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

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 94.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+15} \lor \neg \left(y \leq 3.6 \cdot 10^{+47}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9e+15) (not (<= y 3.6e+47))) (+ 1.0 (* y (sqrt x))) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9e+15) || !(y <= 3.6e+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 <= (-9d+15)) .or. (.not. (y <= 3.6d+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 <= -9e+15) || !(y <= 3.6e+47)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9e+15) or not (y <= 3.6e+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 <= -9e+15) || !(y <= 3.6e+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 <= -9e+15) || ~((y <= 3.6e+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, -9e+15], N[Not[LessEqual[y, 3.6e+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 < -9e15 or 3.60000000000000008e47 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. add-sqr-sqrt 56.0%

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

      3. sqrt-unprod 42.7%

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

      4. sqrt-prod 35.4%

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

      5. pow2 35.4%

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

   4. Applied egg-rr 35.4%

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

   5. Taylor expanded in x around 0 96.0%

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

   if -9e15 < y < 3.60000000000000008e47

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-sqr-sqrt 54.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqrt-prod 100.0%

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

      5. pow2 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 98.6%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+15} \lor \neg \left(y \leq 3.6 \cdot 10^{+47}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+36} \lor \neg \left(y \leq 2.8 \cdot 10^{+89}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e+36) (not (<= y 2.8e+89))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5e+36) || !(y <= 2.8e+89)) {
		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 <= (-5d+36)) .or. (.not. (y <= 2.8d+89))) 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 <= -5e+36) || !(y <= 2.8e+89)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5e+36) or not (y <= 2.8e+89):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5e+36) || !(y <= 2.8e+89))
		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 <= -5e+36) || ~((y <= 2.8e+89)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5e+36], N[Not[LessEqual[y, 2.8e+89]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999977e36 or 2.7999999999999998e89 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. add-sqr-sqrt 53.9%

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

      3. sqrt-unprod 29.7%

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

      4. sqrt-prod 21.7%

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

      5. pow2 21.7%

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

   4. Applied egg-rr 21.7%

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

      1. unpow2 21.7%

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

   6. Applied egg-rr 21.7%

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

      1. flip-+ 9.8%

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

      2. unpow2 9.8%

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

      3. add-sqr-sqrt 9.8%

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

      4. pow2 9.8%

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

      5. clear-num 9.8%

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

      6. sqrt-prod 10.0%

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

      7. sqrt-prod 9.3%

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

      8. add-sqr-sqrt 22.3%

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

      9. associate--r+ 22.3%

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

      10. clear-num 22.3%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in y around inf 93.1%

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

   if -4.99999999999999977e36 < y < 2.7999999999999998e89

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-sqr-sqrt 55.5%

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

      3. sqrt-unprod 99.4%

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

      4. sqrt-prod 98.8%

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

      5. pow2 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in y around 0 94.2%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+36} \lor \neg \left(y \leq 2.8 \cdot 10^{+89}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 27.5:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{\sqrt{x}} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 27.5) (+ 1.0 (* y (sqrt x))) (* x (+ (/ y (sqrt x)) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 27.5) {
		tmp = 1.0 + (y * sqrt(x));
	} else {
		tmp = x * ((y / sqrt(x)) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 27.5d0) then
        tmp = 1.0d0 + (y * sqrt(x))
    else
        tmp = x * ((y / sqrt(x)) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 27.5) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = x * ((y / Math.sqrt(x)) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 27.5:
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = x * ((y / math.sqrt(x)) + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 27.5)
		tmp = Float64(1.0 + Float64(y * sqrt(x)));
	else
		tmp = Float64(x * Float64(Float64(y / sqrt(x)) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 27.5)
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = x * ((y / sqrt(x)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 27.5], N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 27.5

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. add-sqr-sqrt 52.7%

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

      3. sqrt-unprod 72.8%

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

      4. sqrt-prod 72.8%

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

      5. pow2 72.8%

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

   4. Applied egg-rr 72.8%

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

   5. Taylor expanded in x around 0 99.4%

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

   if 27.5 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. add-sqr-sqrt 57.4%

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

      3. sqrt-unprod 74.9%

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

      4. sqrt-prod 67.8%

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

      5. pow2 67.8%

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

   4. Applied egg-rr 67.8%

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

   5. Taylor expanded in x around inf 98.5%

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

      1. *-commutative 98.5%

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

      2. sqrt-div 98.5%

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

      3. metadata-eval 98.5%

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

      4. un-div-inv 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 27.5:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{\sqrt{x}} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 340:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x)))) (if (<= x 340.0) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 340.0) {
		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 <= 340.0d0) 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 <= 340.0) {
		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 <= 340.0:
		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 <= 340.0)
		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 <= 340.0)
		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, 340.0], N[(1.0 + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 340

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. add-sqr-sqrt 52.3%

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

      3. sqrt-unprod 72.3%

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

      4. sqrt-prod 72.3%

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

      5. pow2 72.3%

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

   4. Applied egg-rr 72.3%

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

   5. Taylor expanded in x around 0 99.4%

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

   if 340 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. add-sqr-sqrt 57.8%

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

      3. sqrt-unprod 75.5%

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

      4. sqrt-prod 68.4%

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

      5. pow2 68.4%

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

   4. Applied egg-rr 68.4%

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

   5. Taylor expanded in x around inf 98.5%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. neg-mul-1 98.5%

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

      2. +-commutative 98.5%

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

      3. fma-define 98.5%

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

      4. fma-neg 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 340:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.6% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 - x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. add-sqr-sqrt 54.9%

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

   3. sqrt-unprod 73.8%

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

   4. sqrt-prod 70.5%

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

   5. pow2 70.5%

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

4. Applied egg-rr 70.5%

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

5. Taylor expanded in y around 0 62.4%

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

Alternative 7: 32.3% accurate, 53.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -x

Julia

function code(x, y)
	return Float64(-x)
end

MATLAB

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

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. add-sqr-sqrt 54.9%

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

   3. sqrt-unprod 73.8%

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

   4. sqrt-prod 70.5%

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

   5. pow2 70.5%

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

4. Applied egg-rr 70.5%

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

5. Taylor expanded in x around inf 59.0%

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

6. Taylor expanded in y around 0 28.0%

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

   1. neg-mul-1 28.0%

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

8. Simplified 28.0%

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))