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

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+47}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* y (sqrt x)))))
   (if (<= y -4.3e+24) t_0 (if (<= y 1.25e+47) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y * sqrt(x));
	double tmp;
	if (y <= -4.3e+24) {
		tmp = t_0;
	} else if (y <= 1.25e+47) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y * sqrt(x))
    if (y <= (-4.3d+24)) then
        tmp = t_0
    else if (y <= 1.25d+47) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y * Math.sqrt(x));
	double tmp;
	if (y <= -4.3e+24) {
		tmp = t_0;
	} else if (y <= 1.25e+47) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y * math.sqrt(x))
	tmp = 0
	if y <= -4.3e+24:
		tmp = t_0
	elif y <= 1.25e+47:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y * sqrt(x)))
	tmp = 0.0
	if (y <= -4.3e+24)
		tmp = t_0;
	elseif (y <= 1.25e+47)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y * sqrt(x));
	tmp = 0.0;
	if (y <= -4.3e+24)
		tmp = t_0;
	elseif (y <= 1.25e+47)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.3e+24], t$95$0, If[LessEqual[y, 1.25e+47], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.29999999999999987e24 or 1.25000000000000005e47 < 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 92.2%

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

   5. Simplified 92.2%

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

   if -4.29999999999999987e24 < y < 1.25000000000000005e47

   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 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+24}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+47}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+78}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x))))
   (if (<= y -5.2e+87) t_0 (if (<= y 5.4e+78) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (y <= -5.2e+87) {
		tmp = t_0;
	} else if (y <= 5.4e+78) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(x)
    if (y <= (-5.2d+87)) then
        tmp = t_0
    else if (y <= 5.4d+78) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * Math.sqrt(x);
	double tmp;
	if (y <= -5.2e+87) {
		tmp = t_0;
	} else if (y <= 5.4e+78) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if y <= -5.2e+87:
		tmp = t_0
	elif y <= 5.4e+78:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(x))
	tmp = 0.0
	if (y <= -5.2e+87)
		tmp = t_0;
	elseif (y <= 5.4e+78)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * sqrt(x);
	tmp = 0.0;
	if (y <= -5.2e+87)
		tmp = t_0;
	elseif (y <= 5.4e+78)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.2e+87], t$95$0, If[LessEqual[y, 5.4e+78], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.19999999999999997e87 or 5.40000000000000009e78 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 91.3%

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

   5. Simplified 91.3%

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

   if -5.19999999999999997e87 < y < 5.40000000000000009e78

   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 93.0%

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

   5. Simplified 93.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+87}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+78}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+153}:\\ \;\;\;\;t\_0 \cdot \left(1 + x \cdot \left(x \cdot x + -1\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x (* x x)))))
   (if (<= y -1.2e+153)
     (* t_0 (+ 1.0 (* x (+ (* x x) -1.0))))
     (if (<= y 6.2e+130) (- 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 <= -1.2e+153) {
		tmp = t_0 * (1.0 + (x * ((x * x) + -1.0)));
	} else if (y <= 6.2e+130) {
		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 <= (-1.2d+153)) then
        tmp = t_0 * (1.0d0 + (x * ((x * x) + (-1.0d0))))
    else if (y <= 6.2d+130) 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 <= -1.2e+153) {
		tmp = t_0 * (1.0 + (x * ((x * x) + -1.0)));
	} else if (y <= 6.2e+130) {
		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 <= -1.2e+153:
		tmp = t_0 * (1.0 + (x * ((x * x) + -1.0)))
	elif y <= 6.2e+130:
		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 <= -1.2e+153)
		tmp = Float64(t_0 * Float64(1.0 + Float64(x * Float64(Float64(x * x) + -1.0))));
	elseif (y <= 6.2e+130)
		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 <= -1.2e+153)
		tmp = t_0 * (1.0 + (x * ((x * x) + -1.0)));
	elseif (y <= 6.2e+130)
		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, -1.2e+153], N[(t$95$0 * N[(1.0 + N[(x * N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+130], 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 < -1.19999999999999996e153

   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 3.5%

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

   5. Simplified 3.5%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 5.8%

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

   7. Applied egg-rr 5.8%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 22.3%

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

   10. Simplified 22.3%

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

   if -1.19999999999999996e153 < y < 6.1999999999999999e130

   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 81.2%

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

   5. Simplified 81.2%

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

   if 6.1999999999999999e130 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 3.7%

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

   5. Simplified 3.7%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 3.6%

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

   7. Applied egg-rr 3.6%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 29.3%

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

   10. Simplified 29.3%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+153}:\\ \;\;\;\;\left(1 - x \cdot \left(x \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot x + -1\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := 1 - t\_0\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+152}:\\ \;\;\;\;t\_1 \cdot \left(1 + t\_0\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (- 1.0 t_0)))
   (if (<= y -1.6e+152)
     (* t_1 (+ 1.0 t_0))
     (if (<= y 4.6e+128) (- 1.0 x) (* (- 1.0 x) t_1)))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double t_1 = 1.0 - t_0;
	double tmp;
	if (y <= -1.6e+152) {
		tmp = t_1 * (1.0 + t_0);
	} else if (y <= 4.6e+128) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = 1.0d0 - t_0
    if (y <= (-1.6d+152)) then
        tmp = t_1 * (1.0d0 + t_0)
    else if (y <= 4.6d+128) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 - x) * t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * x);
	double t_1 = 1.0 - t_0;
	double tmp;
	if (y <= -1.6e+152) {
		tmp = t_1 * (1.0 + t_0);
	} else if (y <= 4.6e+128) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 - x) * t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * x)
	t_1 = 1.0 - t_0
	tmp = 0
	if y <= -1.6e+152:
		tmp = t_1 * (1.0 + t_0)
	elif y <= 4.6e+128:
		tmp = 1.0 - x
	else:
		tmp = (1.0 - x) * t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if (y <= -1.6e+152)
		tmp = Float64(t_1 * Float64(1.0 + t_0));
	elseif (y <= 4.6e+128)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 - x) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * x);
	t_1 = 1.0 - t_0;
	tmp = 0.0;
	if (y <= -1.6e+152)
		tmp = t_1 * (1.0 + t_0);
	elseif (y <= 4.6e+128)
		tmp = 1.0 - x;
	else
		tmp = (1.0 - x) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[y, -1.6e+152], N[(t$95$1 * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+128], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.60000000000000003e152

   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 3.5%

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

   5. Simplified 3.5%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 5.8%

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

   7. Applied egg-rr 5.8%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 22.3%

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

   10. Simplified 22.3%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{3}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 22.3%

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

   13. Simplified 22.3%

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

   if -1.60000000000000003e152 < y < 4.59999999999999996e128

   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 81.2%

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

   5. Simplified 81.2%

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

   if 4.59999999999999996e128 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 3.7%

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

   5. Simplified 3.7%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 3.6%

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

   7. Applied egg-rr 3.6%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 29.3%

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

   10. Simplified 29.3%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+152}:\\ \;\;\;\;\left(1 - x \cdot \left(x \cdot x\right)\right) \cdot \left(1 + x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+129}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e+129) (- 1.0 x) (* (- 1.0 x) (- 1.0 (* x (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.2e+129) {
		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 <= 9.2d+129) 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 <= 9.2e+129) {
		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 <= 9.2e+129:
		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 <= 9.2e+129)
		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 <= 9.2e+129)
		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, 9.2e+129], 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 2 regimes

2. if y < 9.19999999999999961e129

   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 71.9%

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

   5. Simplified 71.9%

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

   if 9.19999999999999961e129 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 3.7%

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

   5. Simplified 3.7%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 3.6%

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

   7. Applied egg-rr 3.6%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 29.3%

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

   10. Simplified 29.3%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+129}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(1 - x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+134}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.5e+134) (- 1.0 x) (* x (* x (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.5e+134) {
		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 <= 9.5d+134) 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 <= 9.5e+134) {
		tmp = 1.0 - x;
	} else {
		tmp = x * (x * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.5e+134:
		tmp = 1.0 - x
	else:
		tmp = x * (x * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.5e+134)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(x * Float64(x * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.5e+134)
		tmp = 1.0 - x;
	else
		tmp = x * (x * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.5e+134], N[(1.0 - x), $MachinePrecision], N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5000000000000004e134

   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 71.8%

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

   5. Simplified 71.8%

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

   if 9.5000000000000004e134 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 2.6%

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

   5. Simplified 2.6%

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

      1. flip3-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

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

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

      12. distribute-rgt-out N/A

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

      13. +-commutative N/A

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

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

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

      15. +-lowering-+.f64 2.5%

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

   7. Applied egg-rr 2.5%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 28.8%

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

   10. Simplified 28.8%

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

   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. *-commutative N/A

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

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 27.6%

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

   13. Simplified 27.6%

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

Alternative 8: 61.6% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 115000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 115000.0) 1.0 (- 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 115000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 115000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 115000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 115000.0:
		tmp = 1.0
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 115000.0)
		tmp = 1.0;
	else
		tmp = Float64(0.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 115000.0)
		tmp = 1.0;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 115000.0], 1.0, N[(0.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 115000

   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 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 60.1%

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

   if 115000 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 61.8%

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

   5. Simplified 61.8%

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

   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 61.0%

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

   8. Simplified 61.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 61.0%

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

   10. Applied egg-rr 61.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 115000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.7% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. --lowering--.f64 61.8%

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

5. Simplified 61.8%

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

Alternative 10: 30.9% 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 99.8%

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

3. Taylor expanded in y around 0

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

   1. --lowering--.f64 61.8%

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

5. Simplified 61.8%

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

6. Taylor expanded in x around 0

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

8. Simplified 30.6%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))