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

Percentage Accurate: 99.4% → 99.4%

Time: 7.2s

Alternatives: 7

Speedup: N/A×

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - y), $MachinePrecision] * -3.0 + N[(1.0 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $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. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. +-commutative N/A

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

   13. associate-+r- N/A

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

   14. +-commutative N/A

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

   15. associate-+r- N/A

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

   16. +-commutative N/A

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

   17. distribute-rgt-in N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{1}{x \cdot 3} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) (* (sqrt x) 3.0))))
   (if (<= t_0 -4e+15)
     (* (fma y 3.0 -3.0) (sqrt x))
     (if (<= t_0 2e+153)
       (* (+ (/ 1.0 (* x 3.0)) -3.0) (sqrt x))
       (* (* (sqrt x) y) 3.0)))))

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 <= -4e+15) {
		tmp = fma(y, 3.0, -3.0) * sqrt(x);
	} else if (t_0 <= 2e+153) {
		tmp = ((1.0 / (x * 3.0)) + -3.0) * sqrt(x);
	} else {
		tmp = (sqrt(x) * y) * 3.0;
	}
	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 <= -4e+15)
		tmp = Float64(fma(y, 3.0, -3.0) * sqrt(x));
	elseif (t_0 <= 2e+153)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * 3.0)) + -3.0) * sqrt(x));
	else
		tmp = Float64(Float64(sqrt(x) * y) * 3.0);
	end
	return 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, -4e+15], N[(N[(y * 3.0 + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+153], N[(N[(N[(1.0 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $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))) < -4e15

   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 x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -4e15 < (*.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))) < 2e153

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.1

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

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y + \frac{0.1111111111111111}{x}, 3, -3\right) \cdot \sqrt{x}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-+.f64 99.2

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 85.6

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

   9. Applied rewrites 85.6%

      \[\leadsto \left(\color{blue}{\frac{0.3333333333333333}{x}} + -3\right) \cdot \sqrt{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 85.7%

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

   if 2e153 < (*.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 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{1}{x \cdot 3} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{x} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) (* (sqrt x) 3.0))))
   (if (<= t_0 -4e+15)
     (* (fma y 3.0 -3.0) (sqrt x))
     (if (<= t_0 2e+153)
       (* (+ (/ 0.3333333333333333 x) -3.0) (sqrt x))
       (* (* (sqrt x) y) 3.0)))))

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 <= -4e+15) {
		tmp = fma(y, 3.0, -3.0) * sqrt(x);
	} else if (t_0 <= 2e+153) {
		tmp = ((0.3333333333333333 / x) + -3.0) * sqrt(x);
	} else {
		tmp = (sqrt(x) * y) * 3.0;
	}
	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 <= -4e+15)
		tmp = Float64(fma(y, 3.0, -3.0) * sqrt(x));
	elseif (t_0 <= 2e+153)
		tmp = Float64(Float64(Float64(0.3333333333333333 / x) + -3.0) * sqrt(x));
	else
		tmp = Float64(Float64(sqrt(x) * y) * 3.0);
	end
	return 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, -4e+15], N[(N[(y * 3.0 + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+153], N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $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))) < -4e15

   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 x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -4e15 < (*.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))) < 2e153

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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 85.6

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

   5. Applied rewrites 85.6%

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

   if 2e153 < (*.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 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{x} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;t\_0 \leq -50000000:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) (* (sqrt x) 3.0))))
   (if (<= t_0 -50000000.0)
     (* (fma y 3.0 -3.0) (sqrt x))
     (if (<= t_0 2e+153)
       (* (sqrt (/ 1.0 x)) 0.3333333333333333)
       (* (* (sqrt x) y) 3.0)))))

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 <= -50000000.0) {
		tmp = fma(y, 3.0, -3.0) * sqrt(x);
	} else if (t_0 <= 2e+153) {
		tmp = sqrt((1.0 / x)) * 0.3333333333333333;
	} else {
		tmp = (sqrt(x) * y) * 3.0;
	}
	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 <= -50000000.0)
		tmp = Float64(fma(y, 3.0, -3.0) * sqrt(x));
	elseif (t_0 <= 2e+153)
		tmp = Float64(sqrt(Float64(1.0 / x)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(sqrt(x) * y) * 3.0);
	end
	return 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, -50000000.0], N[(N[(y * 3.0 + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+153], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $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))) < -5e7

   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 x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -5e7 < (*.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))) < 2e153

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{3}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{1}{3} \]

      4. lower-/.f64 82.4

         \[\leadsto \sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.3333333333333333 \]

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.3333333333333333} \]

   if 2e153 < (*.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 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -50000000:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - y), $MachinePrecision] * -3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $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. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. +-commutative N/A

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

   13. associate-+r- N/A

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

   14. +-commutative N/A

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

   15. associate-+r- N/A

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

   16. +-commutative N/A

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

   17. distribute-rgt-in N/A

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 61.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(y * 3.0 + -3.0), $MachinePrecision] * N[Sqrt[x], $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. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-sqrt.f64 59.9

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

5. Applied rewrites 59.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}} \]
6. Add Preprocessing

Alternative 7: 38.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{x} \cdot y\right) \cdot 3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* (sqrt x) y) 3.0))

C

double code(double x, double y) {
	return (sqrt(x) * y) * 3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sqrt(x) * y) * 3.0d0
end function

Java

public static double code(double x, double y) {
	return (Math.sqrt(x) * y) * 3.0;
}

Python

def code(x, y):
	return (math.sqrt(x) * y) * 3.0

Julia

function code(x, y)
	return Float64(Float64(sqrt(x) * y) * 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (sqrt(x) * y) * 3.0;
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $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. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 35.5

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

5. Applied rewrites 35.5%

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

6. Final simplification 35.5%

   \[\leadsto \left(\sqrt{x} \cdot y\right) \cdot 3 \]
7. 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 2024244 
(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)))