Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.9% → 99.9%

Time: 9.7s

Alternatives: 15

Speedup: 5.2×

• 60.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 74.9% accurate, 1.0× speedup

Math

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

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, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + N[(b * b + 12.0), $MachinePrecision]), $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. Initial program 71.0%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 99.9%

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

Alternative 2: 97.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, a \cdot a\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7e-5)
   (fma (* (* (* a a) 2.0) b) b (fma (* a a) (fma (- a 4.0) a 4.0) -1.0))
   (if (<= a 2e-43)
     (fma (* (fma b b 12.0) b) b -1.0)
     (fma
      (* (fma (fma 2.0 a 4.0) a 12.0) b)
      b
      (fma (* a a) (fma 4.0 (- 1.0 a) (* a a)) -1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7e-5) {
		tmp = fma((((a * a) * 2.0) * b), b, fma((a * a), fma((a - 4.0), a, 4.0), -1.0));
	} else if (a <= 2e-43) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((fma(fma(2.0, a, 4.0), a, 12.0) * b), b, fma((a * a), fma(4.0, (1.0 - a), (a * a)), -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7e-5)
		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));
	elseif (a <= 2e-43)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(fma(fma(2.0, a, 4.0), a, 12.0) * b), b, fma(Float64(a * a), fma(4.0, Float64(1.0 - a), Float64(a * a)), -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7e-5], 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], If[LessEqual[a, 2e-43], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(4.0 * N[(1.0 - a), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.9999999999999994e-5

   1. Initial program 66.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 99.9%

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

   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 98.6%

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

   if -6.9999999999999994e-5 < a < 2.00000000000000015e-43

   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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 2.00000000000000015e-43 < 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 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   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 95.5%

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

Alternative 3: 97.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a - 4, a, 4\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, t\_0, -1\right)\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, t\_0, \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b\right) \cdot b + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (- a 4.0) a 4.0)))
   (if (<= a -7e-5)
     (fma (* (* (* a a) 2.0) b) b (fma (* a a) t_0 -1.0))
     (if (<= a 2e-43)
       (fma (* (fma b b 12.0) b) b -1.0)
       (fma (* a a) t_0 (+ (* (* (fma (fma 2.0 a 4.0) a 12.0) b) b) -1.0))))))

C

double code(double a, double b) {
	double t_0 = fma((a - 4.0), a, 4.0);
	double tmp;
	if (a <= -7e-5) {
		tmp = fma((((a * a) * 2.0) * b), b, fma((a * a), t_0, -1.0));
	} else if (a <= 2e-43) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((a * a), t_0, (((fma(fma(2.0, a, 4.0), a, 12.0) * b) * b) + -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a - 4.0), a, 4.0)
	tmp = 0.0
	if (a <= -7e-5)
		tmp = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, fma(Float64(a * a), t_0, -1.0));
	elseif (a <= 2e-43)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(a * a), t_0, Float64(Float64(Float64(fma(fma(2.0, a, 4.0), a, 12.0) * b) * b) + -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision]}, If[LessEqual[a, -7e-5], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2e-43], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * t$95$0 + N[(N[(N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.9999999999999994e-5

   1. Initial program 66.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 99.9%

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

   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 98.6%

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

   if -6.9999999999999994e-5 < a < 2.00000000000000015e-43

   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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 2.00000000000000015e-43 < 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 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   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 95.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 95.5%

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

   13. Applied rewrites 95.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b\right) \cdot b + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, t\_0\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b, b, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)))
   (if (<= a -7e-5)
     (fma (* (* (* a a) 2.0) b) b t_0)
     (if (<= a 2e-43)
       (fma (* (fma b b 12.0) b) b -1.0)
       (fma (* (fma (fma 2.0 a 4.0) a 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 <= -7e-5) {
		tmp = fma((((a * a) * 2.0) * b), b, t_0);
	} else if (a <= 2e-43) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((fma(fma(2.0, a, 4.0), a, 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 <= -7e-5)
		tmp = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, t_0);
	elseif (a <= 2e-43)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(fma(fma(2.0, a, 4.0), a, 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, -7e-5], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + t$95$0), $MachinePrecision], If[LessEqual[a, 2e-43], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.9999999999999994e-5

   1. Initial program 66.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 99.9%

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

   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 98.6%

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

   if -6.9999999999999994e-5 < a < 2.00000000000000015e-43

   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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 2.00000000000000015e-43 < 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 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   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 95.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 95.5%

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

Alternative 5: 97.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (fma
          (* (* (* a a) 2.0) b)
          b
          (fma (* a a) (fma (- a 4.0) a 4.0) -1.0))))
   (if (<= a -7e-5)
     t_0
     (if (<= a 2e-43) (fma (* (fma b b 12.0) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma((((a * a) * 2.0) * b), b, fma((a * a), fma((a - 4.0), a, 4.0), -1.0));
	double tmp;
	if (a <= -7e-5) {
		tmp = t_0;
	} else if (a <= 2e-43) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0))
	tmp = 0.0
	if (a <= -7e-5)
		tmp = t_0;
	elseif (a <= 2e-43)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(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]}, If[LessEqual[a, -7e-5], t$95$0, If[LessEqual[a, 2e-43], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.9999999999999994e-5 or 2.00000000000000015e-43 < a

   1. Initial program 49.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 99.9%

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

   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 97.0%

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

   if -6.9999999999999994e-5 < a < 2.00000000000000015e-43

   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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 6: 96.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(12 \cdot b, b, t\_0\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, a, 12\right) \cdot b, b, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)))
   (if (<= a -7e-5)
     (fma (* 12.0 b) b t_0)
     (if (<= a 2e-43)
       (fma (* (fma b b 12.0) b) b -1.0)
       (fma (* (fma 4.0 a 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 <= -7e-5) {
		tmp = fma((12.0 * b), b, t_0);
	} else if (a <= 2e-43) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((fma(4.0, a, 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 <= -7e-5)
		tmp = fma(Float64(12.0 * b), b, t_0);
	elseif (a <= 2e-43)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(fma(4.0, a, 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, -7e-5], N[(N[(12.0 * b), $MachinePrecision] * b + t$95$0), $MachinePrecision], If[LessEqual[a, 2e-43], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(4.0 * a + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.9999999999999994e-5

   1. Initial program 66.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   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 98.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 98.6%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 96.1%

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

   if -6.9999999999999994e-5 < a < 2.00000000000000015e-43

   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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 2.00000000000000015e-43 < 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 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   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 95.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 95.5%

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

   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 95.2%

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

Alternative 7: 96.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(12 \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(12 \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(4, 1 - a, a \cdot a\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7e-5)
   (fma (* 12.0 b) b (fma (* a a) (fma (- a 4.0) a 4.0) -1.0))
   (if (<= a 2e-43)
     (fma (* (fma b b 12.0) b) b -1.0)
     (fma (* 12.0 b) b (fma (* a a) (fma 4.0 (- 1.0 a) (* a a)) -1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7e-5) {
		tmp = fma((12.0 * b), b, fma((a * a), fma((a - 4.0), a, 4.0), -1.0));
	} else if (a <= 2e-43) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((12.0 * b), b, fma((a * a), fma(4.0, (1.0 - a), (a * a)), -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7e-5)
		tmp = fma(Float64(12.0 * b), b, fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0));
	elseif (a <= 2e-43)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(12.0 * b), b, fma(Float64(a * a), fma(4.0, Float64(1.0 - a), Float64(a * a)), -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7e-5], N[(N[(12.0 * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2e-43], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(12.0 * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(4.0 * N[(1.0 - a), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.9999999999999994e-5

   1. Initial program 66.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   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 98.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 98.6%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 96.1%

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

   if -6.9999999999999994e-5 < a < 2.00000000000000015e-43

   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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 2.00000000000000015e-43 < 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 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   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 95.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 94.0%

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

Alternative 8: 96.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(12 \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* 12.0 b) b (fma (* a a) (fma (- a 4.0) a 4.0) -1.0))))
   (if (<= a -7e-5)
     t_0
     (if (<= a 2e-43) (fma (* (fma b b 12.0) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma((12.0 * b), b, fma((a * a), fma((a - 4.0), a, 4.0), -1.0));
	double tmp;
	if (a <= -7e-5) {
		tmp = t_0;
	} else if (a <= 2e-43) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(12.0 * b), b, fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0))
	tmp = 0.0
	if (a <= -7e-5)
		tmp = t_0;
	elseif (a <= 2e-43)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(12.0 * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7e-5], t$95$0, If[LessEqual[a, 2e-43], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.9999999999999994e-5 or 2.00000000000000015e-43 < a

   1. Initial program 49.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

   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 97.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 97.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 95.0%

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

   if -6.9999999999999994e-5 < a < 2.00000000000000015e-43

   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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 9: 93.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+70}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)))
   (if (<= a -1.1e+32)
     t_0
     (if (<= a 6e+70) (- (* (* (fma b b (fma 4.0 a 12.0)) b) b) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	double tmp;
	if (a <= -1.1e+32) {
		tmp = t_0;
	} else if (a <= 6e+70) {
		tmp = ((fma(b, b, fma(4.0, a, 12.0)) * b) * b) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0)
	tmp = 0.0
	if (a <= -1.1e+32)
		tmp = t_0;
	elseif (a <= 6e+70)
		tmp = Float64(Float64(Float64(fma(b, b, fma(4.0, a, 12.0)) * b) * b) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]}, If[LessEqual[a, -1.1e+32], t$95$0, If[LessEqual[a, 6e+70], N[(N[(N[(N[(b * b + N[(4.0 * a + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1e32 or 5.99999999999999952e70 < a

   1. Initial program 39.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   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 97.6

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

   8. Applied rewrites 97.6%

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

   10. Applied rewrites 97.6%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 97.6%

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

   if -1.1e32 < a < 5.99999999999999952e70

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 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 95.2%

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

Alternative 10: 93.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)))
   (if (<= a -1.1e+32)
     t_0
     (if (<= a 6e+70) (fma (* (fma b b 12.0) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	double tmp;
	if (a <= -1.1e+32) {
		tmp = t_0;
	} else if (a <= 6e+70) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0)
	tmp = 0.0
	if (a <= -1.1e+32)
		tmp = t_0;
	elseif (a <= 6e+70)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision]}, If[LessEqual[a, -1.1e+32], t$95$0, If[LessEqual[a, 6e+70], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1e32 or 5.99999999999999952e70 < a

   1. Initial program 39.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   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 97.6

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

   8. Applied rewrites 97.6%

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

   10. Applied rewrites 97.6%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 97.6%

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

   if -1.1e32 < a < 5.99999999999999952e70

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 95.2

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

   8. Applied rewrites 95.2%

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

Alternative 11: 87.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.6e+91)
   (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 <= -3.6e+91) {
		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 <= -3.6e+91)
		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, -3.6e+91], 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 < -3.6e91

   1. Initial program 67.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 95.1%

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

   if -3.6e91 < a < 6.59999999999999989e153

   1. Initial program 88.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 86.3

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

   8. Applied rewrites 86.3%

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

   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 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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: 72.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.6e+91)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 3.3e+153) (fma (* b b) 12.0 -1.0) (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.6e+91) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 3.3e+153) {
		tmp = fma((b * b), 12.0, -1.0);
	} else {
		tmp = fma((a * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.6e+91)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 3.3e+153)
		tmp = fma(Float64(b * b), 12.0, -1.0);
	else
		tmp = fma(Float64(a * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.6e+91], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 3.3e+153], N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.6e91

   1. Initial program 67.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 95.1%

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

   if -3.6e91 < a < 3.29999999999999994e153

   1. Initial program 88.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

   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 81.6%

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

   9. Taylor expanded in a around 0

      \[\leadsto 12 \cdot {b}^{2} - \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.0%

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

   if 3.29999999999999994e153 < 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 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 13: 69.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+303) (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) <= 4e+303) {
		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) <= 4e+303)
		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], 4e+303], 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) < 4e303

   1. Initial program 75.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

   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 81.1

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

   8. Applied rewrites 81.1%

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

   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 4e303 < (*.f64 b b)

   1. Initial program 58.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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%

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

   9. Taylor expanded in a around 0

      \[\leadsto 12 \cdot {b}^{2} - \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

Alternative 14: 51.6% 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 71.0%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

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 88.1%

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

9. Taylor expanded in a around 0

   \[\leadsto 12 \cdot {b}^{2} - \color{blue}{1} \]
10. Step-by-step derivation

11. Applied rewrites 49.8%

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

Alternative 15: 25.0% 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 71.0%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

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 69.8

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

8. Applied rewrites 69.8%

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

9. Taylor expanded in a around 0

   \[\leadsto -1 \]
10. Step-by-step derivation

11. Applied rewrites 23.0%

    \[\leadsto -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024327 
(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))