Bouland and Aaronson, Equation (25)

Percentage Accurate: 72.6% → 99.2%

Time: 8.4s

Alternatives: 10

Speedup: 5.0×

• 62.2% 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: 72.6% 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: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

3. Taylor expanded in b around 0

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

   1. associate-*r* N/A

      \[\leadsto \left(\left(\color{blue}{\left(2 \cdot {a}^{2}\right) \cdot {b}^{2}} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left({a}^{4} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   3. metadata-eval N/A

      \[\leadsto \left(\left({a}^{\color{blue}{\left(2 \cdot 2\right)}} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   4. pow-sqr N/A

      \[\leadsto \left(\left(\color{blue}{{a}^{2} \cdot {a}^{2}} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   5. *-commutative N/A

      \[\leadsto \left(\left({a}^{2} \cdot {a}^{2} + \color{blue}{\left({a}^{2} \cdot 2\right)} \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   6. associate-*r* N/A

      \[\leadsto \left(\left({a}^{2} \cdot {a}^{2} + \color{blue}{{a}^{2} \cdot \left(2 \cdot {b}^{2}\right)}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   7. distribute-lft-in N/A

      \[\leadsto \left(\color{blue}{{a}^{2} \cdot \left({a}^{2} + 2 \cdot {b}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   8. +-commutative N/A

      \[\leadsto \left({a}^{2} \cdot \color{blue}{\left(2 \cdot {b}^{2} + {a}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   9. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot {a}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   10. unpow2 N/A

       \[\leadsto \left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot \color{blue}{\left(a \cdot a\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   11. associate-*r* N/A

       \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   13. lower-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right)} \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   14. *-commutative N/A

       \[\leadsto \left(\left(\left(\color{blue}{{b}^{2} \cdot 2} + {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   15. lower-fma.f64 N/A

       \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left({b}^{2}, 2, {a}^{2}\right)} \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   16. unpow2 N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, 2, {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   17. lower-*.f64 N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, 2, {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   18. unpow2 N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(b \cdot b, 2, \color{blue}{a \cdot a}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   19. lower-*.f64 53.9

       \[\leadsto \left(\left(\mathsf{fma}\left(b \cdot b, 2, \color{blue}{a \cdot a}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

5. Applied rewrites 53.9%

   \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

6. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 85.9

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

8. Applied rewrites 85.9%

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

9. Taylor expanded in a around 0

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

    1. *-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-fma.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 N/A

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

    12. unpow2 N/A

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

    13. lower-*.f64 N/A

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

    14. lower-pow.f64 99.6

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

11. Applied rewrites 99.6%

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

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(2, b \cdot b, a \cdot a\right) \cdot a\right) \cdot a - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} + 4 \cdot \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+103)
   (fma (* b b) 4.0 (- (* (* (fma 2.0 (* b b) (* a a)) a) a) 1.0))
   (- (+ (pow (fma b b (* a a)) 2.0) (* 4.0 (* (fma a a a) a))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+103) {
		tmp = fma((b * b), 4.0, (((fma(2.0, (b * b), (a * a)) * a) * a) - 1.0));
	} else {
		tmp = (pow(fma(b, b, (a * a)), 2.0) + (4.0 * (fma(a, a, a) * a))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+103)
		tmp = fma(Float64(b * b), 4.0, Float64(Float64(Float64(fma(2.0, Float64(b * b), Float64(a * a)) * a) * a) - 1.0));
	else
		tmp = Float64(Float64((fma(b, b, Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(fma(a, a, a) * a))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+103], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(2.0 * N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e103

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(2 \cdot {a}^{2}\right) \cdot {b}^{2}} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({a}^{4} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      3. metadata-eval N/A

         \[\leadsto \left(\left({a}^{\color{blue}{\left(2 \cdot 2\right)}} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      4. pow-sqr N/A

         \[\leadsto \left(\left(\color{blue}{{a}^{2} \cdot {a}^{2}} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      5. *-commutative N/A

         \[\leadsto \left(\left({a}^{2} \cdot {a}^{2} + \color{blue}{\left({a}^{2} \cdot 2\right)} \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({a}^{2} \cdot {a}^{2} + \color{blue}{{a}^{2} \cdot \left(2 \cdot {b}^{2}\right)}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      7. distribute-lft-in N/A

         \[\leadsto \left(\color{blue}{{a}^{2} \cdot \left({a}^{2} + 2 \cdot {b}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      8. +-commutative N/A

         \[\leadsto \left({a}^{2} \cdot \color{blue}{\left(2 \cdot {b}^{2} + {a}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot {a}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      10. unpow2 N/A

          \[\leadsto \left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot \color{blue}{\left(a \cdot a\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      11. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right)} \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\left(\color{blue}{{b}^{2} \cdot 2} + {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left({b}^{2}, 2, {a}^{2}\right)} \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      16. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, 2, {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, 2, {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      18. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(b \cdot b, 2, \color{blue}{a \cdot a}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      19. lower-*.f64 0.0

          \[\leadsto \left(\left(\mathsf{fma}\left(b \cdot b, 2, \color{blue}{a \cdot a}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   5. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 100.0%

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

   if -2e103 < a

   1. Initial program 86.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lft-mult-inverse N/A

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

      7. *-lft-identity N/A

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

      8. unpow2 N/A

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

      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 98.9

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

   5. Applied rewrites 98.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 98.9

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

   7. Applied rewrites 98.9%

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

Alternative 3: 97.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -950000 \lor \neg \left(a \leq 6.5 \cdot 10^{-13}\right):\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(2, b \cdot b, a \cdot a\right) \cdot a\right) \cdot a - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -950000.0) (not (<= a 6.5e-13)))
   (fma (* b b) 4.0 (- (* (* (fma 2.0 (* b b) (* a a)) a) a) 1.0))
   (- (* (* (fma b b (fma -12.0 a 4.0)) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -950000.0) || !(a <= 6.5e-13)) {
		tmp = fma((b * b), 4.0, (((fma(2.0, (b * b), (a * a)) * a) * a) - 1.0));
	} else {
		tmp = ((fma(b, b, fma(-12.0, a, 4.0)) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -950000.0) || !(a <= 6.5e-13))
		tmp = fma(Float64(b * b), 4.0, Float64(Float64(Float64(fma(2.0, Float64(b * b), Float64(a * a)) * a) * a) - 1.0));
	else
		tmp = Float64(Float64(Float64(fma(b, b, fma(-12.0, a, 4.0)) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -950000.0], N[Not[LessEqual[a, 6.5e-13]], $MachinePrecision]], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(2.0 * N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.5e5 or 6.49999999999999957e-13 < a

   1. Initial program 33.5%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(2 \cdot {a}^{2}\right) \cdot {b}^{2}} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({a}^{4} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      3. metadata-eval N/A

         \[\leadsto \left(\left({a}^{\color{blue}{\left(2 \cdot 2\right)}} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      4. pow-sqr N/A

         \[\leadsto \left(\left(\color{blue}{{a}^{2} \cdot {a}^{2}} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      5. *-commutative N/A

         \[\leadsto \left(\left({a}^{2} \cdot {a}^{2} + \color{blue}{\left({a}^{2} \cdot 2\right)} \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({a}^{2} \cdot {a}^{2} + \color{blue}{{a}^{2} \cdot \left(2 \cdot {b}^{2}\right)}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      7. distribute-lft-in N/A

         \[\leadsto \left(\color{blue}{{a}^{2} \cdot \left({a}^{2} + 2 \cdot {b}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      8. +-commutative N/A

         \[\leadsto \left({a}^{2} \cdot \color{blue}{\left(2 \cdot {b}^{2} + {a}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot {a}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      10. unpow2 N/A

          \[\leadsto \left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot \color{blue}{\left(a \cdot a\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      11. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right)} \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\left(\color{blue}{{b}^{2} \cdot 2} + {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left({b}^{2}, 2, {a}^{2}\right)} \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      16. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, 2, {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, 2, {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      18. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(b \cdot b, 2, \color{blue}{a \cdot a}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      19. lower-*.f64 30.0

          \[\leadsto \left(\left(\mathsf{fma}\left(b \cdot b, 2, \color{blue}{a \cdot a}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   5. Applied rewrites 30.0%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 95.8

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

   8. Applied rewrites 95.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 95.8%

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

   if -9.5e5 < a < 6.49999999999999957e-13

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right)} - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b\right) \cdot b - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -950000 \lor \neg \left(a \leq 6.5 \cdot 10^{-13}\right):\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(2, b \cdot b, a \cdot a\right) \cdot a\right) \cdot a - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(2, b \cdot b, a \cdot a\right) \cdot a\right) \cdot a - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, \left(a \cdot a\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5000000000.0)
   (fma (* b b) 4.0 (- (* (* (fma 2.0 (* b b) (* a a)) a) a) 1.0))
   (-
    (fma (* (fma a (fma 2.0 a -12.0) (fma b b 4.0)) b) b (* (* a a) 4.0))
    1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5000000000.0) {
		tmp = fma((b * b), 4.0, (((fma(2.0, (b * b), (a * a)) * a) * a) - 1.0));
	} else {
		tmp = fma((fma(a, fma(2.0, a, -12.0), fma(b, b, 4.0)) * b), b, ((a * a) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5000000000.0)
		tmp = fma(Float64(b * b), 4.0, Float64(Float64(Float64(fma(2.0, Float64(b * b), Float64(a * a)) * a) * a) - 1.0));
	else
		tmp = Float64(fma(Float64(fma(a, fma(2.0, a, -12.0), fma(b, b, 4.0)) * b), b, Float64(Float64(a * a) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5000000000.0], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(2.0 * N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * N[(2.0 * a + -12.0), $MachinePrecision] + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5e9

   1. Initial program 70.8%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(2 \cdot {a}^{2}\right) \cdot {b}^{2}} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({a}^{4} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      3. metadata-eval N/A

         \[\leadsto \left(\left({a}^{\color{blue}{\left(2 \cdot 2\right)}} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      4. pow-sqr N/A

         \[\leadsto \left(\left(\color{blue}{{a}^{2} \cdot {a}^{2}} + \left(2 \cdot {a}^{2}\right) \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      5. *-commutative N/A

         \[\leadsto \left(\left({a}^{2} \cdot {a}^{2} + \color{blue}{\left({a}^{2} \cdot 2\right)} \cdot {b}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left(\left({a}^{2} \cdot {a}^{2} + \color{blue}{{a}^{2} \cdot \left(2 \cdot {b}^{2}\right)}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      7. distribute-lft-in N/A

         \[\leadsto \left(\color{blue}{{a}^{2} \cdot \left({a}^{2} + 2 \cdot {b}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      8. +-commutative N/A

         \[\leadsto \left({a}^{2} \cdot \color{blue}{\left(2 \cdot {b}^{2} + {a}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot {a}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      10. unpow2 N/A

          \[\leadsto \left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot \color{blue}{\left(a \cdot a\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      11. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 \cdot {b}^{2} + {a}^{2}\right) \cdot a\right)} \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\left(\color{blue}{{b}^{2} \cdot 2} + {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left({b}^{2}, 2, {a}^{2}\right)} \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      16. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, 2, {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{b \cdot b}, 2, {a}^{2}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      18. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(b \cdot b, 2, \color{blue}{a \cdot a}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

      19. lower-*.f64 60.2

          \[\leadsto \left(\left(\mathsf{fma}\left(b \cdot b, 2, \color{blue}{a \cdot a}\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   5. Applied rewrites 60.2%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.7

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

   8. Applied rewrites 88.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 88.7%

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

   if 5e9 < b

   1. Initial program 56.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   4. 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-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, \left(a \cdot a\right) \cdot 4\right)} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+60} \lor \neg \left(a \leq 4.6 \cdot 10^{+47}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.25e+60) (not (<= a 4.6e+47)))
   (- (* (* (fma (* b b) 2.0 4.0) a) a) 1.0)
   (- (* (* (fma b b (fma -12.0 a 4.0)) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.25e+60) || !(a <= 4.6e+47)) {
		tmp = ((fma((b * b), 2.0, 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, fma(-12.0, a, 4.0)) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.25e+60) || !(a <= 4.6e+47))
		tmp = Float64(Float64(Float64(fma(Float64(b * b), 2.0, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, fma(-12.0, a, 4.0)) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.25e+60], N[Not[LessEqual[a, 4.6e+47]], $MachinePrecision]], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.24999999999999994e60 or 4.5999999999999997e47 < a

   1. Initial program 23.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   4. 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-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 78.2%

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

   if -1.24999999999999994e60 < a < 4.5999999999999997e47

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 93.0

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

   5. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right)} - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b\right) \cdot b - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+60} \lor \neg \left(a \leq 4.6 \cdot 10^{+47}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+60} \lor \neg \left(a \leq 4.6 \cdot 10^{+47}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.25e+60) (not (<= a 4.6e+47)))
   (- (* (* (fma (* b b) 2.0 4.0) a) a) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.25e+60) || !(a <= 4.6e+47)) {
		tmp = ((fma((b * b), 2.0, 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.25e+60) || !(a <= 4.6e+47))
		tmp = Float64(Float64(Float64(fma(Float64(b * b), 2.0, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.25e+60], N[Not[LessEqual[a, 4.6e+47]], $MachinePrecision]], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.24999999999999994e60 or 4.5999999999999997e47 < a

   1. Initial program 23.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   4. 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-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 78.2%

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

   if -1.24999999999999994e60 < a < 4.5999999999999997e47

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\left(4 \cdot b\right) \cdot b + {b}^{\color{blue}{3}} \cdot b\right) - 1 \]

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

          \[\leadsto \left({b}^{2} \cdot 4 + \color{blue}{{b}^{3}} \cdot b\right) - 1 \]

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 93.1

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

   5. Applied rewrites 93.1%

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

   7. Applied rewrites 93.0%

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

   9. Applied rewrites 93.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+60} \lor \neg \left(a \leq 4.6 \cdot 10^{+47}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{+136} \lor \neg \left(a \leq 5.4 \cdot 10^{+153}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4.4e+136) (not (<= a 5.4e+153)))
   (- (* (* a a) 4.0) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -4.4e+136) || !(a <= 5.4e+153)) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -4.4e+136) || !(a <= 5.4e+153))
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -4.4e+136], N[Not[LessEqual[a, 5.4e+153]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.3999999999999999e136 or 5.4000000000000001e153 < a

   1. Initial program 13.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   4. 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-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 93.5%

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

   if -4.3999999999999999e136 < a < 5.4000000000000001e153

   1. Initial program 86.6%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\left(4 \cdot b\right) \cdot b + {b}^{\color{blue}{3}} \cdot b\right) - 1 \]

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

          \[\leadsto \left({b}^{2} \cdot 4 + \color{blue}{{b}^{3}} \cdot b\right) - 1 \]

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 83.1

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 83.0%

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

   9. Applied rewrites 83.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{+136} \lor \neg \left(a \leq 5.4 \cdot 10^{+153}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{+136} \lor \neg \left(a \leq 5.4 \cdot 10^{+153}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4.4e+136) (not (<= a 5.4e+153)))
   (- (* (* a a) 4.0) 1.0)
   (- (* (* b b) (fma b b 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -4.4e+136) || !(a <= 5.4e+153)) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * fma(b, b, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -4.4e+136) || !(a <= 5.4e+153))
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -4.4e+136], N[Not[LessEqual[a, 5.4e+153]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.3999999999999999e136 or 5.4000000000000001e153 < a

   1. Initial program 13.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   4. 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-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 93.5%

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

   if -4.3999999999999999e136 < a < 5.4000000000000001e153

   1. Initial program 86.6%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\left(4 \cdot b\right) \cdot b + {b}^{\color{blue}{3}} \cdot b\right) - 1 \]

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

          \[\leadsto \left({b}^{2} \cdot 4 + \color{blue}{{b}^{3}} \cdot b\right) - 1 \]

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 83.1

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 83.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{+136} \lor \neg \left(a \leq 5.4 \cdot 10^{+153}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+150}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.8e+150) (- (* (* a a) 4.0) 1.0) (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.8e+150) {
		tmp = ((a * a) * 4.0) - 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 <= 5.8d+150) then
        tmp = ((a * a) * 4.0d0) - 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 <= 5.8e+150) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.8e+150:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.8e+150)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 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 <= 5.8e+150)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.8e+150], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $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 b < 5.80000000000000022e150

   1. Initial program 69.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + \left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right) + {b}^{4}\right)\right)} - 1 \]
   4. 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-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 56.8%

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

   if 5.80000000000000022e150 < b

   1. Initial program 54.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\left(4 \cdot b\right) \cdot b + {b}^{\color{blue}{3}} \cdot b\right) - 1 \]

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

          \[\leadsto \left({b}^{2} \cdot 4 + \color{blue}{{b}^{3}} \cdot b\right) - 1 \]

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. 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 97.2%

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

Alternative 10: 50.6% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

3. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. pow-plus N/A

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

   8. metadata-eval N/A

      \[\leadsto \left(\left(4 \cdot b\right) \cdot b + {b}^{\color{blue}{3}} \cdot b\right) - 1 \]

   9. cube-unmult N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. cube-unmult N/A

       \[\leadsto \left({b}^{2} \cdot 4 + \color{blue}{{b}^{3}} \cdot b\right) - 1 \]

   16. metadata-eval N/A

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

   17. pow-plus N/A

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

   18. associate-*r* N/A

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

   19. unpow2 N/A

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

   20. pow-sqr N/A

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

   21. metadata-eval N/A

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

   22. lower-fma.f64 N/A

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

   23. unpow2 N/A

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

   24. lower-*.f64 N/A

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

   25. lower-pow.f64 74.1

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

5. Applied rewrites 74.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 53.2%

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

Reproduce

herbie shell --seed 2024343 
(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))