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

Percentage Accurate: 99.9% → 99.9%

Time: 9.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 94.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+84}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* y (sqrt x)))))
   (if (<= y -5e+38) t_0 (if (<= y 6.2e+84) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y * sqrt(x));
	double tmp;
	if (y <= -5e+38) {
		tmp = t_0;
	} else if (y <= 6.2e+84) {
		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 <= (-5d+38)) then
        tmp = t_0
    else if (y <= 6.2d+84) 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 <= -5e+38) {
		tmp = t_0;
	} else if (y <= 6.2e+84) {
		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 <= -5e+38:
		tmp = t_0
	elif y <= 6.2e+84:
		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 <= -5e+38)
		tmp = t_0;
	elseif (y <= 6.2e+84)
		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 <= -5e+38)
		tmp = t_0;
	elseif (y <= 6.2e+84)
		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, -5e+38], t$95$0, If[LessEqual[y, 6.2e+84], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999997e38 or 6.20000000000000006e84 < 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 97.3%

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

   5. Simplified 97.3%

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

   if -4.9999999999999997e38 < y < 6.20000000000000006e84

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 97.1%

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

   5. Simplified 97.1%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+38}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+84}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+39}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+85}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e+39)
   (* y (sqrt x))
   (if (<= y 4.1e+85) (- 1.0 x) (* y (/ x (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.3e+39) {
		tmp = y * sqrt(x);
	} else if (y <= 4.1e+85) {
		tmp = 1.0 - x;
	} else {
		tmp = y * (x / sqrt(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 <= (-3.3d+39)) then
        tmp = y * sqrt(x)
    else if (y <= 4.1d+85) then
        tmp = 1.0d0 - x
    else
        tmp = y * (x / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.3e+39) {
		tmp = y * Math.sqrt(x);
	} else if (y <= 4.1e+85) {
		tmp = 1.0 - x;
	} else {
		tmp = y * (x / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.3e+39:
		tmp = y * math.sqrt(x)
	elif y <= 4.1e+85:
		tmp = 1.0 - x
	else:
		tmp = y * (x / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.3e+39)
		tmp = Float64(y * sqrt(x));
	elseif (y <= 4.1e+85)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * Float64(x / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.3e+39)
		tmp = y * sqrt(x);
	elseif (y <= 4.1e+85)
		tmp = 1.0 - x;
	else
		tmp = y * (x / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.3e+39], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+85], N[(1.0 - x), $MachinePrecision], N[(y * N[(x / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000021e39

   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 89.5%

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

   5. Simplified 89.5%

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

   if -3.30000000000000021e39 < y < 4.09999999999999978e85

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 97.1%

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

   5. Simplified 97.1%

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

   if 4.09999999999999978e85 < 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 95.6%

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

   5. Simplified 95.6%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot x\right), \left(\frac{\frac{1}{2}}{2}\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\frac{1}{2}}{2}\right)\right), y\right) \]

      6. metadata-eval 77.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{4}\right), y\right) \]

   7. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{{\left(x \cdot x\right)}^{0.25}} \cdot y \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{2}\right)}^{\frac{1}{4}}\right), y\right) \]

      2. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot \frac{1}{4}\right)}\right), y\right) \]

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. pow-pow N/A

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

      8. inv-pow N/A

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

      9. pow1/2 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. pow1/2 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. pow-pow N/A

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

      19. inv-pow N/A

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

      20. pow1/2 N/A

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

      21. /-lowering-/.f64 N/A

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

      22. pow1/2 N/A

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

      23. inv-pow N/A

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

      24. pow-pow N/A

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

      25. pow-plus N/A

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

      26. metadata-eval N/A

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

      27. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({x}^{\frac{1}{2}}\right)\right), y\right) \]

      28. pow1/2 N/A

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

      29. sqrt-lowering-sqrt.f64 95.7%

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

   9. Applied egg-rr 95.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+39}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+85}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+84}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x))))
   (if (<= y -3.3e+39) t_0 (if (<= y 9.8e+84) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (y <= -3.3e+39) {
		tmp = t_0;
	} else if (y <= 9.8e+84) {
		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 <= (-3.3d+39)) then
        tmp = t_0
    else if (y <= 9.8d+84) 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 <= -3.3e+39) {
		tmp = t_0;
	} else if (y <= 9.8e+84) {
		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 <= -3.3e+39:
		tmp = t_0
	elif y <= 9.8e+84:
		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 <= -3.3e+39)
		tmp = t_0;
	elseif (y <= 9.8e+84)
		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 <= -3.3e+39)
		tmp = t_0;
	elseif (y <= 9.8e+84)
		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, -3.3e+39], t$95$0, If[LessEqual[y, 9.8e+84], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.30000000000000021e39 or 9.8e84 < 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 92.1%

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

   5. Simplified 92.1%

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

   if -3.30000000000000021e39 < y < 9.8e84

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 97.1%

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

   5. Simplified 97.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+39}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+84}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{x}{1 - x}}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+86}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.9e+155)
   (/ (- 1.0 (* x (* x x))) (+ 1.0 (/ x (- 1.0 x))))
   (if (<= y 2.25e+86) (- 1.0 x) (+ 1.0 (* x (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.9e+155) {
		tmp = (1.0 - (x * (x * x))) / (1.0 + (x / (1.0 - x)));
	} else if (y <= 2.25e+86) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 + (x * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.9d+155)) then
        tmp = (1.0d0 - (x * (x * x))) / (1.0d0 + (x / (1.0d0 - x)))
    else if (y <= 2.25d+86) then
        tmp = 1.0d0 - x
    else
        tmp = 1.0d0 + (x * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.9e+155) {
		tmp = (1.0 - (x * (x * x))) / (1.0 + (x / (1.0 - x)));
	} else if (y <= 2.25e+86) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 + (x * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.9e+155:
		tmp = (1.0 - (x * (x * x))) / (1.0 + (x / (1.0 - x)))
	elif y <= 2.25e+86:
		tmp = 1.0 - x
	else:
		tmp = 1.0 + (x * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.9e+155)
		tmp = Float64(Float64(1.0 - Float64(x * Float64(x * x))) / Float64(1.0 + Float64(x / Float64(1.0 - x))));
	elseif (y <= 2.25e+86)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.9e+155)
		tmp = (1.0 - (x * (x * x))) / (1.0 + (x / (1.0 - x)));
	elseif (y <= 2.25e+86)
		tmp = 1.0 - x;
	else
		tmp = 1.0 + (x * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.9e+155], N[(N[(1.0 - N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e+86], N[(1.0 - x), $MachinePrecision], N[(1.0 + N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8999999999999997e155

   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 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. /-lowering-/.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cube-mult N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

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

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

      13. +-lowering-+.f64 1.5%

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

   7. Applied egg-rr 1.5%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. +-lowering-+.f64 1.5%

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

   9. Applied egg-rr 1.5%

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

   10. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. --lowering--.f64 25.8%

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

   12. Simplified 25.8%

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

   if -4.8999999999999997e155 < y < 2.24999999999999996e86

   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 88.3%

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

   5. Simplified 88.3%

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

   if 2.24999999999999996e86 < 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 5.9%

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

   5. Simplified 5.9%

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

      1. flip3-- N/A

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

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

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

      3. metadata-eval N/A

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

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

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

      5. cube-mult N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

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

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

      13. +-lowering-+.f64 5.8%

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

   7. Applied egg-rr 5.8%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 5.8%

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

   11. Taylor expanded in x around 0

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

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

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

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

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. +-lowering-+.f64 17.3%

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

   13. Simplified 17.3%

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

Alternative 6: 68.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+129}:\\ \;\;\;\;-1 + x \cdot \left(1 - x\right)\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+86}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.2e+129)
   (+ -1.0 (* x (- 1.0 x)))
   (if (<= y 2.25e+86) (- 1.0 x) (+ 1.0 (* x (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.2e+129) {
		tmp = -1.0 + (x * (1.0 - x));
	} else if (y <= 2.25e+86) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 + (x * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.2d+129)) then
        tmp = (-1.0d0) + (x * (1.0d0 - x))
    else if (y <= 2.25d+86) then
        tmp = 1.0d0 - x
    else
        tmp = 1.0d0 + (x * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.2e+129) {
		tmp = -1.0 + (x * (1.0 - x));
	} else if (y <= 2.25e+86) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 + (x * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.2e+129:
		tmp = -1.0 + (x * (1.0 - x))
	elif y <= 2.25e+86:
		tmp = 1.0 - x
	else:
		tmp = 1.0 + (x * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.2e+129)
		tmp = Float64(-1.0 + Float64(x * Float64(1.0 - x)));
	elseif (y <= 2.25e+86)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.2e+129)
		tmp = -1.0 + (x * (1.0 - x));
	elseif (y <= 2.25e+86)
		tmp = 1.0 - x;
	else
		tmp = 1.0 + (x * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.2e+129], N[(-1.0 + N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e+86], N[(1.0 - x), $MachinePrecision], N[(1.0 + N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.20000000000000024e129

   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 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. /-lowering-/.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cube-mult N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

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

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

      13. +-lowering-+.f64 1.5%

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

   7. Applied egg-rr 1.5%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 21.9%

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

   11. Taylor expanded in x around inf

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

       1. associate--r+ N/A

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

       2. sub-neg N/A

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

       3. distribute-rgt-in N/A

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

       4. distribute-lft-neg-out N/A

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

       5. unpow2 N/A

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

       6. associate-/r* N/A

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

       7. *-rgt-identity N/A

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

       8. rgt-mult-inverse N/A

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

       9. associate-*l* N/A

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

       10. unpow2 N/A

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

       11. associate-*r/ N/A

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

       12. *-rgt-identity N/A

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

       13. associate-/r/ N/A

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

       14. associate-/r* N/A

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

       15. unpow2 N/A

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

       16. cube-mult N/A

           \[\leadsto \left(\frac{1}{x} - 1\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(\left(\frac{1}{{x}^{3}} \cdot x\right) \cdot {x}^{2}\right)\right) \]

       17. associate-*r* N/A

           \[\leadsto \left(\frac{1}{x} - 1\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{{x}^{3}} \cdot \left(x \cdot {x}^{2}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \left(\frac{1}{x} - 1\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{{x}^{3}} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

       19. cube-mult N/A

           \[\leadsto \left(\frac{1}{x} - 1\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{{x}^{3}} \cdot {x}^{3}\right)\right) \]

   13. Simplified 24.1%

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

   if -5.20000000000000024e129 < y < 2.24999999999999996e86

   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 89.8%

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

   5. Simplified 89.8%

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

   if 2.24999999999999996e86 < 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 5.9%

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

   5. Simplified 5.9%

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

      1. flip3-- N/A

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

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

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

      3. metadata-eval N/A

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

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

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

      5. cube-mult N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

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

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

      13. +-lowering-+.f64 5.8%

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

   7. Applied egg-rr 5.8%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 5.8%

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

   11. Taylor expanded in x around 0

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

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

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

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

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. +-lowering-+.f64 17.3%

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

   13. Simplified 17.3%

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

Alternative 7: 65.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+131}:\\ \;\;\;\;-1 + x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.5e+131) (+ -1.0 (* x (- 1.0 x))) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.5e+131) {
		tmp = -1.0 + (x * (1.0 - x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.5d+131)) then
        tmp = (-1.0d0) + (x * (1.0d0 - x))
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.5e+131) {
		tmp = -1.0 + (x * (1.0 - x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.5e+131:
		tmp = -1.0 + (x * (1.0 - x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.5e+131)
		tmp = Float64(-1.0 + Float64(x * Float64(1.0 - x)));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.5e+131)
		tmp = -1.0 + (x * (1.0 - x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.5e+131], N[(-1.0 + N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5000000000000001e131

   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 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. /-lowering-/.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cube-mult N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

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

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

      13. +-lowering-+.f64 1.5%

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

   7. Applied egg-rr 1.5%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 21.9%

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

   11. Taylor expanded in x around inf

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

       1. associate--r+ N/A

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

       2. sub-neg N/A

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

       3. distribute-rgt-in N/A

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

       4. distribute-lft-neg-out N/A

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

       5. unpow2 N/A

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

       6. associate-/r* N/A

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

       7. *-rgt-identity N/A

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

       8. rgt-mult-inverse N/A

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

       9. associate-*l* N/A

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

       10. unpow2 N/A

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

       11. associate-*r/ N/A

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

       12. *-rgt-identity N/A

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

       13. associate-/r/ N/A

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

       14. associate-/r* N/A

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

       15. unpow2 N/A

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

       16. cube-mult N/A

           \[\leadsto \left(\frac{1}{x} - 1\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(\left(\frac{1}{{x}^{3}} \cdot x\right) \cdot {x}^{2}\right)\right) \]

       17. associate-*r* N/A

           \[\leadsto \left(\frac{1}{x} - 1\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{{x}^{3}} \cdot \left(x \cdot {x}^{2}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \left(\frac{1}{x} - 1\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{{x}^{3}} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

       19. cube-mult N/A

           \[\leadsto \left(\frac{1}{x} - 1\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{{x}^{3}} \cdot {x}^{3}\right)\right) \]

   13. Simplified 24.1%

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

   if -1.5000000000000001e131 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 72.6%

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

   5. Simplified 72.6%

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

Alternative 8: 65.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+135}:\\ \;\;\;\;0 - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.7e+135) (- 0.0 (* x x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+135) {
		tmp = 0.0 - (x * x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.7d+135)) then
        tmp = 0.0d0 - (x * x)
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+135) {
		tmp = 0.0 - (x * x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.7e+135:
		tmp = 0.0 - (x * x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.7e+135)
		tmp = Float64(0.0 - Float64(x * x));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.7e+135)
		tmp = 0.0 - (x * x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.7e+135], N[(0.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6999999999999998e135

   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 3.0%

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

   5. Simplified 3.0%

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

      1. flip3-- N/A

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

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

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

      3. metadata-eval N/A

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

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

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

      5. cube-mult N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

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

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

      13. +-lowering-+.f64 1.5%

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

   7. Applied egg-rr 1.5%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 22.8%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot {x}^{2}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{{x}^{2}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x}^{2}\right)}\right) \]

       4. unpow2 N/A

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

       5. *-lowering-*.f64 23.6%

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

   13. Simplified 23.6%

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

   if -4.6999999999999998e135 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 72.0%

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

   5. Simplified 72.0%

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

Alternative 9: 62.4% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 59.5%

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

   5. Simplified 59.5%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 58.4%

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

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 63.5%

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

   5. Simplified 63.5%

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

   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 62.7%

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

   8. Simplified 62.7%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 62.7%

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

   10. Applied egg-rr 62.7%

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

4. Final simplification 60.6%

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

Alternative 10: 63.5% 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 61.5%

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

5. Simplified 61.5%

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

Alternative 11: 32.3% 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 61.5%

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

5. Simplified 61.5%

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

6. Taylor expanded in x around 0

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

8. Simplified 29.7%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))