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

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 1.0), $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. lower--.f64 99.9

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{x} + \left(1 - x\right) \leq -10000:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* y (sqrt x)) (- 1.0 x)) -10000.0) (- x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((y * sqrt(x)) + (1.0 - x)) <= -10000.0) {
		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 (((y * sqrt(x)) + (1.0d0 - x)) <= (-10000.0d0)) 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 (((y * Math.sqrt(x)) + (1.0 - x)) <= -10000.0) {
		tmp = -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * math.sqrt(x)) + (1.0 - x)) <= -10000.0:
		tmp = -x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * sqrt(x)) + Float64(1.0 - x)) <= -10000.0)
		tmp = Float64(-x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * sqrt(x)) + (1.0 - x)) <= -10000.0)
		tmp = -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -10000.0], (-x), 1.0]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x))) < -1e4

   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 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 63.1%

      \[\leadsto -x \]

   if -1e4 < (+.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 66.5

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.3%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{x} + \left(1 - x\right) \leq -10000:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, \sqrt{x}, 1\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+103}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+50)
   (fma y (sqrt x) 1.0)
   (if (<= y 2.4e+103) (- 1.0 x) (* y (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+50) {
		tmp = fma(y, sqrt(x), 1.0);
	} else if (y <= 2.4e+103) {
		tmp = 1.0 - x;
	} else {
		tmp = y * sqrt(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+50)
		tmp = fma(y, sqrt(x), 1.0);
	elseif (y <= 2.4e+103)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * sqrt(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+50], N[(y * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y, 2.4e+103], N[(1.0 - x), $MachinePrecision], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8000000000000004e50

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -4.8000000000000004e50 < y < 2.3999999999999998e103

   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 96.7

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

   5. Applied rewrites 96.7%

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

   if 2.3999999999999998e103 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 96.8

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

   5. Applied rewrites 96.8%

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

Alternative 4: 91.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+103}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x))))
   (if (<= y -1.35e+59) t_0 (if (<= y 2.4e+103) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (y <= -1.35e+59) {
		tmp = t_0;
	} else if (y <= 2.4e+103) {
		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 <= (-1.35d+59)) then
        tmp = t_0
    else if (y <= 2.4d+103) 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 <= -1.35e+59) {
		tmp = t_0;
	} else if (y <= 2.4e+103) {
		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 <= -1.35e+59:
		tmp = t_0
	elif y <= 2.4e+103:
		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 <= -1.35e+59)
		tmp = t_0;
	elseif (y <= 2.4e+103)
		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 <= -1.35e+59)
		tmp = t_0;
	elseif (y <= 2.4e+103)
		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, -1.35e+59], t$95$0, If[LessEqual[y, 2.4e+103], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3500000000000001e59 or 2.3999999999999998e103 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if -1.3500000000000001e59 < y < 2.3999999999999998e103

   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 96.7

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

   5. Applied rewrites 96.7%

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

Alternative 5: 98.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 1 < x

   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. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\sqrt{x}}, -x\right) \]
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. +-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 7: 62.1% 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 65.1

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

5. Applied rewrites 65.1%

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

Alternative 8: 31.2% 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 65.1

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

5. Applied rewrites 65.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 30.1%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))