Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.1% → 98.2%

Time: 7.7s

Alternatives: 8

Speedup: 5.3×

• 60.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4 + a, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-11)
   (fma (* a a) (fma (+ 4.0 a) a 4.0) -1.0)
   (- (fma (* (fma b b 4.0) b) b (* (* (fma (* b b) 2.0 4.0) a) a)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-11) {
		tmp = fma((a * a), fma((4.0 + a), a, 4.0), -1.0);
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, ((fma((b * b), 2.0, 4.0) * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-11)
		tmp = fma(Float64(a * a), fma(Float64(4.0 + a), a, 4.0), -1.0);
	else
		tmp = Float64(fma(Float64(fma(b, b, 4.0) * b), b, Float64(Float64(fma(Float64(b * b), 2.0, 4.0) * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-11], N[(N[(a * a), $MachinePrecision] * N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.00000000000000018e-11

   1. Initial program 80.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(1 + \frac{1}{a}\right)\right)}\right) - 1 \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      2. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(1 + \frac{1}{a}\right)\right)\right)}\right) - 1 \]

      4. distribute-lft-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(a \cdot 1 + a \cdot \frac{1}{a}\right)}\right)\right) - 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{a} + a \cdot \frac{1}{a}\right)\right)\right) - 1 \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a + \color{blue}{1}\right)\right)\right) - 1 \]

      7. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(a + 1\right)\right)\right) - 1 \]

      8. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(1 + a\right)}\right)\right) - 1 \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 + a\right)\right)\right)}\right) - 1 \]

      10. *-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      12. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot \color{blue}{\left(a + 1\right)}\right) \cdot a\right)\right) - 1 \]

      13. distribute-lft-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a + a \cdot 1\right)} \cdot a\right)\right) - 1 \]

      14. *-rgt-identity N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a + \color{blue}{a}\right) \cdot a\right)\right) - 1 \]

      15. lower-fma.f64 80.8

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\mathsf{fma}\left(a, a, a\right)} \cdot a\right)\right) - 1 \]

   5. Applied rewrites 80.8%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)}\right) - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot {a}^{\color{blue}{2}}\right) - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 97.7%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \color{blue}{a}\right)\right) - 1 \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right) - 1} \]

   11. Applied rewrites 99.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4 + a, a, 4\right), -1\right)} \]

   if 5.00000000000000018e-11 < (*.f64 b b)

   1. Initial program 68.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(1 + \frac{1}{a}\right)\right)}\right) - 1 \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      2. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(1 + \frac{1}{a}\right)\right)\right)}\right) - 1 \]

      4. distribute-lft-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(a \cdot 1 + a \cdot \frac{1}{a}\right)}\right)\right) - 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{a} + a \cdot \frac{1}{a}\right)\right)\right) - 1 \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a + \color{blue}{1}\right)\right)\right) - 1 \]

      7. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(a + 1\right)\right)\right) - 1 \]

      8. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(1 + a\right)}\right)\right) - 1 \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 + a\right)\right)\right)}\right) - 1 \]

      10. *-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      12. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot \color{blue}{\left(a + 1\right)}\right) \cdot a\right)\right) - 1 \]

      13. distribute-lft-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a + a \cdot 1\right)} \cdot a\right)\right) - 1 \]

      14. *-rgt-identity N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a + \color{blue}{a}\right) \cdot a\right)\right) - 1 \]

      15. lower-fma.f64 82.1

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\mathsf{fma}\left(a, a, a\right)} \cdot a\right)\right) - 1 \]

   5. Applied rewrites 82.1%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)}\right) - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot {a}^{\color{blue}{2}}\right) - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \color{blue}{a}\right)\right) - 1 \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{\left({b}^{4} + a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right)\right)}\right) - 1 \]

       2. distribute-rgt-in N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \color{blue}{\left(\left(-12 \cdot {b}^{2}\right) \cdot a + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)}\right)\right) - 1 \]

       3. associate-*r* N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \left(\color{blue}{-12 \cdot \left({b}^{2} \cdot a\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right)\right) - 1 \]

       4. *-commutative N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \left(-12 \cdot \color{blue}{\left(a \cdot {b}^{2}\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right)\right) - 1 \]

       5. associate-+r+ N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{\left(\left({b}^{4} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)}\right) - 1 \]

       6. metadata-eval N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       7. pow-sqr N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left(\color{blue}{{b}^{2} \cdot {b}^{2}} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       8. associate-*r* N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left({b}^{2} \cdot {b}^{2} + \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}}\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       9. distribute-rgt-in N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\color{blue}{{b}^{2} \cdot \left({b}^{2} + -12 \cdot a\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       10. +-commutative N/A

           \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{2} \cdot \color{blue}{\left(-12 \cdot a + {b}^{2}\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

   11. Applied rewrites 96.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-12, a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{fma}\left(b \cdot \left(4 + {b}^{2}\right), b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 97.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* a a))) 1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (a * a))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (a * a))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (a * a))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (a * a))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(a * a))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (a * a))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 74.5%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(1 + \frac{1}{a}\right)\right)}\right) - 1 \]
4. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

   2. unpow2 N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

   3. associate-*l* N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(1 + \frac{1}{a}\right)\right)\right)}\right) - 1 \]

   4. distribute-lft-in N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(a \cdot 1 + a \cdot \frac{1}{a}\right)}\right)\right) - 1 \]

   5. *-rgt-identity N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{a} + a \cdot \frac{1}{a}\right)\right)\right) - 1 \]

   6. rgt-mult-inverse N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a + \color{blue}{1}\right)\right)\right) - 1 \]

   7. unpow2 N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(a + 1\right)\right)\right) - 1 \]

   8. +-commutative N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(1 + a\right)}\right)\right) - 1 \]

   9. associate-*r* N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 + a\right)\right)\right)}\right) - 1 \]

   10. *-commutative N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

   11. lower-*.f64 N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

   12. +-commutative N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot \color{blue}{\left(a + 1\right)}\right) \cdot a\right)\right) - 1 \]

   13. distribute-lft-in N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a + a \cdot 1\right)} \cdot a\right)\right) - 1 \]

   14. *-rgt-identity N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a + \color{blue}{a}\right) \cdot a\right)\right) - 1 \]

   15. lower-fma.f64 81.4

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\mathsf{fma}\left(a, a, a\right)} \cdot a\right)\right) - 1 \]

5. Applied rewrites 81.4%

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)}\right) - 1 \]

6. Taylor expanded in a around 0

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot {a}^{\color{blue}{2}}\right) - 1 \]
7. Step-by-step derivation

8. Applied rewrites 98.7%

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \color{blue}{a}\right)\right) - 1 \]
9. Add Preprocessing

Alternative 3: 94.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+24)
   (- (* (fma (+ 4.0 a) a (fma (* b b) 2.0 4.0)) (* a a)) 1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+24) {
		tmp = (fma((4.0 + a), a, fma((b * b), 2.0, 4.0)) * (a * a)) - 1.0;
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+24)
		tmp = Float64(Float64(fma(Float64(4.0 + a), a, fma(Float64(b * b), 2.0, 4.0)) * Float64(a * a)) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+24], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e24

   1. Initial program 80.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right)} - 1 \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right) \cdot {a}^{4}} - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right) \cdot {a}^{4}} - 1 \]

   5. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\left(1 - \frac{-4 - \frac{\mathsf{fma}\left(b \cdot b, 2, 4\right)}{a}}{a}\right) \cdot {a}^{4}} - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 75.3%

      \[\leadsto \left(\left(1 - \frac{-4 - \frac{\mathsf{fma}\left(2, b \cdot b, 4\right)}{a}}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} - 1 \]

   8. Taylor expanded in a around 0

      \[\leadsto \left(4 + \left(2 \cdot {b}^{2} + a \cdot \left(4 + a\right)\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.1%

       \[\leadsto \mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) - 1 \]

   if 2e24 < (*.f64 b b)

   1. Initial program 68.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(1 + \frac{1}{a}\right)\right)}\right) - 1 \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      2. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(1 + \frac{1}{a}\right)\right)\right)}\right) - 1 \]

      4. distribute-lft-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(a \cdot 1 + a \cdot \frac{1}{a}\right)}\right)\right) - 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{a} + a \cdot \frac{1}{a}\right)\right)\right) - 1 \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a + \color{blue}{1}\right)\right)\right) - 1 \]

      7. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(a + 1\right)\right)\right) - 1 \]

      8. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(1 + a\right)}\right)\right) - 1 \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 + a\right)\right)\right)}\right) - 1 \]

      10. *-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      12. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot \color{blue}{\left(a + 1\right)}\right) \cdot a\right)\right) - 1 \]

      13. distribute-lft-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a + a \cdot 1\right)} \cdot a\right)\right) - 1 \]

      14. *-rgt-identity N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a + \color{blue}{a}\right) \cdot a\right)\right) - 1 \]

      15. lower-fma.f64 82.4

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\mathsf{fma}\left(a, a, a\right)} \cdot a\right)\right) - 1 \]

   5. Applied rewrites 82.4%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)}\right) - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot {a}^{\color{blue}{2}}\right) - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \color{blue}{a}\right)\right) - 1 \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{\left({b}^{4} + a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right)\right)}\right) - 1 \]

       2. distribute-rgt-in N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \color{blue}{\left(\left(-12 \cdot {b}^{2}\right) \cdot a + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)}\right)\right) - 1 \]

       3. associate-*r* N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \left(\color{blue}{-12 \cdot \left({b}^{2} \cdot a\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right)\right) - 1 \]

       4. *-commutative N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \left(-12 \cdot \color{blue}{\left(a \cdot {b}^{2}\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right)\right) - 1 \]

       5. associate-+r+ N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{\left(\left({b}^{4} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)}\right) - 1 \]

       6. metadata-eval N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       7. pow-sqr N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left(\color{blue}{{b}^{2} \cdot {b}^{2}} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       8. associate-*r* N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left({b}^{2} \cdot {b}^{2} + \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}}\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       9. distribute-rgt-in N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\color{blue}{{b}^{2} \cdot \left({b}^{2} + -12 \cdot a\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       10. +-commutative N/A

           \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{2} \cdot \color{blue}{\left(-12 \cdot a + {b}^{2}\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

   11. Applied rewrites 96.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-12, a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   12. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   13. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto \left(4 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \color{blue}{{b}^{2} \cdot \left(4 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(4 + {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(4 + {b}^{2}\right)\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \color{blue}{\left(b \cdot \left(4 + {b}^{2}\right)\right) \cdot b} + \left(\mathsf{neg}\left(1\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \left(b \cdot \left(4 + {b}^{2}\right)\right) \cdot b + \color{blue}{-1} \]

       9. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \left(4 + {b}^{2}\right), b, -1\right)} \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + {b}^{2}\right) \cdot b}, b, -1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + {b}^{2}\right) \cdot b}, b, -1\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({b}^{2} + 4\right)} \cdot b, b, -1\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{b \cdot b} + 4\right) \cdot b, b, -1\right) \]

       14. lower-fma.f64 93.1

           \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, b, 4\right)} \cdot b, b, -1\right) \]

   14. Applied rewrites 93.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4 + a, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+24)
   (fma (* a a) (fma (+ 4.0 a) a 4.0) -1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+24) {
		tmp = fma((a * a), fma((4.0 + a), a, 4.0), -1.0);
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+24)
		tmp = fma(Float64(a * a), fma(Float64(4.0 + a), a, 4.0), -1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+24], N[(N[(a * a), $MachinePrecision] * N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e24

   1. Initial program 80.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(1 + \frac{1}{a}\right)\right)}\right) - 1 \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      2. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(1 + \frac{1}{a}\right)\right)\right)}\right) - 1 \]

      4. distribute-lft-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(a \cdot 1 + a \cdot \frac{1}{a}\right)}\right)\right) - 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{a} + a \cdot \frac{1}{a}\right)\right)\right) - 1 \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a + \color{blue}{1}\right)\right)\right) - 1 \]

      7. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(a + 1\right)\right)\right) - 1 \]

      8. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(1 + a\right)}\right)\right) - 1 \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 + a\right)\right)\right)}\right) - 1 \]

      10. *-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      12. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot \color{blue}{\left(a + 1\right)}\right) \cdot a\right)\right) - 1 \]

      13. distribute-lft-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a + a \cdot 1\right)} \cdot a\right)\right) - 1 \]

      14. *-rgt-identity N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a + \color{blue}{a}\right) \cdot a\right)\right) - 1 \]

      15. lower-fma.f64 80.5

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\mathsf{fma}\left(a, a, a\right)} \cdot a\right)\right) - 1 \]

   5. Applied rewrites 80.5%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)}\right) - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot {a}^{\color{blue}{2}}\right) - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 97.6%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \color{blue}{a}\right)\right) - 1 \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right) - 1} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4 + a, a, 4\right), -1\right)} \]

   if 2e24 < (*.f64 b b)

   1. Initial program 68.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(1 + \frac{1}{a}\right)\right)}\right) - 1 \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      2. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(1 + \frac{1}{a}\right)\right)\right)}\right) - 1 \]

      4. distribute-lft-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(a \cdot 1 + a \cdot \frac{1}{a}\right)}\right)\right) - 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{a} + a \cdot \frac{1}{a}\right)\right)\right) - 1 \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a + \color{blue}{1}\right)\right)\right) - 1 \]

      7. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(a + 1\right)\right)\right) - 1 \]

      8. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(1 + a\right)}\right)\right) - 1 \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 + a\right)\right)\right)}\right) - 1 \]

      10. *-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      12. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot \color{blue}{\left(a + 1\right)}\right) \cdot a\right)\right) - 1 \]

      13. distribute-lft-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a + a \cdot 1\right)} \cdot a\right)\right) - 1 \]

      14. *-rgt-identity N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a + \color{blue}{a}\right) \cdot a\right)\right) - 1 \]

      15. lower-fma.f64 82.4

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\mathsf{fma}\left(a, a, a\right)} \cdot a\right)\right) - 1 \]

   5. Applied rewrites 82.4%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)}\right) - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot {a}^{\color{blue}{2}}\right) - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \color{blue}{a}\right)\right) - 1 \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{\left({b}^{4} + a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right)\right)}\right) - 1 \]

       2. distribute-rgt-in N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \color{blue}{\left(\left(-12 \cdot {b}^{2}\right) \cdot a + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)}\right)\right) - 1 \]

       3. associate-*r* N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \left(\color{blue}{-12 \cdot \left({b}^{2} \cdot a\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right)\right) - 1 \]

       4. *-commutative N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \left(-12 \cdot \color{blue}{\left(a \cdot {b}^{2}\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right)\right) - 1 \]

       5. associate-+r+ N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{\left(\left({b}^{4} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)}\right) - 1 \]

       6. metadata-eval N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       7. pow-sqr N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left(\color{blue}{{b}^{2} \cdot {b}^{2}} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       8. associate-*r* N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left({b}^{2} \cdot {b}^{2} + \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}}\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       9. distribute-rgt-in N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\color{blue}{{b}^{2} \cdot \left({b}^{2} + -12 \cdot a\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       10. +-commutative N/A

           \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{2} \cdot \color{blue}{\left(-12 \cdot a + {b}^{2}\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

   11. Applied rewrites 96.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-12, a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   12. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   13. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto \left(4 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \color{blue}{{b}^{2} \cdot \left(4 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(4 + {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(4 + {b}^{2}\right)\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \color{blue}{\left(b \cdot \left(4 + {b}^{2}\right)\right) \cdot b} + \left(\mathsf{neg}\left(1\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \left(b \cdot \left(4 + {b}^{2}\right)\right) \cdot b + \color{blue}{-1} \]

       9. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \left(4 + {b}^{2}\right), b, -1\right)} \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + {b}^{2}\right) \cdot b}, b, -1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + {b}^{2}\right) \cdot b}, b, -1\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({b}^{2} + 4\right)} \cdot b, b, -1\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{b \cdot b} + 4\right) \cdot b, b, -1\right) \]

       14. lower-fma.f64 93.1

           \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, b, 4\right)} \cdot b, b, -1\right) \]

   14. Applied rewrites 93.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{+153} \lor \neg \left(a \leq 5.2 \cdot 10^{+128}\right):\\ \;\;\;\;\left(4 \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -6.6e+153) (not (<= a 5.2e+128)))
   (- (* (* 4.0 a) a) 1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -6.6e+153) || !(a <= 5.2e+128)) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -6.6e+153) || !(a <= 5.2e+128))
		tmp = Float64(Float64(Float64(4.0 * a) * a) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -6.6e+153], N[Not[LessEqual[a, 5.2e+128]], $MachinePrecision]], N[(N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.59999999999999989e153 or 5.2e128 < a

   1. Initial program 30.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right)} - 1 \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right) \cdot {a}^{4}} - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right) \cdot {a}^{4}} - 1 \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 - \frac{-4 - \frac{\mathsf{fma}\left(b \cdot b, 2, 4\right)}{a}}{a}\right) \cdot {a}^{4}} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(4 + 2 \cdot {b}^{2}\right)} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 92.4%

      \[\leadsto \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot \color{blue}{a} - 1 \]

   9. Taylor expanded in b around 0

      \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 91.1%

       \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

   if -6.59999999999999989e153 < a < 5.2e128

   1. Initial program 91.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(1 + \frac{1}{a}\right)\right)}\right) - 1 \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      2. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(1 + \frac{1}{a}\right)\right)\right)}\right) - 1 \]

      4. distribute-lft-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(a \cdot 1 + a \cdot \frac{1}{a}\right)}\right)\right) - 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{a} + a \cdot \frac{1}{a}\right)\right)\right) - 1 \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a + \color{blue}{1}\right)\right)\right) - 1 \]

      7. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(a + 1\right)\right)\right) - 1 \]

      8. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(1 + a\right)}\right)\right) - 1 \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 + a\right)\right)\right)}\right) - 1 \]

      10. *-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      12. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot \color{blue}{\left(a + 1\right)}\right) \cdot a\right)\right) - 1 \]

      13. distribute-lft-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a + a \cdot 1\right)} \cdot a\right)\right) - 1 \]

      14. *-rgt-identity N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a + \color{blue}{a}\right) \cdot a\right)\right) - 1 \]

      15. lower-fma.f64 93.7

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\mathsf{fma}\left(a, a, a\right)} \cdot a\right)\right) - 1 \]

   5. Applied rewrites 93.7%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)}\right) - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot {a}^{\color{blue}{2}}\right) - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 98.2%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \color{blue}{a}\right)\right) - 1 \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{\left({b}^{4} + a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right)\right)}\right) - 1 \]

       2. distribute-rgt-in N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \color{blue}{\left(\left(-12 \cdot {b}^{2}\right) \cdot a + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)}\right)\right) - 1 \]

       3. associate-*r* N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \left(\color{blue}{-12 \cdot \left({b}^{2} \cdot a\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right)\right) - 1 \]

       4. *-commutative N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \left(-12 \cdot \color{blue}{\left(a \cdot {b}^{2}\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right)\right) - 1 \]

       5. associate-+r+ N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{\left(\left({b}^{4} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)}\right) - 1 \]

       6. metadata-eval N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       7. pow-sqr N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left(\color{blue}{{b}^{2} \cdot {b}^{2}} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       8. associate-*r* N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left({b}^{2} \cdot {b}^{2} + \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}}\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       9. distribute-rgt-in N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\color{blue}{{b}^{2} \cdot \left({b}^{2} + -12 \cdot a\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       10. +-commutative N/A

           \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{2} \cdot \color{blue}{\left(-12 \cdot a + {b}^{2}\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

   11. Applied rewrites 84.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-12, a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   12. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   13. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto \left(4 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \color{blue}{{b}^{2} \cdot \left(4 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(4 + {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(4 + {b}^{2}\right)\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \color{blue}{\left(b \cdot \left(4 + {b}^{2}\right)\right) \cdot b} + \left(\mathsf{neg}\left(1\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \left(b \cdot \left(4 + {b}^{2}\right)\right) \cdot b + \color{blue}{-1} \]

       9. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \left(4 + {b}^{2}\right), b, -1\right)} \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + {b}^{2}\right) \cdot b}, b, -1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + {b}^{2}\right) \cdot b}, b, -1\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({b}^{2} + 4\right)} \cdot b, b, -1\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{b \cdot b} + 4\right) \cdot b, b, -1\right) \]

       14. lower-fma.f64 82.4

           \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, b, 4\right)} \cdot b, b, -1\right) \]

   14. Applied rewrites 82.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{+153} \lor \neg \left(a \leq 5.2 \cdot 10^{+128}\right):\\ \;\;\;\;\left(4 \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+24)
   (- (* (* a a) (* a a)) 1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+24) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+24)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+24], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e24

   1. Initial program 80.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 96.8

         \[\leadsto \color{blue}{{a}^{4}} - 1 \]

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.8%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} - 1 \]

   if 2e24 < (*.f64 b b)

   1. Initial program 68.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(1 + \frac{1}{a}\right)\right)}\right) - 1 \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      2. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(1 + \frac{1}{a}\right)\right)\right) - 1 \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(1 + \frac{1}{a}\right)\right)\right)}\right) - 1 \]

      4. distribute-lft-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(a \cdot 1 + a \cdot \frac{1}{a}\right)}\right)\right) - 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{a} + a \cdot \frac{1}{a}\right)\right)\right) - 1 \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a + \color{blue}{1}\right)\right)\right) - 1 \]

      7. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(a + 1\right)\right)\right) - 1 \]

      8. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(1 + a\right)}\right)\right) - 1 \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 + a\right)\right)\right)}\right) - 1 \]

      10. *-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(a \cdot \left(1 + a\right)\right) \cdot a\right)}\right) - 1 \]

      12. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot \color{blue}{\left(a + 1\right)}\right) \cdot a\right)\right) - 1 \]

      13. distribute-lft-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(a \cdot a + a \cdot 1\right)} \cdot a\right)\right) - 1 \]

      14. *-rgt-identity N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a + \color{blue}{a}\right) \cdot a\right)\right) - 1 \]

      15. lower-fma.f64 82.4

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\mathsf{fma}\left(a, a, a\right)} \cdot a\right)\right) - 1 \]

   5. Applied rewrites 82.4%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)}\right) - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot {a}^{\color{blue}{2}}\right) - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \color{blue}{a}\right)\right) - 1 \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{\left({b}^{4} + a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right)\right)}\right) - 1 \]

       2. distribute-rgt-in N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \color{blue}{\left(\left(-12 \cdot {b}^{2}\right) \cdot a + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)}\right)\right) - 1 \]

       3. associate-*r* N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \left(\color{blue}{-12 \cdot \left({b}^{2} \cdot a\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right)\right) - 1 \]

       4. *-commutative N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{4} + \left(-12 \cdot \color{blue}{\left(a \cdot {b}^{2}\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right)\right) - 1 \]

       5. associate-+r+ N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{\left(\left({b}^{4} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)}\right) - 1 \]

       6. metadata-eval N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       7. pow-sqr N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left(\color{blue}{{b}^{2} \cdot {b}^{2}} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       8. associate-*r* N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\left({b}^{2} \cdot {b}^{2} + \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}}\right) + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       9. distribute-rgt-in N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \left(\color{blue}{{b}^{2} \cdot \left({b}^{2} + -12 \cdot a\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

       10. +-commutative N/A

           \[\leadsto \left(4 \cdot {b}^{2} + \left({b}^{2} \cdot \color{blue}{\left(-12 \cdot a + {b}^{2}\right)} + \left(a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) \cdot a\right)\right) - 1 \]

   11. Applied rewrites 96.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-12, a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right)} - 1 \]

   12. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   13. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto \left(4 \cdot {b}^{2} + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \left(4 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \color{blue}{{b}^{2} \cdot \left(4 + {b}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(4 + {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(4 + {b}^{2}\right)\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \color{blue}{\left(b \cdot \left(4 + {b}^{2}\right)\right) \cdot b} + \left(\mathsf{neg}\left(1\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \left(b \cdot \left(4 + {b}^{2}\right)\right) \cdot b + \color{blue}{-1} \]

       9. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \left(4 + {b}^{2}\right), b, -1\right)} \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + {b}^{2}\right) \cdot b}, b, -1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(4 + {b}^{2}\right) \cdot b}, b, -1\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({b}^{2} + 4\right)} \cdot b, b, -1\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{b \cdot b} + 4\right) \cdot b, b, -1\right) \]

       14. lower-fma.f64 93.1

           \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, b, 4\right)} \cdot b, b, -1\right) \]

   14. Applied rewrites 93.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2.9 \cdot 10^{+307}:\\ \;\;\;\;\left(4 \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2.9e+307) (- (* (* 4.0 a) a) 1.0) (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2.9e+307) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2.9d+307) then
        tmp = ((4.0d0 * a) * a) - 1.0d0
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2.9e+307) {
		tmp = ((4.0 * a) * a) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2.9e+307:
		tmp = ((4.0 * a) * a) - 1.0
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2.9e+307)
		tmp = Float64(Float64(Float64(4.0 * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2.9e+307)
		tmp = ((4.0 * a) * a) - 1.0;
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2.9e+307], N[(N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.89999999999999997e307

   1. Initial program 78.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right)} - 1 \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right) \cdot {a}^{4}} - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right) \cdot {a}^{4}} - 1 \]

   5. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\left(1 - \frac{-4 - \frac{\mathsf{fma}\left(b \cdot b, 2, 4\right)}{a}}{a}\right) \cdot {a}^{4}} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(4 + 2 \cdot {b}^{2}\right)} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 60.5%

      \[\leadsto \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot \color{blue}{a} - 1 \]

   9. Taylor expanded in b around 0

      \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 57.2%

       \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]

   if 2.89999999999999997e307 < (*.f64 b b)

   1. Initial program 62.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot 4} + {b}^{4}\right) - 1 \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({b}^{2}, 4, {b}^{4}\right)} - 1 \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, 4, {b}^{4}\right) - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, 4, {b}^{4}\right) - 1 \]

      5. lower-pow.f64 100.0

         \[\leadsto \mathsf{fma}\left(b \cdot b, 4, \color{blue}{{b}^{4}}\right) - 1 \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, 4, {b}^{4}\right)} - 1 \]

   6. Taylor expanded in b around 0

      \[\leadsto 4 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{4} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \left(4 \cdot a\right) \cdot a - 1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* 4.0 a) a) 1.0))

C

double code(double a, double b) {
	return ((4.0 * a) * a) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((4.0d0 * a) * a) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return ((4.0 * a) * a) - 1.0;
}

Python

def code(a, b):
	return ((4.0 * a) * a) - 1.0

Julia

function code(a, b)
	return Float64(Float64(Float64(4.0 * a) * a) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((4.0 * a) * a) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 74.5%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around -inf

   \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right)} - 1 \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right) \cdot {a}^{4}} - 1 \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \frac{4 + 2 \cdot {b}^{2}}{a} - 4}{a}\right) \cdot {a}^{4}} - 1 \]

5. Applied rewrites 62.4%

   \[\leadsto \color{blue}{\left(1 - \frac{-4 - \frac{\mathsf{fma}\left(b \cdot b, 2, 4\right)}{a}}{a}\right) \cdot {a}^{4}} - 1 \]

6. Taylor expanded in a around 0

   \[\leadsto {a}^{2} \cdot \color{blue}{\left(4 + 2 \cdot {b}^{2}\right)} - 1 \]
7. Step-by-step derivation

8. Applied rewrites 70.1%

   \[\leadsto \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot \color{blue}{a} - 1 \]

9. Taylor expanded in b around 0

   \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]
10. Step-by-step derivation

11. Applied rewrites 49.9%

    \[\leadsto \left(4 \cdot a\right) \cdot a - 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024299 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))