Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.4% → 99.9%

Time: 7.7s

Alternatives: 14

Speedup: 5.3×

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

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))))
   (if (<= t_0 INFINITY)
     (- t_0 1.0)
     (fma (* (* (* a a) 2.0) b) b (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 - 1.0;
	} else {
		tmp = fma((((a * a) * 2.0) * b), b, fma((a * a), fma((a - 4.0), a, 4.0), -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = 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(3.0 + a)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = 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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 - 1.0), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < +inf.0

   1. Initial program 99.8%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 40.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 100.0%

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

Alternative 2: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{+49}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 4.8e+49)
   (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* (fma (- a) a a) a))) 1.0)
   (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 4.8e+49) {
		tmp = (pow(((a * a) + (b * b)), 2.0) + (4.0 * (fma(-a, a, a) * a))) - 1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 4.8e+49)
		tmp = Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(fma(Float64(-a), a, a) * a))) - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 4.8e+49], N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[((-a) * a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.8e49

   1. Initial program 85.6%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 99.6%

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

   if 4.8e49 < a

   1. Initial program 20.6%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 20.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around inf

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

       1. lower-pow.f64 100.0

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

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{{a}^{4}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \mathbf{if}\;a \leq -1600000000:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, t\_0\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right) \cdot b, b, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)))
   (if (<= a -1600000000.0)
     (fma (* (* (* a a) 2.0) b) b t_0)
     (if (<= a 6.2e-30)
       (- (fma (* b b) 12.0 (pow b 4.0)) 1.0)
       (fma (* (fma a (fma 2.0 a 4.0) 12.0) b) b t_0)))))

C

double code(double a, double b) {
	double t_0 = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	double tmp;
	if (a <= -1600000000.0) {
		tmp = fma((((a * a) * 2.0) * b), b, t_0);
	} else if (a <= 6.2e-30) {
		tmp = fma((b * b), 12.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = fma((fma(a, fma(2.0, a, 4.0), 12.0) * b), b, t_0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0)
	tmp = 0.0
	if (a <= -1600000000.0)
		tmp = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, t_0);
	elseif (a <= 6.2e-30)
		tmp = Float64(fma(Float64(b * b), 12.0, (b ^ 4.0)) - 1.0);
	else
		tmp = fma(Float64(fma(a, fma(2.0, a, 4.0), 12.0) * b), b, t_0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[a, -1600000000.0], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + t$95$0), $MachinePrecision], If[LessEqual[a, 6.2e-30], N[(N[(N[(b * b), $MachinePrecision] * 12.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * N[(2.0 * a + 4.0), $MachinePrecision] + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.6e9

   1. Initial program 53.9%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 98.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 98.5%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 98.5%

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

   if -1.6e9 < a < 6.19999999999999982e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 6.19999999999999982e-30 < a

   1. Initial program 33.2%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 98.4%

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

Alternative 4: 97.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \mathbf{if}\;a \leq -1600000000:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, t\_0\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right) \cdot b, b, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)))
   (if (<= a -1600000000.0)
     (fma (* (* (* a a) 2.0) b) b t_0)
     (if (<= a 6.2e-30)
       (fma (* (fma b b 12.0) b) b -1.0)
       (fma (* (fma a (fma 2.0 a 4.0) 12.0) b) b t_0)))))

C

double code(double a, double b) {
	double t_0 = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	double tmp;
	if (a <= -1600000000.0) {
		tmp = fma((((a * a) * 2.0) * b), b, t_0);
	} else if (a <= 6.2e-30) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((fma(a, fma(2.0, a, 4.0), 12.0) * b), b, t_0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0)
	tmp = 0.0
	if (a <= -1600000000.0)
		tmp = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, t_0);
	elseif (a <= 6.2e-30)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(fma(a, fma(2.0, a, 4.0), 12.0) * b), b, t_0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[a, -1600000000.0], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + t$95$0), $MachinePrecision], If[LessEqual[a, 6.2e-30], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(a * N[(2.0 * a + 4.0), $MachinePrecision] + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.6e9

   1. Initial program 53.9%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 98.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 98.5%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 98.5%

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

   if -1.6e9 < a < 6.19999999999999982e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if 6.19999999999999982e-30 < a

   1. Initial program 33.2%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 98.4%

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

Alternative 5: 97.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1600000000 \lor \neg \left(a \leq 6.2 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1600000000.0) (not (<= a 6.2e-30)))
   (fma (* (* (* a a) 2.0) b) b (fma (* a a) (fma (- a 4.0) a 4.0) -1.0))
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1600000000.0) || !(a <= 6.2e-30)) {
		tmp = fma((((a * a) * 2.0) * b), b, fma((a * a), fma((a - 4.0), a, 4.0), -1.0));
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1600000000.0) || !(a <= 6.2e-30))
		tmp = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0));
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1600000000.0], N[Not[LessEqual[a, 6.2e-30]], $MachinePrecision]], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6e9 or 6.19999999999999982e-30 < a

   1. Initial program 43.6%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 98.4%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 98.4%

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

   if -1.6e9 < a < 6.19999999999999982e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 99.7

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

   8. Applied rewrites 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1600000000 \lor \neg \left(a \leq 6.2 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+43}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+22}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, a, 12\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9e+43)
   (- (* (* a a) (* a a)) 1.0)
   (if (<= a 2.5e+22)
     (- (* (* (fma b b (fma 4.0 a 12.0)) b) b) 1.0)
     (fma (* (fma 4.0 a 12.0) b) b (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9e+43) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else if (a <= 2.5e+22) {
		tmp = ((fma(b, b, fma(4.0, a, 12.0)) * b) * b) - 1.0;
	} else {
		tmp = fma((fma(4.0, a, 12.0) * b), b, fma((a * a), fma((a - 4.0), a, 4.0), -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9e+43)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	elseif (a <= 2.5e+22)
		tmp = Float64(Float64(Float64(fma(b, b, fma(4.0, a, 12.0)) * b) * b) - 1.0);
	else
		tmp = fma(Float64(fma(4.0, a, 12.0) * b), b, fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9e+43], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 2.5e+22], N[(N[(N[(N[(b * b + N[(4.0 * a + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(4.0 * a + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9e43

   1. Initial program 55.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.3%

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

   if -9e43 < a < 2.4999999999999998e22

   1. Initial program 97.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 97.7%

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

   if 2.4999999999999998e22 < a

   1. Initial program 23.5%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 23.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 99.9%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 99.9%

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

Alternative 7: 93.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+43} \lor \neg \left(a \leq 3.05 \cdot 10^{+29}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -9e+43) (not (<= a 3.05e+29)))
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* (fma b b (fma 4.0 a 12.0)) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -9e+43) || !(a <= 3.05e+29)) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((fma(b, b, fma(4.0, a, 12.0)) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -9e+43) || !(a <= 3.05e+29))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, fma(4.0, a, 12.0)) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -9e+43], N[Not[LessEqual[a, 3.05e+29]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + N[(4.0 * a + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9e43 or 3.0499999999999999e29 < a

   1. Initial program 39.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.2%

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

   if -9e43 < a < 3.0499999999999999e29

   1. Initial program 97.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 97.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+43} \lor \neg \left(a \leq 3.05 \cdot 10^{+29}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+43} \lor \neg \left(a \leq 3.05 \cdot 10^{+29}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -9e+43) (not (<= a 3.05e+29)))
   (- (* (* a a) (* a a)) 1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -9e+43) || !(a <= 3.05e+29)) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -9e+43) || !(a <= 3.05e+29))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -9e+43], N[Not[LessEqual[a, 3.05e+29]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9e43 or 3.0499999999999999e29 < a

   1. Initial program 39.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.2%

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

   if -9e43 < a < 3.0499999999999999e29

   1. Initial program 97.1%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 97.7

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

   8. Applied rewrites 97.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+43} \lor \neg \left(a \leq 3.05 \cdot 10^{+29}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.3e+101)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 6.6e+153)
     (fma (* (fma b b 12.0) b) b -1.0)
     (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.3e+101) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 6.6e+153) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((a * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.3e+101)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 6.6e+153)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(a * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.3e+101], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 6.6e+153], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.3000000000000001e101

   1. Initial program 54.3%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 98.1%

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

   if -4.3000000000000001e101 < a < 6.59999999999999989e153

   1. Initial program 88.7%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 85.1

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

   8. Applied rewrites 85.1%

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

   if 6.59999999999999989e153 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

Alternative 10: 83.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+126} \lor \neg \left(a \leq 6.6 \cdot 10^{+153}\right):\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -5.6e+126) (not (<= a 6.6e+153)))
   (fma (* a a) 4.0 -1.0)
   (fma (* (* b b) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -5.6e+126) || !(a <= 6.6e+153)) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = fma(((b * b) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -5.6e+126) || !(a <= 6.6e+153))
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -5.6e+126], N[Not[LessEqual[a, 6.6e+153]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.60000000000000018e126 or 6.59999999999999989e153 < a

   1. Initial program 29.4%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 92.0%

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

   if -5.60000000000000018e126 < a < 6.59999999999999989e153

   1. Initial program 87.6%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 83.1

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

   8. Applied rewrites 83.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 65.2%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 82.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+126} \lor \neg \left(a \leq 6.6 \cdot 10^{+153}\right):\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.3e+101)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 6.6e+153) (fma (* (* b b) b) b -1.0) (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.3e+101) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 6.6e+153) {
		tmp = fma(((b * b) * b), b, -1.0);
	} else {
		tmp = fma((a * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.3e+101)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 6.6e+153)
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	else
		tmp = fma(Float64(a * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.3e+101], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 6.6e+153], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.3000000000000001e101

   1. Initial program 54.3%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 98.1%

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

   if -4.3000000000000001e101 < a < 6.59999999999999989e153

   1. Initial program 88.7%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 85.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, b, 12\right)} \cdot b, b, -1\right) \]

   8. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)} \]

   9. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.4%

       \[\leadsto \mathsf{fma}\left(12 \cdot b, b, -1\right) \]

   12. Taylor expanded in b around inf

       \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 84.7%

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right) \]

   if 6.59999999999999989e153 < a

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(\frac{1}{a} - 1\right)\right)}\right) - 1 \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(\frac{1}{a} - 1\right)\right)\right) - 1 \]

      2. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(\frac{1}{a} - 1\right)\right)\right) - 1 \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(\frac{1}{a} - 1\right)\right)\right)}\right) - 1 \]

      4. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \color{blue}{\left(\frac{1}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right) - 1 \]

      5. remove-double-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) - 1 \]

      6. distribute-neg-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) + 1\right)\right)\right)}\right)\right)\right) - 1 \]

      7. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)}\right)\right)\right)\right)\right) - 1 \]

      8. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{a}\right)}\right)\right)\right)\right)\right) - 1 \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(1 - \frac{1}{a}\right)\right)\right)}\right)\right) - 1 \]

      10. *-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{a}\right) \cdot a}\right)\right)\right)\right) - 1 \]

      11. sub-neg N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)} \cdot a\right)\right)\right)\right) - 1 \]

      12. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) + 1\right)} \cdot a\right)\right)\right)\right) - 1 \]

      13. distribute-lft1-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) \cdot a + a\right)}\right)\right)\right)\right) - 1 \]

      14. distribute-lft-neg-out N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{a} \cdot a\right)\right)} + a\right)\right)\right)\right)\right) - 1 \]

      15. lft-mult-inverse N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\color{blue}{1}\right)\right) + a\right)\right)\right)\right)\right) - 1 \]

      16. metadata-eval N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{-1} + a\right)\right)\right)\right)\right) - 1 \]

      17. distribute-neg-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(a\right)\right)\right)}\right)\right) - 1 \]

      18. metadata-eval N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{1} + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right) - 1 \]

      19. sub-neg N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(1 - a\right)}\right)\right) - 1 \]

   5. Applied rewrites 0.0%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right)}\right) - 1 \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \color{blue}{\left(\left(1 - a\right) \cdot {a}^{2}\right)} + {a}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}} + {a}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2} + {a}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \left(\left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2} + \color{blue}{{a}^{2} \cdot {a}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(4 \cdot \left(1 - a\right) + {a}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(4 \cdot \left(1 - a\right) + {a}^{2}\right) + \color{blue}{-1} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, 4 \cdot \left(1 - a\right) + {a}^{2}, -1\right)} \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a}, 4 \cdot \left(1 - a\right) + {a}^{2}, -1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a}, 4 \cdot \left(1 - a\right) + {a}^{2}, -1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left(4, 1 - a, {a}^{2}\right)}, -1\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, \color{blue}{1 - a}, {a}^{2}\right), -1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, \color{blue}{a \cdot a}\right), -1\right) \]

      14. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, \color{blue}{a \cdot a}\right), -1\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, a \cdot a\right), -1\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(a \cdot a, 4, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(a \cdot a, 4, -1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+296) (fma (* a a) 4.0 -1.0) (fma (* b b) 12.0 -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+296) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = fma((b * b), 12.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+296)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = fma(Float64(b * b), 12.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+296], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.99999999999999996e296

   1. Initial program 77.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(\frac{1}{a} - 1\right)\right)}\right) - 1 \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(\frac{1}{a} - 1\right)\right)\right) - 1 \]

      2. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(\frac{1}{a} - 1\right)\right)\right) - 1 \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(\frac{1}{a} - 1\right)\right)\right)}\right) - 1 \]

      4. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \color{blue}{\left(\frac{1}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right) - 1 \]

      5. remove-double-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) - 1 \]

      6. distribute-neg-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) + 1\right)\right)\right)}\right)\right)\right) - 1 \]

      7. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)}\right)\right)\right)\right)\right) - 1 \]

      8. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{a}\right)}\right)\right)\right)\right)\right) - 1 \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(1 - \frac{1}{a}\right)\right)\right)}\right)\right) - 1 \]

      10. *-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{a}\right) \cdot a}\right)\right)\right)\right) - 1 \]

      11. sub-neg N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)} \cdot a\right)\right)\right)\right) - 1 \]

      12. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) + 1\right)} \cdot a\right)\right)\right)\right) - 1 \]

      13. distribute-lft1-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) \cdot a + a\right)}\right)\right)\right)\right) - 1 \]

      14. distribute-lft-neg-out N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{a} \cdot a\right)\right)} + a\right)\right)\right)\right)\right) - 1 \]

      15. lft-mult-inverse N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\color{blue}{1}\right)\right) + a\right)\right)\right)\right)\right) - 1 \]

      16. metadata-eval N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{-1} + a\right)\right)\right)\right)\right) - 1 \]

      17. distribute-neg-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(a\right)\right)\right)}\right)\right) - 1 \]

      18. metadata-eval N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{1} + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right) - 1 \]

      19. sub-neg N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(1 - a\right)}\right)\right) - 1 \]

   5. Applied rewrites 82.0%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right)}\right) - 1 \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \color{blue}{\left(\left(1 - a\right) \cdot {a}^{2}\right)} + {a}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}} + {a}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2} + {a}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \left(\left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2} + \color{blue}{{a}^{2} \cdot {a}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(4 \cdot \left(1 - a\right) + {a}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(4 \cdot \left(1 - a\right) + {a}^{2}\right) + \color{blue}{-1} \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, 4 \cdot \left(1 - a\right) + {a}^{2}, -1\right)} \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a}, 4 \cdot \left(1 - a\right) + {a}^{2}, -1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a}, 4 \cdot \left(1 - a\right) + {a}^{2}, -1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left(4, 1 - a, {a}^{2}\right)}, -1\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, \color{blue}{1 - a}, {a}^{2}\right), -1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, \color{blue}{a \cdot a}\right), -1\right) \]

      14. lower-*.f64 82.9

          \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, \color{blue}{a \cdot a}\right), -1\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, a \cdot a\right), -1\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(a \cdot a, 4, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 63.3%

       \[\leadsto \mathsf{fma}\left(a \cdot a, 4, -1\right) \]

   if 1.99999999999999996e296 < (*.f64 b b)

   1. Initial program 56.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(\frac{1}{a} - 1\right)\right)}\right) - 1 \]
   4. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(\frac{1}{a} - 1\right)\right)\right) - 1 \]

      2. unpow2 N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(\frac{1}{a} - 1\right)\right)\right) - 1 \]

      3. associate-*l* N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(\frac{1}{a} - 1\right)\right)\right)}\right) - 1 \]

      4. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \color{blue}{\left(\frac{1}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right) - 1 \]

      5. remove-double-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) - 1 \]

      6. distribute-neg-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) + 1\right)\right)\right)}\right)\right)\right) - 1 \]

      7. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)}\right)\right)\right)\right)\right) - 1 \]

      8. sub-neg N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{a}\right)}\right)\right)\right)\right)\right) - 1 \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(1 - \frac{1}{a}\right)\right)\right)}\right)\right) - 1 \]

      10. *-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{a}\right) \cdot a}\right)\right)\right)\right) - 1 \]

      11. sub-neg N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)} \cdot a\right)\right)\right)\right) - 1 \]

      12. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) + 1\right)} \cdot a\right)\right)\right)\right) - 1 \]

      13. distribute-lft1-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) \cdot a + a\right)}\right)\right)\right)\right) - 1 \]

      14. distribute-lft-neg-out N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{a} \cdot a\right)\right)} + a\right)\right)\right)\right)\right) - 1 \]

      15. lft-mult-inverse N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\color{blue}{1}\right)\right) + a\right)\right)\right)\right)\right) - 1 \]

      16. metadata-eval N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{-1} + a\right)\right)\right)\right)\right) - 1 \]

      17. distribute-neg-in N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(a\right)\right)\right)}\right)\right) - 1 \]

      18. metadata-eval N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{1} + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right) - 1 \]

      19. sub-neg N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(1 - a\right)}\right)\right) - 1 \]

   5. Applied rewrites 86.6%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right)}\right) - 1 \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + \left({b}^{2} \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) + {a}^{4}\right)\right) - 1} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) + {a}^{4}\right) + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)} - 1 \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left({b}^{2} \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right)} - 1 \]

      3. +-commutative N/A

         \[\leadsto \left({b}^{2} \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) + \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}\right) - 1 \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) \cdot {b}^{2}} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(2 \cdot {a}^{2} + \color{blue}{\left(4 \cdot 3 + 4 \cdot a\right)}\right) \cdot {b}^{2} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(2 \cdot {a}^{2} + \left(\color{blue}{12} + 4 \cdot a\right)\right) \cdot {b}^{2} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

      8. unpow2 N/A

         \[\leadsto \left(2 \cdot {a}^{2} + \left(12 + 4 \cdot a\right)\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot {a}^{2} + \left(12 + 4 \cdot a\right)\right) \cdot b\right) \cdot b} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot {a}^{2} + \left(12 + 4 \cdot a\right)\right) \cdot b, b, \left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right)} \]

   8. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, a \cdot a\right), -1\right)\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto 12 \cdot {b}^{2} - \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.4%

       \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{12}, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 51.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(b \cdot b, 12, -1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma (* b b) 12.0 -1.0))

C

double code(double a, double b) {
	return fma((b * b), 12.0, -1.0);
}

Julia

function code(a, b)
	return fma(Float64(b * b), 12.0, -1.0)
end

Wolfram

code[a_, b_] := N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]

Derivation

1. Initial program 72.1%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(\frac{1}{a} - 1\right)\right)}\right) - 1 \]
4. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(\frac{1}{a} - 1\right)\right)\right) - 1 \]

   2. unpow2 N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(\frac{1}{a} - 1\right)\right)\right) - 1 \]

   3. associate-*l* N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(\frac{1}{a} - 1\right)\right)\right)}\right) - 1 \]

   4. sub-neg N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \color{blue}{\left(\frac{1}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right) - 1 \]

   5. remove-double-neg N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) - 1 \]

   6. distribute-neg-in N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) + 1\right)\right)\right)}\right)\right)\right) - 1 \]

   7. +-commutative N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)}\right)\right)\right)\right)\right) - 1 \]

   8. sub-neg N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{a}\right)}\right)\right)\right)\right)\right) - 1 \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(1 - \frac{1}{a}\right)\right)\right)}\right)\right) - 1 \]

   10. *-commutative N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{a}\right) \cdot a}\right)\right)\right)\right) - 1 \]

   11. sub-neg N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)} \cdot a\right)\right)\right)\right) - 1 \]

   12. +-commutative N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) + 1\right)} \cdot a\right)\right)\right)\right) - 1 \]

   13. distribute-lft1-in N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) \cdot a + a\right)}\right)\right)\right)\right) - 1 \]

   14. distribute-lft-neg-out N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{a} \cdot a\right)\right)} + a\right)\right)\right)\right)\right) - 1 \]

   15. lft-mult-inverse N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\color{blue}{1}\right)\right) + a\right)\right)\right)\right)\right) - 1 \]

   16. metadata-eval N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{-1} + a\right)\right)\right)\right)\right) - 1 \]

   17. distribute-neg-in N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(a\right)\right)\right)}\right)\right) - 1 \]

   18. metadata-eval N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{1} + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right) - 1 \]

   19. sub-neg N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(1 - a\right)}\right)\right) - 1 \]

5. Applied rewrites 83.2%

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right)}\right) - 1 \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + \left({b}^{2} \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) + {a}^{4}\right)\right) - 1} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left({b}^{2} \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) + {a}^{4}\right) + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)} - 1 \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{\left({b}^{2} \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right)} - 1 \]

   3. +-commutative N/A

      \[\leadsto \left({b}^{2} \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) + \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)}\right) - 1 \]

   4. associate--l+ N/A

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) \cdot {b}^{2}} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

   6. distribute-lft-in N/A

      \[\leadsto \left(2 \cdot {a}^{2} + \color{blue}{\left(4 \cdot 3 + 4 \cdot a\right)}\right) \cdot {b}^{2} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(2 \cdot {a}^{2} + \left(\color{blue}{12} + 4 \cdot a\right)\right) \cdot {b}^{2} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

   8. unpow2 N/A

      \[\leadsto \left(2 \cdot {a}^{2} + \left(12 + 4 \cdot a\right)\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot {a}^{2} + \left(12 + 4 \cdot a\right)\right) \cdot b\right) \cdot b} + \left(\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot {a}^{2} + \left(12 + 4 \cdot a\right)\right) \cdot b, b, \left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1\right)} \]

8. Applied rewrites 87.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(2, a, 4\right), 12\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, a \cdot a\right), -1\right)\right)} \]

9. Taylor expanded in a around 0

   \[\leadsto 12 \cdot {b}^{2} - \color{blue}{1} \]
10. Step-by-step derivation

11. Applied rewrites 54.3%

    \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{12}, -1\right) \]
12. Add Preprocessing

Alternative 14: 24.7% accurate, 155.0× 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 72.1%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{3} \cdot \left(\frac{1}{a} - 1\right)\right)}\right) - 1 \]
4. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot \left(\frac{1}{a} - 1\right)\right)\right) - 1 \]

   2. unpow2 N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{{a}^{2}} \cdot a\right) \cdot \left(\frac{1}{a} - 1\right)\right)\right) - 1 \]

   3. associate-*l* N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(a \cdot \left(\frac{1}{a} - 1\right)\right)\right)}\right) - 1 \]

   4. sub-neg N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \color{blue}{\left(\frac{1}{a} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right) - 1 \]

   5. remove-double-neg N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) - 1 \]

   6. distribute-neg-in N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) + 1\right)\right)\right)}\right)\right)\right) - 1 \]

   7. +-commutative N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)}\right)\right)\right)\right)\right) - 1 \]

   8. sub-neg N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{a}\right)}\right)\right)\right)\right)\right) - 1 \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(1 - \frac{1}{a}\right)\right)\right)}\right)\right) - 1 \]

   10. *-commutative N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{a}\right) \cdot a}\right)\right)\right)\right) - 1 \]

   11. sub-neg N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)} \cdot a\right)\right)\right)\right) - 1 \]

   12. +-commutative N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) + 1\right)} \cdot a\right)\right)\right)\right) - 1 \]

   13. distribute-lft1-in N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right) \cdot a + a\right)}\right)\right)\right)\right) - 1 \]

   14. distribute-lft-neg-out N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{a} \cdot a\right)\right)} + a\right)\right)\right)\right)\right) - 1 \]

   15. lft-mult-inverse N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\color{blue}{1}\right)\right) + a\right)\right)\right)\right)\right) - 1 \]

   16. metadata-eval N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{-1} + a\right)\right)\right)\right)\right) - 1 \]

   17. distribute-neg-in N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(a\right)\right)\right)}\right)\right) - 1 \]

   18. metadata-eval N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \left(\color{blue}{1} + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right) - 1 \]

   19. sub-neg N/A

       \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left({a}^{2} \cdot \color{blue}{\left(1 - a\right)}\right)\right) - 1 \]

5. Applied rewrites 83.2%

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right)}\right) - 1 \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) - 1} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(4 \cdot \color{blue}{\left(\left(1 - a\right) \cdot {a}^{2}\right)} + {a}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2}} + {a}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2} + {a}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

   5. pow-sqr N/A

      \[\leadsto \left(\left(4 \cdot \left(1 - a\right)\right) \cdot {a}^{2} + \color{blue}{{a}^{2} \cdot {a}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(4 \cdot \left(1 - a\right) + {a}^{2}\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto {a}^{2} \cdot \left(4 \cdot \left(1 - a\right) + {a}^{2}\right) + \color{blue}{-1} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, 4 \cdot \left(1 - a\right) + {a}^{2}, -1\right)} \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a}, 4 \cdot \left(1 - a\right) + {a}^{2}, -1\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a}, 4 \cdot \left(1 - a\right) + {a}^{2}, -1\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left(4, 1 - a, {a}^{2}\right)}, -1\right) \]

   12. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, \color{blue}{1 - a}, {a}^{2}\right), -1\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, \color{blue}{a \cdot a}\right), -1\right) \]

   14. lower-*.f64 71.2

       \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, \color{blue}{a \cdot a}\right), -1\right) \]

8. Applied rewrites 71.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, a \cdot a\right), -1\right)} \]

9. Taylor expanded in a around 0

   \[\leadsto -1 \]
10. Step-by-step derivation

11. Applied rewrites 27.6%

    \[\leadsto -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024314 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))