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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - x\right) + \sqrt{x} \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, y, -x\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999996:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 x) (* (sqrt x) y))))
   (if (<= t_0 -2e+15)
     (fma (sqrt x) y (- x))
     (if (<= t_0 0.9999999999996) (- 1.0 x) (fma (sqrt x) y 1.0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) + (sqrt(x) * y);
	double tmp;
	if (t_0 <= -2e+15) {
		tmp = fma(sqrt(x), y, -x);
	} else if (t_0 <= 0.9999999999996) {
		tmp = 1.0 - x;
	} else {
		tmp = fma(sqrt(x), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) + Float64(sqrt(x) * y))
	tmp = 0.0
	if (t_0 <= -2e+15)
		tmp = fma(sqrt(x), y, Float64(-x));
	elseif (t_0 <= 0.9999999999996)
		tmp = Float64(1.0 - x);
	else
		tmp = fma(sqrt(x), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+15], N[(N[Sqrt[x], $MachinePrecision] * y + (-x)), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999996], N[(1.0 - x), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * y + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x))) < -2e15

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. cancel-sign-sub N/A

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

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

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

      7. mul-1-neg N/A

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

      8. rem-square-sqrt N/A

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

      9. unpow2 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-out-- N/A

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

      12. unsub-neg N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-in N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 99.9%

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

   if -2e15 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x))) < 0.999999999999599987

   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. lower--.f64 91.3

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

   5. Applied rewrites 91.3%

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

   if 0.999999999999599987 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x)))

   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

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, y, -x\right)\\ \mathbf{elif}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq 0.9999999999996:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -500:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (- 1.0 x) (* (sqrt x) y)) -500.0) (- x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) + (sqrt(x) * y)) <= -500.0) {
		tmp = -x;
	} else {
		tmp = 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 (((1.0d0 - x) + (sqrt(x) * y)) <= (-500.0d0)) then
        tmp = -x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - x) + (Math.sqrt(x) * y)) <= -500.0) {
		tmp = -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - x) + (math.sqrt(x) * y)) <= -500.0:
		tmp = -x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) + Float64(sqrt(x) * y)) <= -500.0)
		tmp = Float64(-x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - x) + (sqrt(x) * y)) <= -500.0)
		tmp = -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], -500.0], (-x), 1.0]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x))) < -500

   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. lower--.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.9%

      \[\leadsto -x \]

   if -500 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x)))

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 57.5%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -500:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \mathbf{if}\;y \leq -8.6 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+50}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt x) y 1.0)))
   (if (<= y -8.6e+52) t_0 (if (<= y 8e+50) (- 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 <= -8.6e+52) {
		tmp = t_0;
	} else if (y <= 8e+50) {
		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 <= -8.6e+52)
		tmp = t_0;
	elseif (y <= 8e+50)
		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, -8.6e+52], t$95$0, If[LessEqual[y, 8e+50], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5999999999999999e52 or 8.0000000000000006e50 < 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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -8.5999999999999999e52 < y < 8.0000000000000006e50

   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. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

Alternative 5: 93.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+82}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) y)))
   (if (<= y -5.2e+53) t_0 (if (<= y 8.6e+82) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (y <= -5.2e+53) {
		tmp = t_0;
	} else if (y <= 8.6e+82) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * y
    if (y <= (-5.2d+53)) then
        tmp = t_0
    else if (y <= 8.6d+82) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * y;
	double tmp;
	if (y <= -5.2e+53) {
		tmp = t_0;
	} else if (y <= 8.6e+82) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * y
	tmp = 0
	if y <= -5.2e+53:
		tmp = t_0
	elif y <= 8.6e+82:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * y)
	tmp = 0.0
	if (y <= -5.2e+53)
		tmp = t_0;
	elseif (y <= 8.6e+82)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * y;
	tmp = 0.0;
	if (y <= -5.2e+53)
		tmp = t_0;
	elseif (y <= 8.6e+82)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -5.2e+53], t$95$0, If[LessEqual[y, 8.6e+82], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.19999999999999996e53 or 8.60000000000000029e82 < 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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if -5.19999999999999996e53 < y < 8.60000000000000029e82

   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. lower--.f64 94.8

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

   5. Applied rewrites 94.8%

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

Alternative 6: 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. cancel-sign-sub-inv N/A

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

   7. associate-+l- N/A

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

   8. lift--.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 7: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. lower--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 8: 63.6% 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. lower--.f64 60.6

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

5. Applied rewrites 60.6%

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

Alternative 9: 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. lower--.f64 60.6

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

5. Applied rewrites 60.6%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 29.7%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))