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

Percentage Accurate: 99.9% → 99.9%

Time: 9.4s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. fma-define N/A

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

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

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

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

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

   6. --lowering--.f64 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 95.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \sqrt{x} \cdot y\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{+49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+60}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (sqrt x) y))))
   (if (<= y -7.2e+49) t_0 (if (<= y 5e+60) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (sqrt(x) * y);
	double tmp;
	if (y <= -7.2e+49) {
		tmp = t_0;
	} else if (y <= 5e+60) {
		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 + (sqrt(x) * y)
    if (y <= (-7.2d+49)) then
        tmp = t_0
    else if (y <= 5d+60) 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 + (Math.sqrt(x) * y);
	double tmp;
	if (y <= -7.2e+49) {
		tmp = t_0;
	} else if (y <= 5e+60) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (math.sqrt(x) * y)
	tmp = 0
	if y <= -7.2e+49:
		tmp = t_0
	elif y <= 5e+60:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(sqrt(x) * y))
	tmp = 0.0
	if (y <= -7.2e+49)
		tmp = t_0;
	elseif (y <= 5e+60)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (sqrt(x) * y);
	tmp = 0.0;
	if (y <= -7.2e+49)
		tmp = t_0;
	elseif (y <= 5e+60)
		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[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e+49], t$95$0, If[LessEqual[y, 5e+60], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.19999999999999993e49 or 4.99999999999999975e60 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   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 91.5%

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

   5. Simplified 91.5%

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

   if -7.19999999999999993e49 < y < 4.99999999999999975e60

   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 95.9%

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

   5. Simplified 95.9%

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

Alternative 3: 92.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+64}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) y)))
   (if (<= y -7.8e+49) t_0 (if (<= y 2.8e+64) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (y <= -7.8e+49) {
		tmp = t_0;
	} else if (y <= 2.8e+64) {
		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 = sqrt(x) * y
    if (y <= (-7.8d+49)) then
        tmp = t_0
    else if (y <= 2.8d+64) 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 = Math.sqrt(x) * y;
	double tmp;
	if (y <= -7.8e+49) {
		tmp = t_0;
	} else if (y <= 2.8e+64) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * y
	tmp = 0
	if y <= -7.8e+49:
		tmp = t_0
	elif y <= 2.8e+64:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * y)
	tmp = 0.0
	if (y <= -7.8e+49)
		tmp = t_0;
	elseif (y <= 2.8e+64)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * y;
	tmp = 0.0;
	if (y <= -7.8e+49)
		tmp = t_0;
	elseif (y <= 2.8e+64)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -7.8e+49], t$95$0, If[LessEqual[y, 2.8e+64], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8000000000000002e49 or 2.80000000000000024e64 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   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 87.7%

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

   5. Simplified 87.7%

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

   if -7.8000000000000002e49 < y < 2.80000000000000024e64

   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 95.9%

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

   5. Simplified 95.9%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 68.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;\left(1 - x \cdot x\right) \cdot \left(1 + x \cdot \left(x + -1\right)\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+136}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+133)
   (* (- 1.0 (* x x)) (+ 1.0 (* x (+ x -1.0))))
   (if (<= y 4.1e+136) (- 1.0 x) (* (- 1.0 x) (- 1.0 (* x (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+133) {
		tmp = (1.0 - (x * x)) * (1.0 + (x * (x + -1.0)));
	} else if (y <= 4.1e+136) {
		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 <= (-3.8d+133)) then
        tmp = (1.0d0 - (x * x)) * (1.0d0 + (x * (x + (-1.0d0))))
    else if (y <= 4.1d+136) 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 <= -3.8e+133) {
		tmp = (1.0 - (x * x)) * (1.0 + (x * (x + -1.0)));
	} else if (y <= 4.1e+136) {
		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 <= -3.8e+133:
		tmp = (1.0 - (x * x)) * (1.0 + (x * (x + -1.0)))
	elif y <= 4.1e+136:
		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 <= -3.8e+133)
		tmp = Float64(Float64(1.0 - Float64(x * x)) * Float64(1.0 + Float64(x * Float64(x + -1.0))));
	elseif (y <= 4.1e+136)
		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 <= -3.8e+133)
		tmp = (1.0 - (x * x)) * (1.0 + (x * (x + -1.0)));
	elseif (y <= 4.1e+136)
		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, -3.8e+133], 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, 4.1e+136], 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 < -3.8000000000000002e133

   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 3.1%

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

   5. Simplified 3.1%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 23.1%

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

   7. Applied egg-rr 23.1%

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

   8. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. +-lowering-+.f64 29.2%

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

   10. Simplified 29.2%

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

   if -3.8000000000000002e133 < y < 4.0999999999999998e136

   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 86.5%

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

   5. Simplified 86.5%

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

   if 4.0999999999999998e136 < y

   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 4.4%

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

   5. Simplified 4.4%

      \[\leadsto \color{blue}{1 - x} \]
   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. +-commutative N/A

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

      16. +-lowering-+.f64 4.3%

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

   7. Applied egg-rr 4.3%

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

   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 22.9%

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

   10. Simplified 22.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;\left(1 - x \cdot x\right) \cdot \left(1 + x \cdot \left(x + -1\right)\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+136}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+137}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x (* x x)))))
   (if (<= y -3.8e+133) t_0 (if (<= y 3e+137) (- 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 <= -3.8e+133) {
		tmp = t_0;
	} else if (y <= 3e+137) {
		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 <= (-3.8d+133)) then
        tmp = t_0
    else if (y <= 3d+137) 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 <= -3.8e+133) {
		tmp = t_0;
	} else if (y <= 3e+137) {
		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 <= -3.8e+133:
		tmp = t_0
	elif y <= 3e+137:
		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 <= -3.8e+133)
		tmp = t_0;
	elseif (y <= 3e+137)
		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 <= -3.8e+133)
		tmp = t_0;
	elseif (y <= 3e+137)
		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, -3.8e+133], t$95$0, If[LessEqual[y, 3e+137], 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 < -3.8000000000000002e133

   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 3.1%

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

   5. Simplified 3.1%

      \[\leadsto \color{blue}{1 - x} \]
   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. +-commutative N/A

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

      16. +-lowering-+.f64 4.4%

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

   7. Applied egg-rr 4.4%

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

   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 26.4%

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

   if -3.8000000000000002e133 < y < 3.0000000000000001e137

   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 86.5%

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

   5. Simplified 86.5%

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

   if 3.0000000000000001e137 < y

   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 4.4%

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

   5. Simplified 4.4%

      \[\leadsto \color{blue}{1 - x} \]
   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. +-commutative N/A

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

      16. +-lowering-+.f64 4.3%

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

   7. Applied egg-rr 4.3%

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

   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 22.9%

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

   10. Simplified 22.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;1 - x \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+137}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;1 - x \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+138}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+133)
   (- 1.0 (* x (* x x)))
   (if (<= y 1.3e+138) (- 1.0 x) (* (- 1.0 x) (- 1.0 (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+133) {
		tmp = 1.0 - (x * (x * x));
	} else if (y <= 1.3e+138) {
		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 <= (-3.8d+133)) then
        tmp = 1.0d0 - (x * (x * x))
    else if (y <= 1.3d+138) 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 <= -3.8e+133) {
		tmp = 1.0 - (x * (x * x));
	} else if (y <= 1.3e+138) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * (1.0 - (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+133:
		tmp = 1.0 - (x * (x * x))
	elif y <= 1.3e+138:
		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 <= -3.8e+133)
		tmp = Float64(1.0 - Float64(x * Float64(x * x)));
	elseif (y <= 1.3e+138)
		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 <= -3.8e+133)
		tmp = 1.0 - (x * (x * x));
	elseif (y <= 1.3e+138)
		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, -3.8e+133], N[(1.0 - N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+138], 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 < -3.8000000000000002e133

   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 3.1%

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

   5. Simplified 3.1%

      \[\leadsto \color{blue}{1 - x} \]
   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. +-commutative N/A

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

      16. +-lowering-+.f64 4.4%

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

   7. Applied egg-rr 4.4%

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

   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 26.4%

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

   if -3.8000000000000002e133 < y < 1.3e138

   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 86.4%

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

   5. Simplified 86.4%

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

   if 1.3e138 < y

   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 2.4%

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

   5. Simplified 2.4%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 2.4%

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

   7. Applied egg-rr 2.4%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 21.1%

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

   10. Simplified 21.1%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;1 - x \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+138}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{x \cdot x}{0 - x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+138}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.5e+124)
   (/ (* x x) (- 0.0 x))
   (if (<= y 3.2e+138) (- 1.0 x) (* (- 1.0 x) (- 1.0 (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.5e+124) {
		tmp = (x * x) / (0.0 - x);
	} else if (y <= 3.2e+138) {
		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 <= (-5.5d+124)) then
        tmp = (x * x) / (0.0d0 - x)
    else if (y <= 3.2d+138) 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 <= -5.5e+124) {
		tmp = (x * x) / (0.0 - x);
	} else if (y <= 3.2e+138) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * (1.0 - (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.5e+124:
		tmp = (x * x) / (0.0 - x)
	elif y <= 3.2e+138:
		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 <= -5.5e+124)
		tmp = Float64(Float64(x * x) / Float64(0.0 - x));
	elseif (y <= 3.2e+138)
		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 <= -5.5e+124)
		tmp = (x * x) / (0.0 - x);
	elseif (y <= 3.2e+138)
		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, -5.5e+124], N[(N[(x * x), $MachinePrecision] / N[(0.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+138], 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 < -5.49999999999999977e124

   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 3.1%

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

   5. Simplified 3.1%

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

   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 4.4%

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

   8. Simplified 4.4%

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

      1. flip-- N/A

         \[\leadsto \frac{0 \cdot 0 - x \cdot x}{\color{blue}{0 + x}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{0 - x \cdot x}{0 + x} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(x \cdot x\right)}{\color{blue}{0} + x} \]

      4. +-lft-identity N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x \cdot x\right), x\right) \]

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

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

      8. *-lowering-*.f64 23.2%

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

   10. Applied egg-rr 23.2%

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

   if -5.49999999999999977e124 < y < 3.2000000000000001e138

   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 87.3%

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

   5. Simplified 87.3%

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

   if 3.2000000000000001e138 < y

   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 2.4%

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

   5. Simplified 2.4%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 2.4%

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

   7. Applied egg-rr 2.4%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 21.1%

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

   10. Simplified 21.1%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{x \cdot x}{0 - x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+138}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{x \cdot x}{0 - x}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+138}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.5e+124)
   (/ (* x x) (- 0.0 x))
   (if (<= y 4.6e+138) (- 1.0 x) (* (* x x) (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.5e+124) {
		tmp = (x * x) / (0.0 - x);
	} else if (y <= 4.6e+138) {
		tmp = 1.0 - x;
	} else {
		tmp = (x * 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 <= (-5.5d+124)) then
        tmp = (x * x) / (0.0d0 - x)
    else if (y <= 4.6d+138) then
        tmp = 1.0d0 - x
    else
        tmp = (x * x) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.5e+124) {
		tmp = (x * x) / (0.0 - x);
	} else if (y <= 4.6e+138) {
		tmp = 1.0 - x;
	} else {
		tmp = (x * x) * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.5e+124:
		tmp = (x * x) / (0.0 - x)
	elif y <= 4.6e+138:
		tmp = 1.0 - x
	else:
		tmp = (x * x) * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.5e+124)
		tmp = Float64(Float64(x * x) / Float64(0.0 - x));
	elseif (y <= 4.6e+138)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(x * x) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.5e+124)
		tmp = (x * x) / (0.0 - x);
	elseif (y <= 4.6e+138)
		tmp = 1.0 - x;
	else
		tmp = (x * x) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.5e+124], N[(N[(x * x), $MachinePrecision] / N[(0.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+138], N[(1.0 - x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.49999999999999977e124

   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 3.1%

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

   5. Simplified 3.1%

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

   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 4.4%

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

   8. Simplified 4.4%

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

      1. flip-- N/A

         \[\leadsto \frac{0 \cdot 0 - x \cdot x}{\color{blue}{0 + x}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{0 - x \cdot x}{0 + x} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(x \cdot x\right)}{\color{blue}{0} + x} \]

      4. +-lft-identity N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x \cdot x\right), x\right) \]

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

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

      8. *-lowering-*.f64 23.2%

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

   10. Applied egg-rr 23.2%

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

   if -5.49999999999999977e124 < y < 4.60000000000000015e138

   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 87.3%

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

   5. Simplified 87.3%

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

   if 4.60000000000000015e138 < y

   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 2.4%

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

   5. Simplified 2.4%

      \[\leadsto \color{blue}{1 - x} \]
   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. +-commutative N/A

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

      16. +-lowering-+.f64 2.3%

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

   7. Applied egg-rr 2.3%

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

   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 21.5%

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

   10. Simplified 21.5%

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

   11. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

          \[\leadsto {x}^{3} \cdot \color{blue}{x} \]

       3. unpow3 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot x \]

       4. unpow2 N/A

          \[\leadsto \left({x}^{2} \cdot x\right) \cdot x \]

       5. associate-*l* N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

       6. unpow2 N/A

          \[\leadsto {x}^{2} \cdot {x}^{\color{blue}{2}} \]

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

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 20.4%

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

   13. Simplified 20.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{x \cdot x}{0 - x}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+138}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+126}:\\ \;\;\;\;1 - x \cdot x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+138}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+126)
   (- 1.0 (* x x))
   (if (<= y 1.2e+138) (- 1.0 x) (* (* x x) (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+126) {
		tmp = 1.0 - (x * x);
	} else if (y <= 1.2e+138) {
		tmp = 1.0 - x;
	} else {
		tmp = (x * 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 <= (-1.45d+126)) then
        tmp = 1.0d0 - (x * x)
    else if (y <= 1.2d+138) then
        tmp = 1.0d0 - x
    else
        tmp = (x * x) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+126) {
		tmp = 1.0 - (x * x);
	} else if (y <= 1.2e+138) {
		tmp = 1.0 - x;
	} else {
		tmp = (x * x) * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+126:
		tmp = 1.0 - (x * x)
	elif y <= 1.2e+138:
		tmp = 1.0 - x
	else:
		tmp = (x * x) * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+126)
		tmp = Float64(1.0 - Float64(x * x));
	elseif (y <= 1.2e+138)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(x * x) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+126)
		tmp = 1.0 - (x * x);
	elseif (y <= 1.2e+138)
		tmp = 1.0 - x;
	else
		tmp = (x * x) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+126], N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+138], N[(1.0 - x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.44999999999999993e126

   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 3.2%

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

   5. Simplified 3.2%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 22.6%

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

   7. Applied egg-rr 22.6%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 23.6%

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

   if -1.44999999999999993e126 < y < 1.2e138

   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 86.8%

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

   5. Simplified 86.8%

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

   if 1.2e138 < y

   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 2.4%

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

   5. Simplified 2.4%

      \[\leadsto \color{blue}{1 - x} \]
   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. +-commutative N/A

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

      16. +-lowering-+.f64 2.3%

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

   7. Applied egg-rr 2.3%

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

   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 21.5%

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

   10. Simplified 21.5%

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

   11. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

          \[\leadsto {x}^{3} \cdot \color{blue}{x} \]

       3. unpow3 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot x \]

       4. unpow2 N/A

          \[\leadsto \left({x}^{2} \cdot x\right) \cdot x \]

       5. associate-*l* N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

       6. unpow2 N/A

          \[\leadsto {x}^{2} \cdot {x}^{\color{blue}{2}} \]

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

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 20.4%

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

   13. Simplified 20.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+126}:\\ \;\;\;\;1 - x \cdot x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+138}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+138}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.2e+138) (- 1.0 x) (* (* x x) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.2e+138) {
		tmp = 1.0 - x;
	} else {
		tmp = (x * 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 <= 4.2d+138) then
        tmp = 1.0d0 - x
    else
        tmp = (x * x) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.2e+138) {
		tmp = 1.0 - x;
	} else {
		tmp = (x * x) * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.2e+138:
		tmp = 1.0 - x
	else:
		tmp = (x * x) * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.2e+138)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(x * x) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.2e+138)
		tmp = 1.0 - x;
	else
		tmp = (x * x) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.2e+138], N[(1.0 - x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.20000000000000014e138

   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 73.9%

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

   5. Simplified 73.9%

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

   if 4.20000000000000014e138 < y

   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 2.4%

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

   5. Simplified 2.4%

      \[\leadsto \color{blue}{1 - x} \]
   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. +-commutative N/A

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

      16. +-lowering-+.f64 2.3%

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

   7. Applied egg-rr 2.3%

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

   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 21.5%

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

   10. Simplified 21.5%

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

   11. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

          \[\leadsto {x}^{3} \cdot \color{blue}{x} \]

       3. unpow3 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot x \]

       4. unpow2 N/A

          \[\leadsto \left({x}^{2} \cdot x\right) \cdot x \]

       5. associate-*l* N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

       6. unpow2 N/A

          \[\leadsto {x}^{2} \cdot {x}^{\color{blue}{2}} \]

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

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 20.4%

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

   13. Simplified 20.4%

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

Alternative 12: 61.1% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00185:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 0.00185) 1.0 (- 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00185) {
		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 <= 0.00185d0) 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 <= 0.00185) {
		tmp = 1.0;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00185:
		tmp = 1.0
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00185)
		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 <= 0.00185)
		tmp = 1.0;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00185], 1.0, N[(0.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0018500000000000001

   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 61.0%

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

   5. Simplified 61.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 59.9%

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

   if 0.0018500000000000001 < 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 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 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 66.3%

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

   8. Simplified 66.3%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 66.3%

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

   10. Applied egg-rr 66.3%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00185:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.1% 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 63.9%

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

5. Simplified 63.9%

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

Alternative 14: 32.1% 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 63.9%

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

5. Simplified 63.9%

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

6. Taylor expanded in x around 0

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

8. Simplified 29.3%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))