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

Percentage Accurate: 99.9% → 99.9%

Time: 10.2s

Alternatives: 15

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: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+32}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* y (sqrt x)))))
   (if (<= y -8.5e+28) t_0 (if (<= y 2.75e+32) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y * sqrt(x));
	double tmp;
	if (y <= -8.5e+28) {
		tmp = t_0;
	} else if (y <= 2.75e+32) {
		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 <= (-8.5d+28)) then
        tmp = t_0
    else if (y <= 2.75d+32) 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 <= -8.5e+28) {
		tmp = t_0;
	} else if (y <= 2.75e+32) {
		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 <= -8.5e+28:
		tmp = t_0
	elif y <= 2.75e+32:
		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 <= -8.5e+28)
		tmp = t_0;
	elseif (y <= 2.75e+32)
		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 <= -8.5e+28)
		tmp = t_0;
	elseif (y <= 2.75e+32)
		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, -8.5e+28], t$95$0, If[LessEqual[y, 2.75e+32], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.49999999999999954e28 or 2.74999999999999992e32 < y

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   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 94.9%

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

   5. Simplified 94.9%

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

   if -8.49999999999999954e28 < y < 2.74999999999999992e32

   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 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+28}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+32}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+97}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x))))
   (if (<= y -5.2e+73) t_0 (if (<= y 1.15e+97) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (y <= -5.2e+73) {
		tmp = t_0;
	} else if (y <= 1.15e+97) {
		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 <= (-5.2d+73)) then
        tmp = t_0
    else if (y <= 1.15d+97) 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 <= -5.2e+73) {
		tmp = t_0;
	} else if (y <= 1.15e+97) {
		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 <= -5.2e+73:
		tmp = t_0
	elif y <= 1.15e+97:
		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 <= -5.2e+73)
		tmp = t_0;
	elseif (y <= 1.15e+97)
		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 <= -5.2e+73)
		tmp = t_0;
	elseif (y <= 1.15e+97)
		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, -5.2e+73], t$95$0, If[LessEqual[y, 1.15e+97], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2000000000000001e73 or 1.15000000000000003e97 < y

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   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 91.9%

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

   5. Simplified 91.9%

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

   if -5.2000000000000001e73 < y < 1.15000000000000003e97

   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 96.8%

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

   5. Simplified 96.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+73}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+97}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(x)
    if (x <= 1.0d0) then
        tmp = 1.0d0 + t_0
    else
        tmp = t_0 - x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + t_0
	else:
		tmp = t_0 - x
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   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 97.8%

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

   5. Simplified 97.8%

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

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right)}\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot y\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

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

      15. rem-square-sqrt N/A

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \sqrt{x} \cdot y + \left(\mathsf{neg}\left(x\right)\right) \]

      3. sub-neg N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 98.8%

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

   8. Simplified 98.8%

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

4. Final simplification 98.4%

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

Alternative 5: 69.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{+152}:\\ \;\;\;\;1 - t\_0 \cdot \left(t\_0 \cdot t\_0\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot \left(x \cdot \left(1 - x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= y -6e+152)
     (- 1.0 (* t_0 (* t_0 t_0)))
     (if (<= y 4.8e+125)
       (- 1.0 x)
       (* (- 1.0 t_0) (+ 1.0 (* x (+ -1.0 (* x (* x (- 1.0 x)))))))))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if (y <= -6e+152) {
		tmp = 1.0 - (t_0 * (t_0 * t_0));
	} else if (y <= 4.8e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - t_0) * (1.0 + (x * (-1.0 + (x * (x * (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 = x * (x * x)
    if (y <= (-6d+152)) then
        tmp = 1.0d0 - (t_0 * (t_0 * t_0))
    else if (y <= 4.8d+125) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 - t_0) * (1.0d0 + (x * ((-1.0d0) + (x * (x * (1.0d0 - x))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if (y <= -6e+152) {
		tmp = 1.0 - (t_0 * (t_0 * t_0));
	} else if (y <= 4.8e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - t_0) * (1.0 + (x * (-1.0 + (x * (x * (1.0 - x))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * x)
	tmp = 0
	if y <= -6e+152:
		tmp = 1.0 - (t_0 * (t_0 * t_0))
	elif y <= 4.8e+125:
		tmp = 1.0 - x
	else:
		tmp = (1.0 - t_0) * (1.0 + (x * (-1.0 + (x * (x * (1.0 - x))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (y <= -6e+152)
		tmp = Float64(1.0 - Float64(t_0 * Float64(t_0 * t_0)));
	elseif (y <= 4.8e+125)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 - t_0) * Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(x * Float64(1.0 - x)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (y <= -6e+152)
		tmp = 1.0 - (t_0 * (t_0 * t_0));
	elseif (y <= 4.8e+125)
		tmp = 1.0 - x;
	else
		tmp = (1.0 - t_0) * (1.0 + (x * (-1.0 + (x * (x * (1.0 - x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e+152], N[(1.0 - N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+125], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.99999999999999981e152

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 3.9%

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

   5. Simplified 3.9%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. flip3-- N/A

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

      5. frac-times N/A

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

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

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

   7. Applied egg-rr 1.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 21.7%

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

   if -5.99999999999999981e152 < y < 4.7999999999999999e125

   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 86.2%

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

   5. Simplified 86.2%

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

   if 4.7999999999999999e125 < y

   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 2.0%

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

   5. Simplified 2.0%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 1.9%

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

   7. Applied egg-rr 1.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(1 + x \cdot \left({x}^{2} \cdot \left(1 + -1 \cdot x\right) - 1\right)\right)}\right) \]
   9. Step-by-step derivation

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + -1 \cdot x\right) + -1\right)\right)\right)\right) \]

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 28.3%

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

   10. Simplified 28.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+152}:\\ \;\;\;\;1 - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x \cdot \left(x \cdot x\right)\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot \left(x \cdot \left(1 - x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;y \leq -9 \cdot 10^{+152}:\\ \;\;\;\;t\_0 \cdot \left(1 + x \cdot \left(x \cdot x + -1\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + x \cdot \left(-1 + x \cdot \left(x \cdot \left(1 - x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x (* x x)))))
   (if (<= y -9e+152)
     (* t_0 (+ 1.0 (* x (+ (* x x) -1.0))))
     (if (<= y 2e+125)
       (- 1.0 x)
       (* t_0 (+ 1.0 (* x (+ -1.0 (* x (* x (- 1.0 x)))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x * (x * x));
	double tmp;
	if (y <= -9e+152) {
		tmp = t_0 * (1.0 + (x * ((x * x) + -1.0)));
	} else if (y <= 2e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0 * (1.0 + (x * (-1.0 + (x * (x * (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 * x))
    if (y <= (-9d+152)) then
        tmp = t_0 * (1.0d0 + (x * ((x * x) + (-1.0d0))))
    else if (y <= 2d+125) then
        tmp = 1.0d0 - x
    else
        tmp = t_0 * (1.0d0 + (x * ((-1.0d0) + (x * (x * (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 * x));
	double tmp;
	if (y <= -9e+152) {
		tmp = t_0 * (1.0 + (x * ((x * x) + -1.0)));
	} else if (y <= 2e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0 * (1.0 + (x * (-1.0 + (x * (x * (1.0 - x))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x * (x * x))
	tmp = 0
	if y <= -9e+152:
		tmp = t_0 * (1.0 + (x * ((x * x) + -1.0)))
	elif y <= 2e+125:
		tmp = 1.0 - x
	else:
		tmp = t_0 * (1.0 + (x * (-1.0 + (x * (x * (1.0 - x))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x * Float64(x * x)))
	tmp = 0.0
	if (y <= -9e+152)
		tmp = Float64(t_0 * Float64(1.0 + Float64(x * Float64(Float64(x * x) + -1.0))));
	elseif (y <= 2e+125)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(x * Float64(1.0 - x)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x * (x * x));
	tmp = 0.0;
	if (y <= -9e+152)
		tmp = t_0 * (1.0 + (x * ((x * x) + -1.0)));
	elseif (y <= 2e+125)
		tmp = 1.0 - x;
	else
		tmp = t_0 * (1.0 + (x * (-1.0 + (x * (x * (1.0 - x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e+152], N[(t$95$0 * N[(1.0 + N[(x * N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+125], N[(1.0 - x), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.0000000000000002e152

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 3.9%

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

   5. Simplified 3.9%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 13.1%

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

   7. Applied egg-rr 13.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(1 + x \cdot \left({x}^{2} - 1\right)\right)}\right) \]
   9. Step-by-step derivation

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 21.7%

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

   10. Simplified 21.7%

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

   if -9.0000000000000002e152 < y < 1.9999999999999998e125

   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 86.2%

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

   5. Simplified 86.2%

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

   if 1.9999999999999998e125 < y

   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 2.0%

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

   5. Simplified 2.0%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 1.9%

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

   7. Applied egg-rr 1.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(1 + x \cdot \left({x}^{2} \cdot \left(1 + -1 \cdot x\right) - 1\right)\right)}\right) \]
   9. Step-by-step derivation

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(1 + -1 \cdot x\right) + -1\right)\right)\right)\right) \]

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 28.3%

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

   10. Simplified 28.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+152}:\\ \;\;\;\;\left(1 - x \cdot \left(x \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot x + -1\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x \cdot \left(x \cdot x\right)\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot \left(x \cdot \left(1 - x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+151}:\\ \;\;\;\;\left(1 - x \cdot \left(x \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot x + -1\right)\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+124}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(-1 + x \cdot \left(1 - x\right)\right)\right) \cdot \left(1 - x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.26e+151)
   (* (- 1.0 (* x (* x x))) (+ 1.0 (* x (+ (* x x) -1.0))))
   (if (<= y 4.2e+124)
     (- 1.0 x)
     (* (+ 1.0 (* x (+ -1.0 (* x (- 1.0 x))))) (- 1.0 (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.26e+151) {
		tmp = (1.0 - (x * (x * x))) * (1.0 + (x * ((x * x) + -1.0)));
	} else if (y <= 4.2e+124) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 + (x * (-1.0 + (x * (1.0 - x))))) * (1.0 - (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.26d+151)) then
        tmp = (1.0d0 - (x * (x * x))) * (1.0d0 + (x * ((x * x) + (-1.0d0))))
    else if (y <= 4.2d+124) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 + (x * ((-1.0d0) + (x * (1.0d0 - x))))) * (1.0d0 - (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.26e+151) {
		tmp = (1.0 - (x * (x * x))) * (1.0 + (x * ((x * x) + -1.0)));
	} else if (y <= 4.2e+124) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 + (x * (-1.0 + (x * (1.0 - x))))) * (1.0 - (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.26e+151:
		tmp = (1.0 - (x * (x * x))) * (1.0 + (x * ((x * x) + -1.0)))
	elif y <= 4.2e+124:
		tmp = 1.0 - x
	else:
		tmp = (1.0 + (x * (-1.0 + (x * (1.0 - x))))) * (1.0 - (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.26e+151)
		tmp = Float64(Float64(1.0 - Float64(x * Float64(x * x))) * Float64(1.0 + Float64(x * Float64(Float64(x * x) + -1.0))));
	elseif (y <= 4.2e+124)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(1.0 - x))))) * Float64(1.0 - Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.26e+151)
		tmp = (1.0 - (x * (x * x))) * (1.0 + (x * ((x * x) + -1.0)));
	elseif (y <= 4.2e+124)
		tmp = 1.0 - x;
	else
		tmp = (1.0 + (x * (-1.0 + (x * (1.0 - x))))) * (1.0 - (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.26e+151], N[(N[(1.0 - N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+124], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.26000000000000006e151

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 3.9%

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

   5. Simplified 3.9%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 13.1%

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

   7. Applied egg-rr 13.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \color{blue}{\left(1 + x \cdot \left({x}^{2} - 1\right)\right)}\right) \]
   9. Step-by-step derivation

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 21.7%

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

   10. Simplified 21.7%

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

   if -1.26000000000000006e151 < y < 4.20000000000000023e124

   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 86.2%

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

   5. Simplified 86.2%

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

   if 4.20000000000000023e124 < y

   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 2.0%

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

   5. Simplified 2.0%

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

      1. flip-- N/A

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

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

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

      3. metadata-eval N/A

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

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

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

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

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

      6. +-lowering-+.f64 1.9%

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

   7. Applied egg-rr 1.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 1.9%

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

   9. Applied egg-rr 1.9%

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

   10. Taylor expanded in x around 0

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

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

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

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

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

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

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

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

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

       8. mul-1-neg N/A

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

       9. unsub-neg N/A

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

       10. --lowering--.f64 28.2%

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

   12. Simplified 28.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+151}:\\ \;\;\;\;\left(1 - x \cdot \left(x \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot x + -1\right)\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+124}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(-1 + x \cdot \left(1 - x\right)\right)\right) \cdot \left(1 - x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot x\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{+151}:\\ \;\;\;\;t\_0 \cdot \left(1 + x \cdot \left(x + -1\right)\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+124}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(-1 + x \cdot \left(1 - x\right)\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x x))))
   (if (<= y -2.7e+151)
     (* t_0 (+ 1.0 (* x (+ x -1.0))))
     (if (<= y 7.5e+124)
       (- 1.0 x)
       (* (+ 1.0 (* x (+ -1.0 (* x (- 1.0 x))))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= -2.7e+151) {
		tmp = t_0 * (1.0 + (x * (x + -1.0)));
	} else if (y <= 7.5e+124) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 + (x * (-1.0 + (x * (1.0 - x))))) * 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 - (x * x)
    if (y <= (-2.7d+151)) then
        tmp = t_0 * (1.0d0 + (x * (x + (-1.0d0))))
    else if (y <= 7.5d+124) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 + (x * ((-1.0d0) + (x * (1.0d0 - x))))) * t_0
    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 <= -2.7e+151) {
		tmp = t_0 * (1.0 + (x * (x + -1.0)));
	} else if (y <= 7.5e+124) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 + (x * (-1.0 + (x * (1.0 - x))))) * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x * x)
	tmp = 0
	if y <= -2.7e+151:
		tmp = t_0 * (1.0 + (x * (x + -1.0)))
	elif y <= 7.5e+124:
		tmp = 1.0 - x
	else:
		tmp = (1.0 + (x * (-1.0 + (x * (1.0 - x))))) * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x * x))
	tmp = 0.0
	if (y <= -2.7e+151)
		tmp = Float64(t_0 * Float64(1.0 + Float64(x * Float64(x + -1.0))));
	elseif (y <= 7.5e+124)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(1.0 - x))))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x * x);
	tmp = 0.0;
	if (y <= -2.7e+151)
		tmp = t_0 * (1.0 + (x * (x + -1.0)));
	elseif (y <= 7.5e+124)
		tmp = 1.0 - x;
	else
		tmp = (1.0 + (x * (-1.0 + (x * (1.0 - x))))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.7e+151], N[(t$95$0 * N[(1.0 + N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+124], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7000000000000001e151

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 3.9%

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

   5. Simplified 3.9%

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

      1. flip-- N/A

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

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

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

      3. metadata-eval N/A

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

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

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

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

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

      6. +-lowering-+.f64 10.4%

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

   7. Applied egg-rr 10.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 10.4%

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

   9. Applied egg-rr 10.4%

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

   10. Taylor expanded in x around 0

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

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

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

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

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

       6. +-lowering-+.f64 21.5%

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

   12. Simplified 21.5%

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

   if -2.7000000000000001e151 < y < 7.50000000000000038e124

   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 86.2%

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

   5. Simplified 86.2%

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

   if 7.50000000000000038e124 < y

   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 2.0%

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

   5. Simplified 2.0%

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

      1. flip-- N/A

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

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

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

      3. metadata-eval N/A

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

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

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

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

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

      6. +-lowering-+.f64 1.9%

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

   7. Applied egg-rr 1.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 1.9%

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

   9. Applied egg-rr 1.9%

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

   10. Taylor expanded in x around 0

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

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

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

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

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

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

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

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

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

       8. mul-1-neg N/A

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

       9. unsub-neg N/A

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

       10. --lowering--.f64 28.2%

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

   12. Simplified 28.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+151}:\\ \;\;\;\;\left(1 - x \cdot x\right) \cdot \left(1 + x \cdot \left(x + -1\right)\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+124}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(-1 + x \cdot \left(1 - x\right)\right)\right) \cdot \left(1 - x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+152}:\\ \;\;\;\;\left(1 - x \cdot x\right) \cdot \left(1 + x \cdot \left(x + -1\right)\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e+152)
   (* (- 1.0 (* x x)) (+ 1.0 (* x (+ x -1.0))))
   (if (<= y 4.8e+125) (- 1.0 x) (* (- 1.0 x) (- 1.0 (* x (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+152) {
		tmp = (1.0 - (x * x)) * (1.0 + (x * (x + -1.0)));
	} else if (y <= 4.8e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * (1.0 - (x * (x * 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 <= (-2.6d+152)) then
        tmp = (1.0d0 - (x * x)) * (1.0d0 + (x * (x + (-1.0d0))))
    else if (y <= 4.8d+125) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 - x) * (1.0d0 - (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+152) {
		tmp = (1.0 - (x * x)) * (1.0 + (x * (x + -1.0)));
	} else if (y <= 4.8e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * (1.0 - (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.6e+152:
		tmp = (1.0 - (x * x)) * (1.0 + (x * (x + -1.0)))
	elif y <= 4.8e+125:
		tmp = 1.0 - x
	else:
		tmp = (1.0 - x) * (1.0 - (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e+152)
		tmp = Float64(Float64(1.0 - Float64(x * x)) * Float64(1.0 + Float64(x * Float64(x + -1.0))));
	elseif (y <= 4.8e+125)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 - x) * Float64(1.0 - Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.6e+152)
		tmp = (1.0 - (x * x)) * (1.0 + (x * (x + -1.0)));
	elseif (y <= 4.8e+125)
		tmp = 1.0 - x;
	else
		tmp = (1.0 - x) * (1.0 - (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e+152], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+125], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6000000000000001e152

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 3.9%

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

   5. Simplified 3.9%

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

      1. flip-- N/A

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

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

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

      3. metadata-eval N/A

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

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

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

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

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

      6. +-lowering-+.f64 10.4%

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

   7. Applied egg-rr 10.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 10.4%

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

   9. Applied egg-rr 10.4%

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

   10. Taylor expanded in x around 0

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

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

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

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

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

       6. +-lowering-+.f64 21.5%

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

   12. Simplified 21.5%

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

   if -2.6000000000000001e152 < y < 4.7999999999999999e125

   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 86.2%

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

   5. Simplified 86.2%

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

   if 4.7999999999999999e125 < y

   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 2.0%

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

   5. Simplified 2.0%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 1.9%

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

   7. Applied egg-rr 1.9%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 26.3%

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

   10. Simplified 26.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+152}:\\ \;\;\;\;\left(1 - x \cdot x\right) \cdot \left(1 + x \cdot \left(x + -1\right)\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x (* x x)))))
   (if (<= y -2.5e+151) t_0 (if (<= y 3.2e+125) (- 1.0 x) (* (- 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x * (x * x));
	double tmp;
	if (y <= -2.5e+151) {
		tmp = t_0;
	} else if (y <= 3.2e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * 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 - (x * (x * x))
    if (y <= (-2.5d+151)) then
        tmp = t_0
    else if (y <= 3.2d+125) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 - x) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x * (x * x));
	double tmp;
	if (y <= -2.5e+151) {
		tmp = t_0;
	} else if (y <= 3.2e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x * (x * x))
	tmp = 0
	if y <= -2.5e+151:
		tmp = t_0
	elif y <= 3.2e+125:
		tmp = 1.0 - x
	else:
		tmp = (1.0 - x) * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x * Float64(x * x)))
	tmp = 0.0
	if (y <= -2.5e+151)
		tmp = t_0;
	elseif (y <= 3.2e+125)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 - x) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x * (x * x));
	tmp = 0.0;
	if (y <= -2.5e+151)
		tmp = t_0;
	elseif (y <= 3.2e+125)
		tmp = 1.0 - x;
	else
		tmp = (1.0 - x) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e+151], t$95$0, If[LessEqual[y, 3.2e+125], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5000000000000001e151

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 3.9%

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

   5. Simplified 3.9%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 13.1%

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

   7. Applied egg-rr 13.1%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 21.2%

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

   if -2.5000000000000001e151 < y < 3.19999999999999983e125

   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 86.2%

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

   5. Simplified 86.2%

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

   if 3.19999999999999983e125 < y

   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 2.0%

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

   5. Simplified 2.0%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 1.9%

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

   7. Applied egg-rr 1.9%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 26.3%

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

   10. Simplified 26.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+151}:\\ \;\;\;\;1 - x \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+151}:\\ \;\;\;\;1 - x \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8e+151)
   (- 1.0 (* x (* x x)))
   (if (<= y 4.8e+125) (- 1.0 x) (* (- 1.0 x) (- 1.0 (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8e+151) {
		tmp = 1.0 - (x * (x * x));
	} else if (y <= 4.8e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * (1.0 - (x * 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 <= (-8d+151)) then
        tmp = 1.0d0 - (x * (x * x))
    else if (y <= 4.8d+125) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 - x) * (1.0d0 - (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8e+151) {
		tmp = 1.0 - (x * (x * x));
	} else if (y <= 4.8e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * (1.0 - (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8e+151:
		tmp = 1.0 - (x * (x * x))
	elif y <= 4.8e+125:
		tmp = 1.0 - x
	else:
		tmp = (1.0 - x) * (1.0 - (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8e+151)
		tmp = Float64(1.0 - Float64(x * Float64(x * x)));
	elseif (y <= 4.8e+125)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 - x) * Float64(1.0 - Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8e+151)
		tmp = 1.0 - (x * (x * x));
	elseif (y <= 4.8e+125)
		tmp = 1.0 - x;
	else
		tmp = (1.0 - x) * (1.0 - (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8e+151], N[(1.0 - N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+125], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000014e151

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 3.9%

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

   5. Simplified 3.9%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 13.1%

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

   7. Applied egg-rr 13.1%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 21.2%

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

   if -8.00000000000000014e151 < y < 4.7999999999999999e125

   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 86.2%

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

   5. Simplified 86.2%

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

   if 4.7999999999999999e125 < y

   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 2.0%

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

   5. Simplified 2.0%

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

      1. flip-- N/A

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

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

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

      3. metadata-eval N/A

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

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

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

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

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

      6. +-lowering-+.f64 1.9%

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

   7. Applied egg-rr 1.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 1.9%

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

   9. Applied egg-rr 1.9%

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

   10. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. --lowering--.f64 25.9%

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

   12. Simplified 25.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+151}:\\ \;\;\;\;1 - x \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+125}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e+125) (- 1.0 x) (* (- 1.0 x) (- 1.0 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * (1.0 - (x * 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 <= 4.4d+125) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 - x) * (1.0d0 - (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.4e+125) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * (1.0 - (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.4e+125:
		tmp = 1.0 - x
	else:
		tmp = (1.0 - x) * (1.0 - (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e+125)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 - x) * Float64(1.0 - Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.4e+125)
		tmp = 1.0 - x;
	else
		tmp = (1.0 - x) * (1.0 - (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e+125], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.39999999999999982e125

   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 75.0%

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

   5. Simplified 75.0%

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

   if 4.39999999999999982e125 < y

   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 2.0%

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

   5. Simplified 2.0%

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

      1. flip-- N/A

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

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

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

      3. metadata-eval N/A

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

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

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

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

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

      6. +-lowering-+.f64 1.9%

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

   7. Applied egg-rr 1.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 1.9%

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

   9. Applied egg-rr 1.9%

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

   10. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. --lowering--.f64 25.9%

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

   12. Simplified 25.9%

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

Alternative 13: 62.4% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2600.0) 1.0 (- 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2600.0) {
		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 <= 2600.0d0) 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 <= 2600.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2600.0:
		tmp = 1.0
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2600.0)
		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 <= 2600.0)
		tmp = 1.0;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2600.0], 1.0, N[(0.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2600

   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 66.4%

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

   5. Simplified 66.4%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 64.3%

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

   if 2600 < 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 59.7%

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

   5. Simplified 59.7%

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

   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 58.5%

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

   8. Simplified 58.5%

      \[\leadsto \color{blue}{0 - x} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 58.5%

         \[\leadsto \mathsf{neg.f64}\left(x\right) \]

   10. Applied egg-rr 58.5%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.7% 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 62.8%

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

5. Simplified 62.8%

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

Alternative 15: 31.8% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. --lowering--.f64 62.8%

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

5. Simplified 62.8%

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

6. Taylor expanded in x around 0

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

8. Simplified 30.5%

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))