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

Percentage Accurate: 99.9% → 99.9%

Time: 13.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* y (sqrt x)))))
   (if (<= y -5.5e+33) t_0 (if (<= y 1.15e+20) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y * sqrt(x));
	double tmp;
	if (y <= -5.5e+33) {
		tmp = t_0;
	} else if (y <= 1.15e+20) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	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 = 1.0d0 + (y * sqrt(x))
    if (y <= (-5.5d+33)) then
        tmp = t_0
    else if (y <= 1.15d+20) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y * Math.sqrt(x));
	double tmp;
	if (y <= -5.5e+33) {
		tmp = t_0;
	} else if (y <= 1.15e+20) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y * math.sqrt(x))
	tmp = 0
	if y <= -5.5e+33:
		tmp = t_0
	elif y <= 1.15e+20:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y * sqrt(x)))
	tmp = 0.0
	if (y <= -5.5e+33)
		tmp = t_0;
	elseif (y <= 1.15e+20)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y * sqrt(x));
	tmp = 0.0;
	if (y <= -5.5e+33)
		tmp = t_0;
	elseif (y <= 1.15e+20)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+33], t$95$0, If[LessEqual[y, 1.15e+20], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000006e33 or 1.15e20 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\sqrt{x} \cdot y\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{y}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 94.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right)\right) \]

   5. Simplified 94.0%

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

   if -5.5000000000000006e33 < y < 1.15e20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+33}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+44}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x))))
   (if (<= y -1.45e+46) t_0 (if (<= y 2.9e+44) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (y <= -1.45e+46) {
		tmp = t_0;
	} else if (y <= 2.9e+44) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	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 (y <= (-1.45d+46)) then
        tmp = t_0
    else if (y <= 2.9d+44) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    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 (y <= -1.45e+46) {
		tmp = t_0;
	} else if (y <= 2.9e+44) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if y <= -1.45e+46:
		tmp = t_0
	elif y <= 2.9e+44:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(x))
	tmp = 0.0
	if (y <= -1.45e+46)
		tmp = t_0;
	elseif (y <= 2.9e+44)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * sqrt(x);
	tmp = 0.0;
	if (y <= -1.45e+46)
		tmp = t_0;
	elseif (y <= 2.9e+44)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.45e+46], t$95$0, If[LessEqual[y, 2.9e+44], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4500000000000001e46 or 2.9000000000000002e44 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{y}\right) \]

      2. sqrt-lowering-sqrt.f64 91.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right) \]

   5. Simplified 91.4%

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

   if -1.4500000000000001e46 < y < 2.9000000000000002e44

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 97.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 97.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+44}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot x\\ \mathbf{if}\;y \leq 2.9 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - x\right) + \left(x \cdot t\_0\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot \left(1 - x\right)\right)\right)}{\left(1 - x\right) \cdot \frac{1 - x}{t\_0}}}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x x))))
   (if (<= y 2.9e+44)
     (/ (* (- 1.0 x) y) y)
     (/
      (/
       (+ (- 1.0 x) (* (* x t_0) (+ 1.0 (* x (+ -1.0 (* x (- 1.0 x)))))))
       (* (- 1.0 x) (/ (- 1.0 x) t_0)))
      (- 1.0 x)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= 2.9e+44) {
		tmp = ((1.0 - x) * y) / y;
	} else {
		tmp = (((1.0 - x) + ((x * t_0) * (1.0 + (x * (-1.0 + (x * (1.0 - x))))))) / ((1.0 - x) * ((1.0 - x) / t_0))) / (1.0 - 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 = 1.0d0 - (x * x)
    if (y <= 2.9d+44) then
        tmp = ((1.0d0 - x) * y) / y
    else
        tmp = (((1.0d0 - x) + ((x * t_0) * (1.0d0 + (x * ((-1.0d0) + (x * (1.0d0 - x))))))) / ((1.0d0 - x) * ((1.0d0 - x) / t_0))) / (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= 2.9e+44) {
		tmp = ((1.0 - x) * y) / y;
	} else {
		tmp = (((1.0 - x) + ((x * t_0) * (1.0 + (x * (-1.0 + (x * (1.0 - x))))))) / ((1.0 - x) * ((1.0 - x) / t_0))) / (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x * x)
	tmp = 0
	if y <= 2.9e+44:
		tmp = ((1.0 - x) * y) / y
	else:
		tmp = (((1.0 - x) + ((x * t_0) * (1.0 + (x * (-1.0 + (x * (1.0 - x))))))) / ((1.0 - x) * ((1.0 - x) / t_0))) / (1.0 - x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x * x))
	tmp = 0.0
	if (y <= 2.9e+44)
		tmp = Float64(Float64(Float64(1.0 - x) * y) / y);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - x) + Float64(Float64(x * t_0) * Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(1.0 - x))))))) / Float64(Float64(1.0 - x) * Float64(Float64(1.0 - x) / t_0))) / Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x * x);
	tmp = 0.0;
	if (y <= 2.9e+44)
		tmp = ((1.0 - x) * y) / y;
	else
		tmp = (((1.0 - x) + ((x * t_0) * (1.0 + (x * (-1.0 + (x * (1.0 - x))))))) / ((1.0 - x) * ((1.0 - x) / t_0))) / (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.9e+44], N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x * t$95$0), $MachinePrecision] * N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - x), $MachinePrecision] * N[(N[(1.0 - x), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9000000000000002e44

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 75.2%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 75.2%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + x}{1 \cdot 1 - x \cdot x}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + x\right), \color{blue}{\left(1 \cdot 1 - x \cdot x\right)}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + 1\right), \left(\color{blue}{1 \cdot 1} - x \cdot x\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\color{blue}{1 \cdot 1} - x \cdot x\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(1 - \color{blue}{x} \cdot x\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 54.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   7. Applied egg-rr 54.3%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1 \cdot 1 - x \cdot x}{x + 1} \]

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. *-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. inv-pow N/A

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

      10. associate-*l* N/A

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

      11. div-inv N/A

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

      12. associate-*l/ N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 - x\right) \cdot y\right), \color{blue}{y}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 - x\right), y\right), y\right) \]

      15. --lowering--.f64 75.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right), y\right) \]

   9. Applied egg-rr 75.8%

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

   if 2.9000000000000002e44 < y

   1. Initial program 99.8%

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

   4. Applied egg-rr 16.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{2}}{1 - x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(1 + x\right)}^{2}\right), \color{blue}{\left(1 - x\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right), \left(\color{blue}{1} - x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + x\right), \left(1 + x\right)\right), \left(\color{blue}{1} - x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x + 1\right), \left(1 + x\right)\right), \left(1 - x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(1 + x\right)\right), \left(1 - x\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(x + 1\right)\right), \left(1 - x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(1 - x\right)\right) \]

      8. --lowering--.f64 10.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right) \]

   7. Simplified 10.5%

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

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) \cdot x + \left(x + 1\right) \cdot 1\right), \mathsf{\_.f64}\left(\color{blue}{1}, x\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) \cdot x + \left(x + 1\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) \cdot x + \left(x + 1\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

      4. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot 1 - x \cdot x}{1 - x} \cdot x + \left(x + 1\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 \cdot 1 - x \cdot x\right) \cdot x}{1 - x} + \left(x + 1\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 \cdot 1 - x \cdot x\right) \cdot x}{1 - x} + \left(1 + x\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

      7. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 \cdot 1 - x \cdot x\right) \cdot x}{1 - x} + \frac{1 \cdot 1 - x \cdot x}{1 - x}\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 \cdot 1 - x \cdot x\right) \cdot x}{1 - x} + \frac{1}{\frac{1 - x}{1 \cdot 1 - x \cdot x}}\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

      9. frac-add N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\left(1 \cdot 1 - x \cdot x\right) \cdot x\right) \cdot \frac{1 - x}{1 \cdot 1 - x \cdot x} + \left(1 - x\right) \cdot 1}{\left(1 - x\right) \cdot \frac{1 - x}{1 \cdot 1 - x \cdot x}}\right), \mathsf{\_.f64}\left(\color{blue}{1}, x\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(1 \cdot 1 - x \cdot x\right) \cdot x\right) \cdot \frac{1 - x}{1 \cdot 1 - x \cdot x} + \left(1 - x\right) \cdot 1\right), \left(\left(1 - x\right) \cdot \frac{1 - x}{1 \cdot 1 - x \cdot x}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, x\right)\right) \]

   9. Applied egg-rr 10.2%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \color{blue}{\left(1 + x \cdot \left(x \cdot \left(1 + -1 \cdot x\right) - 1\right)\right)}\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(1 + -1 \cdot x\right) - 1\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + -1 \cdot x\right) - 1\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + -1 \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(1 + -1 \cdot x\right) + -1\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + x \cdot \left(1 + -1 \cdot x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(x \cdot \left(1 + -1 \cdot x\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(1 + -1 \cdot x\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

       8. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

       9. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(1 - x\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

       10. --lowering--.f64 21.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), 1\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(1, x\right)\right) \]

   12. Simplified 21.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - x\right) + \left(x \cdot \left(1 - x \cdot x\right)\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot \left(1 - x\right)\right)\right)}{\left(1 - x\right) \cdot \frac{1 - x}{1 - x \cdot x}}}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(3 + x \cdot \left(\left(1 + x\right) \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.9e+44)
   (/ (* (- 1.0 x) y) y)
   (+ 1.0 (* x (+ 3.0 (* x (* (+ 1.0 x) 4.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.9e+44) {
		tmp = ((1.0 - x) * y) / y;
	} else {
		tmp = 1.0 + (x * (3.0 + (x * ((1.0 + x) * 4.0))));
	}
	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+44) then
        tmp = ((1.0d0 - x) * y) / y
    else
        tmp = 1.0d0 + (x * (3.0d0 + (x * ((1.0d0 + x) * 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.9e+44) {
		tmp = ((1.0 - x) * y) / y;
	} else {
		tmp = 1.0 + (x * (3.0 + (x * ((1.0 + x) * 4.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.9e+44:
		tmp = ((1.0 - x) * y) / y
	else:
		tmp = 1.0 + (x * (3.0 + (x * ((1.0 + x) * 4.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.9e+44)
		tmp = Float64(Float64(Float64(1.0 - x) * y) / y);
	else
		tmp = Float64(1.0 + Float64(x * Float64(3.0 + Float64(x * Float64(Float64(1.0 + x) * 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.9e+44)
		tmp = ((1.0 - x) * y) / y;
	else
		tmp = 1.0 + (x * (3.0 + (x * ((1.0 + x) * 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.9e+44], N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(1.0 + N[(x * N[(3.0 + N[(x * N[(N[(1.0 + x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9000000000000002e44

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 75.2%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 75.2%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + x}{1 \cdot 1 - x \cdot x}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + x\right), \color{blue}{\left(1 \cdot 1 - x \cdot x\right)}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + 1\right), \left(\color{blue}{1 \cdot 1} - x \cdot x\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\color{blue}{1 \cdot 1} - x \cdot x\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(1 - \color{blue}{x} \cdot x\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 54.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   7. Applied egg-rr 54.3%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1 \cdot 1 - x \cdot x}{x + 1} \]

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. *-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. inv-pow N/A

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

      10. associate-*l* N/A

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

      11. div-inv N/A

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

      12. associate-*l/ N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 - x\right) \cdot y\right), \color{blue}{y}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 - x\right), y\right), y\right) \]

      15. --lowering--.f64 75.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right), y\right) \]

   9. Applied egg-rr 75.8%

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

   if 2.9000000000000002e44 < y

   1. Initial program 99.8%

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

   4. Applied egg-rr 16.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{2}}{1 - x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(1 + x\right)}^{2}\right), \color{blue}{\left(1 - x\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right), \left(\color{blue}{1} - x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + x\right), \left(1 + x\right)\right), \left(\color{blue}{1} - x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x + 1\right), \left(1 + x\right)\right), \left(1 - x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(1 + x\right)\right), \left(1 - x\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(x + 1\right)\right), \left(1 - x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(1 - x\right)\right) \]

      8. --lowering--.f64 10.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right) \]

   7. Simplified 10.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x \cdot \left(4 + 4 \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(3 + x \cdot \left(4 + 4 \cdot x\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(3 + x \cdot \left(4 + 4 \cdot x\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \color{blue}{\left(x \cdot \left(4 + 4 \cdot x\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \mathsf{*.f64}\left(x, \color{blue}{\left(4 + 4 \cdot x\right)}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \mathsf{*.f64}\left(x, \left(4 + x \cdot \color{blue}{4}\right)\right)\right)\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{4}\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \mathsf{*.f64}\left(x, \left(\left(1 + x\right) \cdot 4\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(1 + x\right), \color{blue}{4}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x + 1\right), 4\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 21.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), 4\right)\right)\right)\right)\right) \]

   10. Simplified 21.2%

       \[\leadsto \color{blue}{1 + x \cdot \left(3 + x \cdot \left(\left(x + 1\right) \cdot 4\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(3 + x \cdot \left(\left(1 + x\right) \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.6% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.0) 1.0 (- 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0 - 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 = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0
	else:
		tmp = 0.0 - x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, 1.0], 1.0, N[(0.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 59.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 59.6%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 58.3%

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

   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 0

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

      1. --lowering--.f64 69.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 69.6%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x} \]

      3. --lowering--.f64 67.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{x}\right) \]

   8. Simplified 67.8%

      \[\leadsto \color{blue}{0 - x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.6% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-3 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.0) 1.0 (- -3.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -3.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = (-3.0d0) - 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 = -3.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0
	else:
		tmp = -3.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 1.0;
	else
		tmp = Float64(-3.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0;
	else
		tmp = -3.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.0], 1.0, N[(-3.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 59.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 59.6%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 58.3%

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

   if 1 < x

   1. Initial program 99.9%

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

   4. Applied egg-rr 18.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{2}}{1 - x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(1 + x\right)}^{2}\right), \color{blue}{\left(1 - x\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right), \left(\color{blue}{1} - x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + x\right), \left(1 + x\right)\right), \left(\color{blue}{1} - x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x + 1\right), \left(1 + x\right)\right), \left(1 - x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(1 + x\right)\right), \left(1 - x\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(x + 1\right)\right), \left(1 - x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(1 - x\right)\right) \]

      8. --lowering--.f64 34.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right) \]

   7. Simplified 34.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(1 + 3 \cdot \frac{1}{x}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(1 + 3 \cdot \frac{1}{x}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(-1 \cdot x\right) \cdot \left(3 \cdot \frac{1}{x} + \color{blue}{1}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) + \color{blue}{1 \cdot \left(-1 \cdot x\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) + -1 \cdot \color{blue}{x} \]

      5. mul-1-neg N/A

         \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) - \color{blue}{x} \]

      7. mul-1-neg N/A

         \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - x \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(3 \cdot \frac{1}{x}\right) \cdot x\right)\right) - x \]

      9. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(3 \cdot \left(\frac{1}{x} \cdot x\right)\right)\right) - x \]

      10. lft-mult-inverse N/A

          \[\leadsto \left(\mathsf{neg}\left(3 \cdot 1\right)\right) - x \]

      11. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(3\right)\right) - x \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(3\right)\right), \color{blue}{x}\right) \]

      13. metadata-eval 67.7%

          \[\leadsto \mathsf{\_.f64}\left(-3, x\right) \]

   10. Simplified 67.7%

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

Alternative 8: 63.6% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. --lowering--.f64 64.8%

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

5. Simplified 64.8%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + x}{1 \cdot 1 - x \cdot x}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + x\right), \color{blue}{\left(1 \cdot 1 - x \cdot x\right)}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + 1\right), \left(\color{blue}{1 \cdot 1} - x \cdot x\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\color{blue}{1 \cdot 1} - x \cdot x\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(1 - \color{blue}{x} \cdot x\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 47.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

7. Applied egg-rr 47.1%

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

   1. clear-num N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{1 \cdot 1 - x \cdot x}{x + 1} \]

   3. +-commutative N/A

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

   4. flip-- N/A

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

   5. *-rgt-identity N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. pow-plus N/A

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

   9. inv-pow N/A

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

   10. associate-*l* N/A

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

   11. div-inv N/A

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

   12. associate-*l/ N/A

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

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(1 - x\right) \cdot y\right), \color{blue}{y}\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 - x\right), y\right), y\right) \]

   15. --lowering--.f64 65.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right), y\right) \]

9. Applied egg-rr 65.0%

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

Alternative 9: 32.9% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.0) 1.0 -3.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = -3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0
	else:
		tmp = -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 1.0;
	else
		tmp = -3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0;
	else
		tmp = -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.0], 1.0, -3.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 59.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 59.6%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 58.3%

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

   if 1 < x

   1. Initial program 99.9%

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

   4. Applied egg-rr 18.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{2}}{1 - x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(1 + x\right)}^{2}\right), \color{blue}{\left(1 - x\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right), \left(\color{blue}{1} - x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + x\right), \left(1 + x\right)\right), \left(\color{blue}{1} - x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x + 1\right), \left(1 + x\right)\right), \left(1 - x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(1 + x\right)\right), \left(1 - x\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(x + 1\right)\right), \left(1 - x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(1 - x\right)\right) \]

      8. --lowering--.f64 34.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right) \]

   7. Simplified 34.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(1 + 3 \cdot \frac{1}{x}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(1 + 3 \cdot \frac{1}{x}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(-1 \cdot x\right) \cdot \left(3 \cdot \frac{1}{x} + \color{blue}{1}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) + \color{blue}{1 \cdot \left(-1 \cdot x\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) + -1 \cdot \color{blue}{x} \]

      5. mul-1-neg N/A

         \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) - \color{blue}{x} \]

      7. mul-1-neg N/A

         \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - x \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(3 \cdot \frac{1}{x}\right) \cdot x\right)\right) - x \]

      9. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(3 \cdot \left(\frac{1}{x} \cdot x\right)\right)\right) - x \]

      10. lft-mult-inverse N/A

          \[\leadsto \left(\mathsf{neg}\left(3 \cdot 1\right)\right) - x \]

      11. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(3\right)\right) - x \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(3\right)\right), \color{blue}{x}\right) \]

      13. metadata-eval 67.7%

          \[\leadsto \mathsf{\_.f64}\left(-3, x\right) \]

   10. Simplified 67.7%

       \[\leadsto \color{blue}{-3 - x} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{-3} \]
   12. Step-by-step derivation

   13. Simplified 4.7%

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

Alternative 10: 62.7% 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. Taylor expanded in y around 0

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

   1. --lowering--.f64 64.8%

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{x}\right) \]

5. Simplified 64.8%

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

Alternative 11: 3.2% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -3.0)

C

double code(double x, double y) {
	return -3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -3.0d0
end function

Java

public static double code(double x, double y) {
	return -3.0;
}

Python

def code(x, y):
	return -3.0

Julia

function code(x, y)
	return -3.0
end

MATLAB

function tmp = code(x, y)
	tmp = -3.0;
end

Wolfram

code[x_, y_] := -3.0

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 40.7%

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

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{2}}{1 - x}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(1 + x\right)}^{2}\right), \color{blue}{\left(1 - x\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right), \left(\color{blue}{1} - x\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + x\right), \left(1 + x\right)\right), \left(\color{blue}{1} - x\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x + 1\right), \left(1 + x\right)\right), \left(1 - x\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(1 + x\right)\right), \left(1 - x\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(x + 1\right)\right), \left(1 - x\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, 1\right)\right), \left(1 - x\right)\right) \]

   8. --lowering--.f64 45.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right) \]

7. Simplified 45.5%

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

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(1 + 3 \cdot \frac{1}{x}\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(1 + 3 \cdot \frac{1}{x}\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(-1 \cdot x\right) \cdot \left(3 \cdot \frac{1}{x} + \color{blue}{1}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) + \color{blue}{1 \cdot \left(-1 \cdot x\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) + -1 \cdot \color{blue}{x} \]

   5. mul-1-neg N/A

      \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(-1 \cdot x\right) - \color{blue}{x} \]

   7. mul-1-neg N/A

      \[\leadsto \left(3 \cdot \frac{1}{x}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - x \]

   8. distribute-rgt-neg-out N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(3 \cdot \frac{1}{x}\right) \cdot x\right)\right) - x \]

   9. associate-*l* N/A

      \[\leadsto \left(\mathsf{neg}\left(3 \cdot \left(\frac{1}{x} \cdot x\right)\right)\right) - x \]

   10. lft-mult-inverse N/A

       \[\leadsto \left(\mathsf{neg}\left(3 \cdot 1\right)\right) - x \]

   11. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left(3\right)\right) - x \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(3\right)\right), \color{blue}{x}\right) \]

   13. metadata-eval 36.7%

       \[\leadsto \mathsf{\_.f64}\left(-3, x\right) \]

10. Simplified 36.7%

    \[\leadsto \color{blue}{-3 - x} \]

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{-3} \]
12. Step-by-step derivation

13. Simplified 3.7%

    \[\leadsto \color{blue}{-3} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))