Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Percentage Accurate: 99.4% → 99.5%

Time: 9.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.3333333333333333, \sqrt{x} \cdot \mathsf{fma}\left(y, 3, -3\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma (sqrt (/ 1.0 x)) 0.3333333333333333 (* (sqrt x) (fma y 3.0 -3.0))))

C

double code(double x, double y) {
	return fma(sqrt((1.0 / x)), 0.3333333333333333, (sqrt(x) * fma(y, 3.0, -3.0)));
}

Julia

function code(x, y)
	return fma(sqrt(Float64(1.0 / x)), 0.3333333333333333, Float64(sqrt(x) * fma(y, 3.0, -3.0)))
end

Wolfram

code[x_, y_] := N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333 + N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

   3. lift-+.f64 N/A

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

   4. +-commutative N/A

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} - 1\right) \]

   5. associate--l+ N/A

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(y - 1\right)\right)} \]

   6. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot 9} \cdot \left(3 \cdot \sqrt{x}\right) + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot 9}, 3 \cdot \sqrt{x}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right)} \]

   8. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{x \cdot 9}}, 3 \cdot \sqrt{x}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{x \cdot 9}}, 3 \cdot \sqrt{x}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{9 \cdot x}}, 3 \cdot \sqrt{x}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   11. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{9}}{x}}, 3 \cdot \sqrt{x}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{9}}}{x}, 3 \cdot \sqrt{x}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{9}^{-1}}}{x}, 3 \cdot \sqrt{x}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   14. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{9}^{-1}}{x}}, 3 \cdot \sqrt{x}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{9}}}{x}, 3 \cdot \sqrt{x}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9}}{x}, \color{blue}{3 \cdot \sqrt{x}}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9}}{x}, \color{blue}{\sqrt{x} \cdot 3}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9}}{x}, \color{blue}{\sqrt{x} \cdot 3}, \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9}}{x}, \sqrt{x} \cdot 3, \color{blue}{\left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)}\right) \]

   20. lower--.f64 99.4

       \[\leadsto \mathsf{fma}\left(\frac{0.1111111111111111}{x}, \sqrt{x} \cdot 3, \color{blue}{\left(y - 1\right)} \cdot \left(3 \cdot \sqrt{x}\right)\right) \]

   21. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9}}{x}, \sqrt{x} \cdot 3, \left(y - 1\right) \cdot \color{blue}{\left(3 \cdot \sqrt{x}\right)}\right) \]

   22. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{9}}{x}, \sqrt{x} \cdot 3, \left(y - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 3\right)}\right) \]

   23. lower-*.f64 99.4

       \[\leadsto \mathsf{fma}\left(\frac{0.1111111111111111}{x}, \sqrt{x} \cdot 3, \left(y - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 3\right)}\right) \]

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.1111111111111111}{x}, \sqrt{x} \cdot 3, \left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\right)} \]

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-3 \cdot \sqrt{x} + \left(\frac{1}{3} \cdot \sqrt{\frac{1}{x}} + 3 \cdot \left(\sqrt{x} \cdot y\right)\right)} \]
6. Step-by-step derivation

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, -3 \cdot \sqrt{x} + 3 \cdot \left(\sqrt{x} \cdot y\right)\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{3}, -3 \cdot \sqrt{x} + 3 \cdot \left(\sqrt{x} \cdot y\right)\right) \]

   7. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{3}, -3 \cdot \sqrt{x} + 3 \cdot \left(\sqrt{x} \cdot y\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, -3 \cdot \sqrt{x} + 3 \cdot \color{blue}{\left(y \cdot \sqrt{x}\right)}\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, -3 \cdot \sqrt{x} + \color{blue}{\left(3 \cdot y\right) \cdot \sqrt{x}}\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, \color{blue}{\sqrt{x} \cdot \left(-3 + 3 \cdot y\right)}\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, \sqrt{x} \cdot \color{blue}{\left(3 \cdot y + -3\right)}\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, \sqrt{x} \cdot \left(3 \cdot y + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, \sqrt{x} \cdot \color{blue}{\left(3 \cdot y - 3\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, \color{blue}{\left(3 \cdot y - 3\right) \cdot \sqrt{x}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, \color{blue}{\left(3 \cdot y - 3\right) \cdot \sqrt{x}}\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, \color{blue}{\left(3 \cdot y + \left(\mathsf{neg}\left(3\right)\right)\right)} \cdot \sqrt{x}\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, \left(3 \cdot y + \color{blue}{-3}\right) \cdot \sqrt{x}\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, \left(\color{blue}{y \cdot 3} + -3\right) \cdot \sqrt{x}\right) \]

   19. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{3}, \color{blue}{\mathsf{fma}\left(y, 3, -3\right)} \cdot \sqrt{x}\right) \]

   20. lower-sqrt.f64 99.5

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.3333333333333333, \mathsf{fma}\left(y, 3, -3\right) \cdot \color{blue}{\sqrt{x}}\right) \]

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.3333333333333333, \mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\right)} \]

8. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.3333333333333333, \sqrt{x} \cdot \mathsf{fma}\left(y, 3, -3\right)\right) \]
9. Add Preprocessing

Alternative 2: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \mathbf{elif}\;t\_0 \leq 10^{+153}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{x} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 3\right) \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) (* (sqrt x) 3.0))))
   (if (<= t_0 -1000000.0)
     (* (* (- y 1.0) (sqrt x)) 3.0)
     (if (<= t_0 1e+153)
       (* (+ (/ 0.3333333333333333 x) -3.0) (sqrt x))
       (* (* y 3.0) (sqrt x))))))

C

double code(double x, double y) {
	double t_0 = (((1.0 / (9.0 * x)) + y) - 1.0) * (sqrt(x) * 3.0);
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = ((y - 1.0) * sqrt(x)) * 3.0;
	} else if (t_0 <= 1e+153) {
		tmp = ((0.3333333333333333 / x) + -3.0) * sqrt(x);
	} else {
		tmp = (y * 3.0) * sqrt(x);
	}
	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 / (9.0d0 * x)) + y) - 1.0d0) * (sqrt(x) * 3.0d0)
    if (t_0 <= (-1000000.0d0)) then
        tmp = ((y - 1.0d0) * sqrt(x)) * 3.0d0
    else if (t_0 <= 1d+153) then
        tmp = ((0.3333333333333333d0 / x) + (-3.0d0)) * sqrt(x)
    else
        tmp = (y * 3.0d0) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((1.0 / (9.0 * x)) + y) - 1.0) * (Math.sqrt(x) * 3.0);
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = ((y - 1.0) * Math.sqrt(x)) * 3.0;
	} else if (t_0 <= 1e+153) {
		tmp = ((0.3333333333333333 / x) + -3.0) * Math.sqrt(x);
	} else {
		tmp = (y * 3.0) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((1.0 / (9.0 * x)) + y) - 1.0) * (math.sqrt(x) * 3.0)
	tmp = 0
	if t_0 <= -1000000.0:
		tmp = ((y - 1.0) * math.sqrt(x)) * 3.0
	elif t_0 <= 1e+153:
		tmp = ((0.3333333333333333 / x) + -3.0) * math.sqrt(x)
	else:
		tmp = (y * 3.0) * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * Float64(sqrt(x) * 3.0))
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = Float64(Float64(Float64(y - 1.0) * sqrt(x)) * 3.0);
	elseif (t_0 <= 1e+153)
		tmp = Float64(Float64(Float64(0.3333333333333333 / x) + -3.0) * sqrt(x));
	else
		tmp = Float64(Float64(y * 3.0) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((1.0 / (9.0 * x)) + y) - 1.0) * (sqrt(x) * 3.0);
	tmp = 0.0;
	if (t_0 <= -1000000.0)
		tmp = ((y - 1.0) * sqrt(x)) * 3.0;
	elseif (t_0 <= 1e+153)
		tmp = ((0.3333333333333333 / x) + -3.0) * sqrt(x);
	else
		tmp = (y * 3.0) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], N[(N[(N[(y - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+153], N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(y * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -1e6

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} - 1\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 57.8

         \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\frac{0.1111111111111111}{x}} - 1\right) \]

   5. Applied rewrites 57.8%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\frac{0.1111111111111111}{x}} - 1\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{\frac{1}{9}}{x} - 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \left(\frac{\frac{1}{9}}{x} - 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\frac{\frac{1}{9}}{x} - 1\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(\frac{\frac{1}{9}}{x} - 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(\frac{\frac{1}{9}}{x} - 1\right)\right) \cdot 3} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{9}}{x} - 1\right) \cdot \sqrt{x}\right)} \cdot 3 \]

      7. lower-*.f64 57.8

         \[\leadsto \color{blue}{\left(\left(\frac{0.1111111111111111}{x} - 1\right) \cdot \sqrt{x}\right)} \cdot 3 \]

   7. Applied rewrites 57.8%

      \[\leadsto \color{blue}{\left(\left(\frac{0.1111111111111111}{x} - 1\right) \cdot \sqrt{x}\right) \cdot 3} \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(y - 1\right)} \cdot \sqrt{x}\right) \cdot 3 \]
   9. Step-by-step derivation

      1. lower--.f64 98.9

         \[\leadsto \left(\color{blue}{\left(y - 1\right)} \cdot \sqrt{x}\right) \cdot 3 \]

   10. Applied rewrites 98.9%

       \[\leadsto \left(\color{blue}{\left(y - 1\right)} \cdot \sqrt{x}\right) \cdot 3 \]

   if -1e6 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 1e153

   1. Initial program 99.2%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{9} \cdot \frac{1}{x} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\left(\frac{1}{9} \cdot \frac{1}{x} - 1\right) \cdot \sqrt{x}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot \left(\frac{1}{9} \cdot \frac{1}{x} - 1\right)\right) \cdot \sqrt{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(3 \cdot \left(\frac{1}{9} \cdot \frac{1}{x} - 1\right)\right) \cdot \sqrt{x}} \]

      4. sub-neg N/A

         \[\leadsto \left(3 \cdot \color{blue}{\left(\frac{1}{9} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot \sqrt{x} \]

      5. metadata-eval N/A

         \[\leadsto \left(3 \cdot \left(\frac{1}{9} \cdot \frac{1}{x} + \color{blue}{-1}\right)\right) \cdot \sqrt{x} \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{9} \cdot \frac{1}{x}\right) \cdot 3 + -1 \cdot 3\right)} \cdot \sqrt{x} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{9} \cdot \frac{1}{x}\right) \cdot 3 + \color{blue}{-3}\right) \cdot \sqrt{x} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{9} \cdot \frac{1}{x}\right) \cdot 3 + -3\right)} \cdot \sqrt{x} \]

      9. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1}{9} \cdot 1}{x}} \cdot 3 + -3\right) \cdot \sqrt{x} \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{\color{blue}{\frac{1}{9}}}{x} \cdot 3 + -3\right) \cdot \sqrt{x} \]

      11. associate-*l/ N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{1}{9} \cdot 3}{x}} + -3\right) \cdot \sqrt{x} \]

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 85.4

          \[\leadsto \left(\frac{0.3333333333333333}{x} + -3\right) \cdot \color{blue}{\sqrt{x}} \]

   5. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333}{x} + -3\right) \cdot \sqrt{x}} \]

   if 1e153 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot y \]

      4. lower-sqrt.f64 98.8

         \[\leadsto \left(3 \cdot \color{blue}{\sqrt{x}}\right) \cdot y \]

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.7%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -1000000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 10^{+153}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{x} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 3\right) \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x)))))

C

double code(double x, double y) {
	return 3.0 * ((y * sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 3.0d0 * ((y * sqrt(x)) + (((1.0d0 / (x * 9.0d0)) - 1.0d0) * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return 3.0 * ((y * Math.sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * Math.sqrt(x)));
}

Python

def code(x, y):
	return 3.0 * ((y * math.sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(3.0 * Float64(Float64(y * sqrt(x)) + Float64(Float64(Float64(1.0 / Float64(x * 9.0)) - 1.0) * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = 3.0 * ((y * sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(3.0 * N[(N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x)))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))