Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.6% → 99.9%

Time: 7.6s

Alternatives: 10

Speedup: 5.3×

• 61.1% 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: 73.6% 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: 99.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, -12\right), a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, \mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (fma
  (* (fma (fma 2.0 a -12.0) a (fma b b 4.0)) b)
  b
  (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)))

C

double code(double a, double b) {
	return fma((fma(fma(2.0, a, -12.0), a, fma(b, b, 4.0)) * b), b, fma((fma((4.0 + a), a, 4.0) * a), a, -1.0));
}

Julia

function code(a, b)
	return fma(Float64(fma(fma(2.0, a, -12.0), a, fma(b, b, 4.0)) * b), b, fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0))
end

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. *-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

   17. distribute-lft-neg-in N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f64 89.8

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 90.9%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 99.9%

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

Alternative 2: 98.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, -12\right), a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5.0)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (fma (* (fma (fma 2.0 a -12.0) a (fma b b 4.0)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5.0) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma(fma(2.0, a, -12.0), a, fma(b, b, 4.0)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5.0)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(fma(2.0, a, -12.0), a, fma(b, b, 4.0)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5.0], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(N[(2.0 * a + -12.0), $MachinePrecision] * a + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5

   1. Initial program 78.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 81.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 100.0%

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

   if 5 < (*.f64 b b)

   1. Initial program 68.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 99.9%

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

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, -12\right), a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.7%

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

Alternative 3: 99.8% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. *-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

   17. distribute-lft-neg-in N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f64 89.8

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 90.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 99.4%

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

Alternative 4: 93.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{if}\;a \leq -1.55 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)))
   (if (<= a -1.55e+35)
     t_0
     (if (<= a 2.05e+71) (fma (* (fma -12.0 a (fma b b 4.0)) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	double tmp;
	if (a <= -1.55e+35) {
		tmp = t_0;
	} else if (a <= 2.05e+71) {
		tmp = fma((fma(-12.0, a, fma(b, b, 4.0)) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0)
	tmp = 0.0
	if (a <= -1.55e+35)
		tmp = t_0;
	elseif (a <= 2.05e+71)
		tmp = fma(Float64(fma(-12.0, a, fma(b, b, 4.0)) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]}, If[LessEqual[a, -1.55e+35], t$95$0, If[LessEqual[a, 2.05e+71], N[(N[(N[(-12.0 * a + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.54999999999999993e35 or 2.0500000000000001e71 < a

   1. Initial program 40.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 78.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 100.0%

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

   if -1.54999999999999993e35 < a < 2.0500000000000001e71

   1. Initial program 95.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 94.8%

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

Alternative 5: 93.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{if}\;a \leq -1.55 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)))
   (if (<= a -1.55e+35)
     t_0
     (if (<= a 2.05e+71) (fma (* (fma b b 4.0) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	double tmp;
	if (a <= -1.55e+35) {
		tmp = t_0;
	} else if (a <= 2.05e+71) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0)
	tmp = 0.0
	if (a <= -1.55e+35)
		tmp = t_0;
	elseif (a <= 2.05e+71)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]}, If[LessEqual[a, -1.55e+35], t$95$0, If[LessEqual[a, 2.05e+71], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.54999999999999993e35 or 2.0500000000000001e71 < a

   1. Initial program 40.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 78.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 100.0%

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

   if -1.54999999999999993e35 < a < 2.0500000000000001e71

   1. Initial program 95.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 94.6

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

   8. Applied rewrites 94.6%

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

Alternative 6: 87.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(4, a, 4\right) \cdot a\right) \cdot a - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.8e+153)
   (- (* (* a a) 4.0) 1.0)
   (if (<= a 1.1e+99)
     (fma (* (fma b b 4.0) b) b -1.0)
     (- (* (* (fma 4.0 a 4.0) a) a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.8e+153) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (a <= 1.1e+99) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = ((fma(4.0, a, 4.0) * a) * a) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.8e+153)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	elseif (a <= 1.1e+99)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = Float64(Float64(Float64(fma(4.0, a, 4.0) * a) * a) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.8e+153], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 1.1e+99], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(4.0 * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.79999999999999985e153

   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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-pow.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.7%

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

   if -4.79999999999999985e153 < a < 1.09999999999999989e99

   1. Initial program 89.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 84.3

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

   8. Applied rewrites 84.3%

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

   if 1.09999999999999989e99 < a

   1. Initial program 66.7%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 96.0%

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

Alternative 7: 84.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{if}\;a \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* (* a a) 4.0) 1.0)))
   (if (<= a -4.8e+153)
     t_0
     (if (<= a 6.8e+153) (fma (* (fma b b 4.0) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = ((a * a) * 4.0) - 1.0;
	double tmp;
	if (a <= -4.8e+153) {
		tmp = t_0;
	} else if (a <= 6.8e+153) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * a) * 4.0) - 1.0)
	tmp = 0.0
	if (a <= -4.8e+153)
		tmp = t_0;
	elseif (a <= 6.8e+153)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -4.8e+153], t$95$0, If[LessEqual[a, 6.8e+153], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.79999999999999985e153 or 6.7999999999999995e153 < a

   1. Initial program 30.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-pow.f64 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.8%

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

   if -4.79999999999999985e153 < a < 6.7999999999999995e153

   1. Initial program 89.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 81.4

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

   8. Applied rewrites 81.4%

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

Alternative 8: 69.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+291) (- (* (* a a) 4.0) 1.0) (* (* b b) 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+291) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (b * b) * 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+291) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (b * b) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+291) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (b * b) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+291:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (b * b) * 4.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+291)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(b * b) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+291)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (b * b) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+291], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e291

   1. Initial program 75.7%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-pow.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 60.1%

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

   if 5.0000000000000001e291 < (*.f64 b b)

   1. Initial program 65.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

   8. Applied rewrites 97.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 97.3%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 97.3%

       \[\leadsto \left(b \cdot b\right) \cdot 4 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.7% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. *-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

   17. distribute-lft-neg-in N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f64 89.8

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate--l+ N/A

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

8. Applied rewrites 75.2%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 47.1%

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

Alternative 10: 28.1% accurate, 14.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. *-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

   17. distribute-lft-neg-in N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f64 89.8

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate--l+ N/A

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

8. Applied rewrites 75.2%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 47.1%

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

12. Taylor expanded in b around inf

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

14. Applied rewrites 26.0%

    \[\leadsto \left(b \cdot b\right) \cdot 4 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(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))