:
		tmp = (y * -x) / y
	elif y <= 2.65e+227:
		tmp = 1.0 - x
	else:
		tmp = (y * x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.85e+115)
		tmp = Float64(Float64(y * Float64(-x)) / y);
	elseif (y <= 2.65e+227)
		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 <= -1.85e+115)
		tmp = (y * -x) / y;
	elseif (y <= 2.65e+227)
		tmp = 1.0 - x;
	else
		tmp = (y * x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.85e+115], N[(N[(y * (-x)), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 2.65e+227], N[(1.0 - x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.85000000000000003e115

   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. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. div-sub 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 3.7%

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

      1. associate-*r/ 25.7%

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

   8. Applied egg-rr 25.7%

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

   9. Taylor expanded in x around inf 26.8%

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

       1. associate-*r* 26.8%

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

       2. mul-1-neg 26.8%

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

   11. Simplified 26.8%

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

   if -1.85000000000000003e115 < y < 2.65000000000000004e227

   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 56.3%

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

      2. sqrt-unprod 87.4%

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

      3. *-commutative 87.4%

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

      4. *-commutative 87.4%

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

      5. swap-sqr 83.8%

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

      6. add-sqr-sqrt 83.9%

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

      7. pow2 83.9%

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

   4. Applied egg-rr 83.9%

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

   5. Taylor expanded in y around 0 82.0%

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

   if 2.65000000000000004e227 < y

   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. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. div-sub 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 2.5%

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

   7. Taylor expanded in x around inf 1.2%

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

      1. neg-mul-1 99.7%

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

   9. Simplified 1.2%

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

       1. associate-*r/ 0.9%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 31.9%

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

       4. sqr-neg 31.9%

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

       5. sqrt-unprod 32.0%

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

       6. add-sqr-sqrt 32.0%

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

   11. Applied egg-rr 32.0%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+115}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{y}\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+227}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{+227}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.65e+227) (- 1.0 x) (/ (* y x) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.65e+227) {
		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 <= 2.65d+227) 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 <= 2.65e+227) {
		tmp = 1.0 - x;
	} else {
		tmp = (y * x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.65e+227:
		tmp = 1.0 - x
	else:
		tmp = (y * x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.65e+227)
		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 <= 2.65e+227)
		tmp = 1.0 - x;
	else
		tmp = (y * x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.65e+227], N[(1.0 - x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.65000000000000004e227

   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 45.0%

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

      2. sqrt-unprod 70.0%

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

      3. *-commutative 70.0%

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

      4. *-commutative 70.0%

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

      5. swap-sqr 67.1%

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

      6. add-sqr-sqrt 67.1%

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

      7. pow2 67.1%

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

   4. Applied egg-rr 67.1%

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

   5. Taylor expanded in y around 0 66.4%

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

   if 2.65000000000000004e227 < y

   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. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. div-sub 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 2.5%

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

   7. Taylor expanded in x around inf 1.2%

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

      1. neg-mul-1 99.7%

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

   9. Simplified 1.2%

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

       1. associate-*r/ 0.9%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 31.9%

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

       4. sqr-neg 31.9%

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

       5. sqrt-unprod 32.0%

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

       6. add-sqr-sqrt 32.0%

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

   11. Applied egg-rr 32.0%

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

Alternative 8: 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 99.8%

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

   3. Taylor expanded in x around 0 99.3%

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

   4. Taylor expanded in y around 0 56.5%

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

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.2%

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

      1. associate--l+ 76.2%

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

      2. div-sub 76.2%

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

   5. Simplified 76.2%

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

   6. Taylor expanded in y around 0 41.6%

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

   7. Taylor expanded in x around inf 64.8%

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

      1. neg-mul-1 64.8%

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

   9. Simplified 64.8%

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

Alternative 9: 62.8% 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 49.5%

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

   2. sqrt-unprod 67.7%

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

   3. *-commutative 67.7%

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

   4. *-commutative 67.7%

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

   5. swap-sqr 64.4%

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

   6. add-sqr-sqrt 64.4%

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

   7. pow2 64.4%

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

4. Applied egg-rr 64.4%

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

5. Taylor expanded in y around 0 61.1%

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

Alternative 10: 31.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 67.3%

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

4. Taylor expanded in y around 0 29.2%

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

Reproduce

herbie shell --seed 2024144 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))