Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.9% → 99.9%

Time: 8.5s

Alternatives: 14

Speedup: 5.3×

• 59.9% 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: 73.9% 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, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a + 4, 4\right)\\ \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(a, 2, -12\right), a, 4\right)\right), b \cdot b, \mathsf{fma}\left(a \cdot a, t\_0, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a (+ a 4.0) 4.0)))
   (if (<= (* b b) 2e-49)
     (fma (* t_0 a) a -1.0)
     (fma
      (fma b b (fma (fma a 2.0 -12.0) a 4.0))
      (* b b)
      (fma (* a a) t_0 -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, (a + 4.0), 4.0);
	double tmp;
	if ((b * b) <= 2e-49) {
		tmp = fma((t_0 * a), a, -1.0);
	} else {
		tmp = fma(fma(b, b, fma(fma(a, 2.0, -12.0), a, 4.0)), (b * b), fma((a * a), t_0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, Float64(a + 4.0), 4.0)
	tmp = 0.0
	if (Float64(b * b) <= 2e-49)
		tmp = fma(Float64(t_0 * a), a, -1.0);
	else
		tmp = fma(fma(b, b, fma(fma(a, 2.0, -12.0), a, 4.0)), Float64(b * b), fma(Float64(a * a), t_0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 2e-49], N[(N[(t$95$0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * b + N[(N[(a * 2.0 + -12.0), $MachinePrecision] * a + 4.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.99999999999999987e-49

   1. Initial program 83.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 83.8

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

   if 1.99999999999999987e-49 < (*.f64 b b)

   1. Initial program 67.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 70.7

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. *-lft-identity N/A

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

   11. lower-fma.f64 76.5

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 99.9%

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

Alternative 3: 98.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a + 4, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, -12\right), a, b \cdot b\right) \cdot b + 4 \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e-18)
   (fma (* (fma a (+ a 4.0) 4.0) a) a -1.0)
   (fma (+ (* (fma (fma 2.0 a -12.0) a (* b b)) b) (* 4.0 b)) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e-18) {
		tmp = fma((fma(a, (a + 4.0), 4.0) * a), a, -1.0);
	} else {
		tmp = fma(((fma(fma(2.0, a, -12.0), a, (b * b)) * b) + (4.0 * b)), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e-18)
		tmp = fma(Float64(fma(a, Float64(a + 4.0), 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(Float64(fma(fma(2.0, a, -12.0), a, Float64(b * b)) * b) + Float64(4.0 * b)), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e-18], N[(N[(N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * a + -12.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] + N[(4.0 * b), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e-18

   1. Initial program 82.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 82.3

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

   if 4.0000000000000003e-18 < (*.f64 b b)

   1. Initial program 67.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 70.9

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.0%

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

   12. Applied rewrites 97.0%

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

Alternative 4: 98.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a + 4, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, -12 + 2 \cdot a, 4\right)\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e-18)
   (fma (* (fma a (+ a 4.0) 4.0) a) a -1.0)
   (fma (* (fma b b (fma a (+ -12.0 (* 2.0 a)) 4.0)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e-18) {
		tmp = fma((fma(a, (a + 4.0), 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, fma(a, (-12.0 + (2.0 * a)), 4.0)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e-18)
		tmp = fma(Float64(fma(a, Float64(a + 4.0), 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, fma(a, Float64(-12.0 + Float64(2.0 * a)), 4.0)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e-18], N[(N[(N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + N[(a * N[(-12.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e-18

   1. Initial program 82.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 82.3

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

   if 4.0000000000000003e-18 < (*.f64 b b)

   1. Initial program 67.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 70.9

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.0%

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

Alternative 5: 98.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a + 4, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, -12\right), a, \mathsf{fma}\left(b, b, 4\right)\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e-18)
   (fma (* (fma a (+ a 4.0) 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) <= 4e-18) {
		tmp = fma((fma(a, (a + 4.0), 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) <= 4e-18)
		tmp = fma(Float64(fma(a, Float64(a + 4.0), 4.0) * a), a, -1.0);
	else
		tmp = fma(fma(fma(2.0, a, -12.0), a, fma(b, b, 4.0)), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e-18], N[(N[(N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(2.0 * a + -12.0), $MachinePrecision] * a + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e-18

   1. Initial program 82.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 82.3

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

   if 4.0000000000000003e-18 < (*.f64 b b)

   1. Initial program 67.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 70.9

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.0%

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

   12. Applied rewrites 96.9%

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

Alternative 6: 98.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a + 4, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, 2 \cdot a, 4\right)\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e-18)
   (fma (* (fma a (+ a 4.0) 4.0) a) a -1.0)
   (fma (* (fma b b (fma a (* 2.0 a) 4.0)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e-18) {
		tmp = fma((fma(a, (a + 4.0), 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, fma(a, (2.0 * a), 4.0)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e-18)
		tmp = fma(Float64(fma(a, Float64(a + 4.0), 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, fma(a, Float64(2.0 * a), 4.0)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e-18], N[(N[(N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + N[(a * N[(2.0 * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e-18

   1. Initial program 82.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 82.3

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

   if 4.0000000000000003e-18 < (*.f64 b b)

   1. Initial program 67.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 70.9

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.0%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 96.9%

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

Alternative 7: 94.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a + 4, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(-12, a, \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1000000.0)
   (fma (* (fma a (+ a 4.0) 4.0) a) a -1.0)
   (- (* (* b b) (fma -12.0 a (fma b b 4.0))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000.0) {
		tmp = fma((fma(a, (a + 4.0), 4.0) * a), a, -1.0);
	} else {
		tmp = ((b * b) * fma(-12.0, a, fma(b, b, 4.0))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1000000.0)
		tmp = fma(Float64(fma(a, Float64(a + 4.0), 4.0) * a), a, -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * fma(-12.0, a, fma(b, b, 4.0))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1000000.0], N[(N[(N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(-12.0 * a + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e6

   1. Initial program 83.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 83.0

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 99.4%

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

   9. Applied rewrites 99.5%

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

   if 1e6 < (*.f64 b b)

   1. Initial program 66.5%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 93.4

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

   5. Applied rewrites 93.4%

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

Alternative 8: 94.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a + 4, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1000000.0)
   (fma (* (fma a (+ a 4.0) 4.0) a) a -1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000.0) {
		tmp = fma((fma(a, (a + 4.0), 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1000000.0)
		tmp = fma(Float64(fma(a, Float64(a + 4.0), 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1000000.0], N[(N[(N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e6

   1. Initial program 83.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 83.0

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 99.4%

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

   9. Applied rewrites 99.5%

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

   if 1e6 < (*.f64 b b)

   1. Initial program 66.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 69.7

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.8%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 91.2%

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

Alternative 9: 84.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+131} \lor \neg \left(a \leq 3.5 \cdot 10^{+153}\right):\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -8.6e+131) (not (<= a 3.5e+153)))
   (fma (* a a) 4.0 -1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -8.6e+131) || !(a <= 3.5e+153)) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -8.6e+131) || !(a <= 3.5e+153))
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -8.6e+131], N[Not[LessEqual[a, 3.5e+153]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.6000000000000003e131 or 3.4999999999999999e153 < a

   1. Initial program 21.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 21.0

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 21.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 91.4%

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

   if -8.6000000000000003e131 < a < 3.4999999999999999e153

   1. Initial program 92.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 94.2

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 85.7%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 85.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+131} \lor \neg \left(a \leq 3.5 \cdot 10^{+153}\right):\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+131} \lor \neg \left(a \leq 3.5 \cdot 10^{+153}\right):\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -8.6e+131) (not (<= a 3.5e+153)))
   (fma (* a a) 4.0 -1.0)
   (fma (fma b b 4.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -8.6e+131) || !(a <= 3.5e+153)) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -8.6e+131) || !(a <= 3.5e+153))
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -8.6e+131], N[Not[LessEqual[a, 3.5e+153]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.6000000000000003e131 or 3.4999999999999999e153 < a

   1. Initial program 21.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 21.0

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 21.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 91.4%

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

   if -8.6000000000000003e131 < a < 3.4999999999999999e153

   1. Initial program 92.1%

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

   3. Taylor expanded in a around 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. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

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

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in 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. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 85.4

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

   8. Applied rewrites 85.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+131} \lor \neg \left(a \leq 3.5 \cdot 10^{+153}\right):\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, a, 4\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6e+131)
   (fma (* a a) 4.0 -1.0)
   (if (<= a 2.5e+101)
     (fma (* (fma b b 4.0) b) b -1.0)
     (fma (* a a) (fma 4.0 a 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.6e+131) {
		tmp = fma((a * a), 4.0, -1.0);
	} else if (a <= 2.5e+101) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma((a * a), fma(4.0, a, 4.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.6e+131)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	elseif (a <= 2.5e+101)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(Float64(a * a), fma(4.0, a, 4.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.6e+131], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], If[LessEqual[a, 2.5e+101], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.6000000000000003e131

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 0.0

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 88.7%

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

   if -8.6000000000000003e131 < a < 2.49999999999999994e101

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 96.1

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 87.2%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 86.9%

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

   if 2.49999999999999994e101 < a

   1. Initial program 58.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 58.1

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 58.1%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.2%

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

Alternative 12: 69.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1.5 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1.5e+298) (fma (* a a) 4.0 -1.0) (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1.5e+298) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1.5e+298)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1.5e+298], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.49999999999999994e298

   1. Initial program 78.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 79.0

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 81.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 62.9%

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

   if 1.49999999999999994e298 < (*.f64 b b)

   1. Initial program 65.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 96.3%

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

Alternative 13: 51.1% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. *-lft-identity N/A

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

   11. lower-fma.f64 76.5

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. associate-*r* N/A

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

   6. distribute-lft-in N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. distribute-rgt-out N/A

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

   10. metadata-eval N/A

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

   11. distribute-lft-in N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-out N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

7. Applied rewrites 68.5%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 51.9%

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

Alternative 14: 25.4% accurate, 160.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. *-lft-identity N/A

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

   11. lower-fma.f64 76.5

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. associate-*r* N/A

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

   6. distribute-lft-in N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. distribute-rgt-out N/A

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

   10. metadata-eval N/A

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

   11. distribute-lft-in N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-out N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

7. Applied rewrites 68.5%

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

8. Taylor expanded in a around 0

   \[\leadsto -1 \]
9. Step-by-step derivation

10. Applied rewrites 28.8%

    \[\leadsto -1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024323 
(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))