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

Percentage Accurate: 99.9% → 99.9%

Time: 5.3s

Alternatives: 7

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, 1.1× 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 99.8%

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

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - x\right) + y \cdot \sqrt{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, y, -x\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999808:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 x) (* y (sqrt x)))))
   (if (<= t_0 -2e+16)
     (fma (sqrt x) y (- x))
     (if (<= t_0 0.9999999999999808) (- 1.0 x) (fma (sqrt x) y 1.0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) + (y * sqrt(x));
	double tmp;
	if (t_0 <= -2e+16) {
		tmp = fma(sqrt(x), y, -x);
	} else if (t_0 <= 0.9999999999999808) {
		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(y * sqrt(x)))
	tmp = 0.0
	if (t_0 <= -2e+16)
		tmp = fma(sqrt(x), y, Float64(-x));
	elseif (t_0 <= 0.9999999999999808)
		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[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+16], N[(N[Sqrt[x], $MachinePrecision] * y + (-x)), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999808], 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))) < -2e16

   1. Initial program 99.8%

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -2e16 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x))) < 0.999999999999980793

   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 95.5

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

   5. Applied rewrites 95.5%

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

   if 0.999999999999980793 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x)))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \sqrt{x} \cdot y} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 3: 61.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + y \cdot \sqrt{x} \leq -50:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (- 1.0 x) (* y (sqrt x))) -50.0) (- x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) + (y * sqrt(x))) <= -50.0) {
		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) + (y * sqrt(x))) <= (-50.0d0)) 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) + (y * Math.sqrt(x))) <= -50.0) {
		tmp = -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - x) + (y * math.sqrt(x))) <= -50.0:
		tmp = -x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) + Float64(y * sqrt(x))) <= -50.0)
		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) + (y * sqrt(x))) <= -50.0)
		tmp = -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -50.0], (-x), 1.0]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x))) < -50

   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 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 52.7%

      \[\leadsto -x \]

   if -50 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 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. 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 59.0%

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

Alternative 4: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+37} \lor \neg \left(y \leq 1.2 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.4e+37) (not (<= y 1.2e+14))) (fma (sqrt x) y 1.0) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.4e+37) || !(y <= 1.2e+14)) {
		tmp = fma(sqrt(x), y, 1.0);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.4e+37) || !(y <= 1.2e+14))
		tmp = fma(sqrt(x), y, 1.0);
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.4e+37], N[Not[LessEqual[y, 1.2e+14]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * y + 1.0), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e37 or 1.2e14 < 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 93.9

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

   5. Applied rewrites 93.9%

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

   if -2.4e37 < y < 1.2e14

   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 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+37} \lor \neg \left(y \leq 1.2 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+64} \lor \neg \left(y \leq 1.4 \cdot 10^{+41}\right):\\ \;\;\;\;\sqrt{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6e+64) (not (<= y 1.4e+41))) (* (sqrt x) y) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+64) || !(y <= 1.4e+41)) {
		tmp = sqrt(x) * y;
	} 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 <= (-1.6d+64)) .or. (.not. (y <= 1.4d+41))) then
        tmp = sqrt(x) * y
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+64) || !(y <= 1.4e+41)) {
		tmp = Math.sqrt(x) * y;
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.6e+64) or not (y <= 1.4e+41):
		tmp = math.sqrt(x) * y
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.6e+64) || !(y <= 1.4e+41))
		tmp = Float64(sqrt(x) * y);
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.6e+64) || ~((y <= 1.4e+41)))
		tmp = sqrt(x) * y;
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.6e+64], N[Not[LessEqual[y, 1.4e+41]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.60000000000000009e64 or 1.4e41 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   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. remove-double-neg N/A

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

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

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out-- N/A

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

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

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. fp-cancel-sub-sign N/A

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

      13. mul-1-neg N/A

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 91.1%

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

   if -1.60000000000000009e64 < y < 1.4e41

   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 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+64} \lor \neg \left(y \leq 1.4 \cdot 10^{+41}\right):\\ \;\;\;\;\sqrt{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.2% 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 99.8%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 57.1

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

5. Applied rewrites 57.1%

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

Alternative 7: 31.3% 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 99.8%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 57.1

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

5. Applied rewrites 57.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 26.9%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))