Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.8% → 99.9%

Time: 7.9s

Alternatives: 14

Speedup: 5.2×

• 61.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.8% 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, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in a around 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 99.1

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

5. Applied rewrites 99.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 99.1

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 99.9%

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

12. Applied rewrites 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a - 4, a, 4\right) \cdot a\\ \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;t\_0 \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, a, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, \mathsf{fma}\left(b, b, 12\right)\right), b \cdot b, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (fma (- a 4.0) a 4.0) a)))
   (if (<= (* b b) 0.0)
     (- (* t_0 a) 1.0)
     (fma t_0 a (fma (fma (fma 2.0 a 4.0) a (fma b b 12.0)) (* b b) -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma((a - 4.0), a, 4.0) * a;
	double tmp;
	if ((b * b) <= 0.0) {
		tmp = (t_0 * a) - 1.0;
	} else {
		tmp = fma(t_0, a, fma(fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)), (b * b), -1.0));
	}
	return tmp;
}

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 0.0], N[(N[(t$95$0 * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(t$95$0 * a + N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0

   1. Initial program 79.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. unpow2 N/A

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

      9. unpow3 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. mul-1-neg N/A

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

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

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

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

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

      16. metadata-eval N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

   10. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 b b)

   1. Initial program 77.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 99.9%

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

Alternative 3: 98.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 82.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. unpow2 N/A

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

      9. unpow3 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. mul-1-neg N/A

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

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

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

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

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

      16. metadata-eval N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

   10. Applied rewrites 100.0%

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

   if 5 < (*.f64 b b)

   1. Initial program 73.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 97.8%

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

Alternative 4: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in a around 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 99.1

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

5. Applied rewrites 99.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 99.1

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 99.9%

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

Alternative 5: 97.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(a - 4, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a - 1\\ \mathbf{if}\;a \leq -0.45:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+18}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* (* (fma (- a 4.0) a (fma (* b b) 2.0 4.0)) a) a) 1.0)))
   (if (<= a -0.45)
     t_0
     (if (<= a 6e+18) (- (* (* (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((a - 4.0), a, fma((b * b), 2.0, 4.0)) * a) * a) - 1.0;
	double tmp;
	if (a <= -0.45) {
		tmp = t_0;
	} else if (a <= 6e+18) {
		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 = Float64(Float64(Float64(fma(Float64(a - 4.0), a, fma(Float64(b * b), 2.0, 4.0)) * a) * a) - 1.0)
	tmp = 0.0
	if (a <= -0.45)
		tmp = t_0;
	elseif (a <= 6e+18)
		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[(N[(a - 4.0), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -0.45], t$95$0, If[LessEqual[a, 6e+18], 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 < -0.450000000000000011 or 6e18 < a

   1. Initial program 52.4%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.8%

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

   if -0.450000000000000011 < a < 6e18

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left(a \cdot {b}^{2}\right) + \left(12 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   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 98.5%

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

Alternative 6: 93.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a - 4, a, 4\right)\\ \mathbf{if}\;a \leq -1.08 \cdot 10^{+36}:\\ \;\;\;\;\left(t\_0 \cdot a\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{+70}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (- a 4.0) a 4.0)))
   (if (<= a -1.08e+36)
     (- (* (* t_0 a) a) 1.0)
     (if (<= a 1.02e+70)
       (- (* (* (fma b b (fma 4.0 a 12.0)) b) b) 1.0)
       (fma t_0 (* a a) -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma((a - 4.0), a, 4.0);
	double tmp;
	if (a <= -1.08e+36) {
		tmp = ((t_0 * a) * a) - 1.0;
	} else if (a <= 1.02e+70) {
		tmp = ((fma(b, b, fma(4.0, a, 12.0)) * b) * b) - 1.0;
	} else {
		tmp = fma(t_0, (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a - 4.0), a, 4.0)
	tmp = 0.0
	if (a <= -1.08e+36)
		tmp = Float64(Float64(Float64(t_0 * a) * a) - 1.0);
	elseif (a <= 1.02e+70)
		tmp = Float64(Float64(Float64(fma(b, b, fma(4.0, a, 12.0)) * b) * b) - 1.0);
	else
		tmp = fma(t_0, Float64(a * a), -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, -1.08e+36], N[(N[(N[(t$95$0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 1.02e+70], N[(N[(N[(N[(b * b + N[(4.0 * a + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], N[(t$95$0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.08000000000000001e36

   1. Initial program 76.7%

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

   3. Taylor expanded in a around 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. unpow2 N/A

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

      9. unpow3 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. mul-1-neg N/A

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

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

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

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

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

      16. metadata-eval N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

   10. Applied rewrites 99.9%

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

   if -1.08000000000000001e36 < a < 1.02e70

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left(a \cdot {b}^{2}\right) + \left(12 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   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 94.8%

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

   if 1.02e70 < a

   1. Initial program 12.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 100.0%

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

Alternative 7: 93.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a - 4, a, 4\right)\\ \mathbf{if}\;a \leq -1.08 \cdot 10^{+36}:\\ \;\;\;\;\left(t\_0 \cdot a\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, a, \mathsf{fma}\left(b, b, 12\right)\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (- a 4.0) a 4.0)))
   (if (<= a -1.08e+36)
     (- (* (* t_0 a) a) 1.0)
     (if (<= a 1.02e+70)
       (fma (fma 4.0 a (fma b b 12.0)) (* b b) -1.0)
       (fma t_0 (* a a) -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma((a - 4.0), a, 4.0);
	double tmp;
	if (a <= -1.08e+36) {
		tmp = ((t_0 * a) * a) - 1.0;
	} else if (a <= 1.02e+70) {
		tmp = fma(fma(4.0, a, fma(b, b, 12.0)), (b * b), -1.0);
	} else {
		tmp = fma(t_0, (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a - 4.0), a, 4.0)
	tmp = 0.0
	if (a <= -1.08e+36)
		tmp = Float64(Float64(Float64(t_0 * a) * a) - 1.0);
	elseif (a <= 1.02e+70)
		tmp = fma(fma(4.0, a, fma(b, b, 12.0)), Float64(b * b), -1.0);
	else
		tmp = fma(t_0, Float64(a * a), -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, -1.08e+36], N[(N[(N[(t$95$0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 1.02e+70], N[(N[(4.0 * a + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(t$95$0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.08000000000000001e36

   1. Initial program 76.7%

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

   3. Taylor expanded in a around 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. unpow2 N/A

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

      9. unpow3 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. mul-1-neg N/A

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

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

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

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

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

      16. metadata-eval N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

   10. Applied rewrites 99.9%

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

   if -1.08000000000000001e36 < a < 1.02e70

   1. Initial program 99.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied rewrites 94.8%

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

   if 1.02e70 < a

   1. Initial program 12.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 100.0%

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

Alternative 8: 93.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a - 4, a, 4\right)\\ \mathbf{if}\;a \leq -1.08 \cdot 10^{+36}:\\ \;\;\;\;\left(t\_0 \cdot a\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (- a 4.0) a 4.0)))
   (if (<= a -1.08e+36)
     (- (* (* t_0 a) a) 1.0)
     (if (<= a 1.02e+70)
       (fma (* (fma b b 12.0) b) b -1.0)
       (fma t_0 (* a a) -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma((a - 4.0), a, 4.0);
	double tmp;
	if (a <= -1.08e+36) {
		tmp = ((t_0 * a) * a) - 1.0;
	} else if (a <= 1.02e+70) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma(t_0, (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a - 4.0), a, 4.0)
	tmp = 0.0
	if (a <= -1.08e+36)
		tmp = Float64(Float64(Float64(t_0 * a) * a) - 1.0);
	elseif (a <= 1.02e+70)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(t_0, Float64(a * a), -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, -1.08e+36], N[(N[(N[(t$95$0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 1.02e+70], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(t$95$0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.08000000000000001e36

   1. Initial program 76.7%

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

   3. Taylor expanded in a around 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. unpow2 N/A

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

      9. unpow3 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. mul-1-neg N/A

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

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

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

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

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

      16. metadata-eval N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

   10. Applied rewrites 99.9%

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

   if -1.08000000000000001e36 < a < 1.02e70

   1. Initial program 99.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   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. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 94.7

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

   8. Applied rewrites 94.7%

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

   if 1.02e70 < a

   1. Initial program 12.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 100.0%

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

Alternative 9: 93.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right), a \cdot a, -1\right)\\ \mathbf{if}\;a \leq -1.08 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma (- a 4.0) a 4.0) (* a a) -1.0)))
   (if (<= a -1.08e+36)
     t_0
     (if (<= a 1.02e+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 <= -1.08e+36) {
		tmp = t_0;
	} else if (a <= 1.02e+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(fma(Float64(a - 4.0), a, 4.0), Float64(a * a), -1.0)
	tmp = 0.0
	if (a <= -1.08e+36)
		tmp = t_0;
	elseif (a <= 1.02e+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[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[a, -1.08e+36], t$95$0, If[LessEqual[a, 1.02e+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 < -1.08000000000000001e36 or 1.02e70 < a

   1. Initial program 46.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 100.0%

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

   if -1.08000000000000001e36 < a < 1.02e70

   1. Initial program 99.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   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. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 94.7

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

   8. Applied rewrites 94.7%

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

Alternative 10: 85.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e+38)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 6.8e+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 <= -3.3e+38) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 6.8e+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 <= -3.3e+38)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 6.8e+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, -3.3e+38], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 6.8e+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 < -3.2999999999999999e38

   1. Initial program 76.7%

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

   3. Taylor expanded in a around 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 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 80.2%

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

   if -3.2999999999999999e38 < a < 6.7999999999999995e153

   1. Initial program 94.4%

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

   3. Taylor expanded in a around 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 98.6

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

   5. Applied rewrites 98.6%

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

   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. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 90.3

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

   8. Applied rewrites 90.3%

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

   if 6.7999999999999995e153 < 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 11: 71.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-4, a, 4\right), -1\right)\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e+38)
   (fma (* a a) (fma -4.0 a 4.0) -1.0)
   (if (<= a 1.3e+145) (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 <= -3.3e+38) {
		tmp = fma((a * a), fma(-4.0, a, 4.0), -1.0);
	} else if (a <= 1.3e+145) {
		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 <= -3.3e+38)
		tmp = fma(Float64(a * a), fma(-4.0, a, 4.0), -1.0);
	elseif (a <= 1.3e+145)
		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, -3.3e+38], N[(N[(a * a), $MachinePrecision] * N[(-4.0 * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 1.3e+145], 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 < -3.2999999999999999e38

   1. Initial program 76.7%

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

   3. Taylor expanded in a around 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 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 80.2%

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

   if -3.2999999999999999e38 < a < 1.30000000000000001e145

   1. Initial program 96.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

   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. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 91.3

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 66.5%

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

   if 1.30000000000000001e145 < 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 92.9%

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

Alternative 12: 69.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+291) (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) <= 5e+291) {
		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) <= 5e+291)
		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], 5e+291], 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) < 5.0000000000000001e291

   1. Initial program 77.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in 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 78.3

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

   8. Applied rewrites 78.3%

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

   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 60.1%

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

   if 5.0000000000000001e291 < (*.f64 b b)

   1. Initial program 77.0%

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

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 98.6%

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

Alternative 13: 51.8% 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 77.6%

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

3. Taylor expanded in a around 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 99.1

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

5. Applied rewrites 99.1%

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

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. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. distribute-rgt-in N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 67.7

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

8. Applied rewrites 67.7%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 47.5%

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

Alternative 14: 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 77.6%

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

3. Taylor expanded in a around 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 99.1

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

5. Applied rewrites 99.1%

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

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 66.1

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

8. Applied rewrites 66.1%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 22.6%

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

Reproduce

herbie shell --seed 2024296 
(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))