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

Percentage Accurate: 99.9% → 99.9%

Time: 8.8s

Alternatives: 8

Speedup: 1.1×

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+17}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+29}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.12e+17)
   (+ 1.0 (* y (sqrt x)))
   (if (<= y 4.1e+29) (- 1.0 x) (fma (sqrt x) y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.12e+17) {
		tmp = 1.0 + (y * sqrt(x));
	} else if (y <= 4.1e+29) {
		tmp = 1.0 - x;
	} else {
		tmp = fma(sqrt(x), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.12e+17)
		tmp = Float64(1.0 + Float64(y * sqrt(x)));
	elseif (y <= 4.1e+29)
		tmp = Float64(1.0 - x);
	else
		tmp = fma(sqrt(x), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.12e+17], N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+29], N[(1.0 - x), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * y + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.12e17

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 90.3%

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

   if -1.12e17 < y < 4.1000000000000003e29

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 100.0%

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

   5. Simplified 100.0%

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

   if 4.1000000000000003e29 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 92.8%

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

   5. Simplified 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, y, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+29}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt x) y 1.0)))
   (if (<= y -1.12e+17) t_0 (if (<= y 4.9e+29) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(x), y, 1.0);
	double tmp;
	if (y <= -1.12e+17) {
		tmp = t_0;
	} else if (y <= 4.9e+29) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(x), y, 1.0)
	tmp = 0.0
	if (y <= -1.12e+17)
		tmp = t_0;
	elseif (y <= 4.9e+29)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * y + 1.0), $MachinePrecision]}, If[LessEqual[y, -1.12e+17], t$95$0, If[LessEqual[y, 4.9e+29], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.12e17 or 4.9000000000000001e29 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 91.5%

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

   5. Simplified 91.5%

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

   if -1.12e17 < y < 4.9000000000000001e29

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 100.0%

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

   5. Simplified 100.0%

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

Alternative 4: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+55}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x))))
   (if (<= y -1.4e+116) t_0 (if (<= y 1.5e+55) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (y <= -1.4e+116) {
		tmp = t_0;
	} else if (y <= 1.5e+55) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(x)
    if (y <= (-1.4d+116)) then
        tmp = t_0
    else if (y <= 1.5d+55) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * Math.sqrt(x);
	double tmp;
	if (y <= -1.4e+116) {
		tmp = t_0;
	} else if (y <= 1.5e+55) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if y <= -1.4e+116:
		tmp = t_0
	elif y <= 1.5e+55:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(x))
	tmp = 0.0
	if (y <= -1.4e+116)
		tmp = t_0;
	elseif (y <= 1.5e+55)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * sqrt(x);
	tmp = 0.0;
	if (y <= -1.4e+116)
		tmp = t_0;
	elseif (y <= 1.5e+55)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4e+116], t$95$0, If[LessEqual[y, 1.5e+55], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000002e116 or 1.50000000000000008e55 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lft-mult-inverse N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. div-sub N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. distribute-rgt-in N/A

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

      19. +-commutative N/A

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

      20. accelerator-lowering-fma.f64 N/A

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

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

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

      22. *-commutative N/A

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

      23. div-sub N/A

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

      24. sub-neg N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\sqrt{x} \cdot y} \]
   7. 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 94.5%

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

   8. Simplified 94.5%

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

   if -1.40000000000000002e116 < y < 1.50000000000000008e55

   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 92.1%

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

   5. Simplified 92.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+116}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+55}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

   4. *-inverses N/A

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

   5. associate-/l* N/A

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

   6. associate-*l/ N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac N/A

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

   9. +-commutative N/A

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

   10. associate-+r+ N/A

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

   11. lft-mult-inverse N/A

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

   12. distribute-rgt-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. div-sub N/A

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

   16. *-commutative N/A

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

   17. distribute-lft-in N/A

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

   18. distribute-rgt-in N/A

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

   19. +-commutative N/A

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

   20. accelerator-lowering-fma.f64 N/A

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

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

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

   22. *-commutative N/A

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

   23. div-sub N/A

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

   24. sub-neg N/A

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

5. Simplified 99.9%

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

Alternative 6: 61.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.3e-6) 1.0 (- 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.3e-6) {
		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 <= 1.3d-6) 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 <= 1.3e-6) {
		tmp = 1.0;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.3e-6:
		tmp = 1.0
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.3e-6)
		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 <= 1.3e-6)
		tmp = 1.0;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.3e-6], 1.0, N[(0.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000005e-6

   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 58.1%

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

   5. Simplified 58.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 57.9%

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

   if 1.30000000000000005e-6 < 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 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 61.5%

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

   8. Simplified 61.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 61.5%

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

   10. Applied egg-rr 61.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 5.5× 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 60.7%

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

5. Simplified 60.7%

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

Alternative 8: 31.9% accurate, 22.0× 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 60.7%

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

5. Simplified 60.7%

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

6. Taylor expanded in x around 0

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

8. Simplified 28.2%

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

Reproduce

herbie shell --seed 2024193 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))