Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.0% → 99.0%

Time: 7.9s

Alternatives: 12

Speedup: 5.5×

• 60.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% 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: 99.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 94.4%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 99.0%

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

Alternative 2: 94.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, 4, 12\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 100000000000.0)
   (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)
   (- (* (* (fma b b (fma a 4.0 12.0)) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 100000000000.0) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = ((fma(b, b, fma(a, 4.0, 12.0)) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 100000000000.0)
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, fma(a, 4.0, 12.0)) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 100000000000.0], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(N[(b * b + N[(a * 4.0 + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e11

   1. Initial program 80.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in b around 0

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 98.7

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

   8. Applied rewrites 98.7%

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

   10. Applied rewrites 98.8%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 99.6%

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

   if 1e11 < (*.f64 b b)

   1. Initial program 54.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left(a \cdot {b}^{2}\right) + \left(12 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   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 92.2%

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

Alternative 3: 98.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 94.4%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 99.0%

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

12. Taylor expanded in a around inf

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

14. Applied rewrites 98.0%

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

Alternative 4: 94.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 100000000000.0)
   (fma (* (fma (- a 4.0) a 4.0) a) a -1.0)
   (fma (* (* b b) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 100000000000.0) {
		tmp = fma((fma((a - 4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma(((b * b) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 100000000000.0)
		tmp = fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 100000000000.0], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e11

   1. Initial program 80.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in b around 0

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 98.7

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

   8. Applied rewrites 98.7%

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

   10. Applied rewrites 98.8%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 99.6%

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

   if 1e11 < (*.f64 b b)

   1. Initial program 54.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 91.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 91.0%

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

Alternative 5: 94.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 100000000000.0)
   (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)
   (fma (* (* b b) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 100000000000.0) {
		tmp = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	} else {
		tmp = fma(((b * b) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 100000000000.0)
		tmp = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0);
	else
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 100000000000.0], N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e11

   1. Initial program 80.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in b around 0

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 99.5%

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

   if 1e11 < (*.f64 b b)

   1. Initial program 54.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 91.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 91.0%

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

Alternative 6: 83.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (* a a) 4.0 -1.0)))
   (if (<= a -2.5e+117)
     t_0
     (if (<= a 6.8e+153) (fma (* b b) (* b b) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma((a * a), 4.0, -1.0);
	double tmp;
	if (a <= -2.5e+117) {
		tmp = t_0;
	} else if (a <= 6.8e+153) {
		tmp = fma((b * b), (b * b), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(a * a), 4.0, -1.0)
	tmp = 0.0
	if (a <= -2.5e+117)
		tmp = t_0;
	elseif (a <= 6.8e+153)
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]}, If[LessEqual[a, -2.5e+117], t$95$0, If[LessEqual[a, 6.8e+153], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.49999999999999992e117 or 6.7999999999999995e153 < a

   1. Initial program 25.3%

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 98.8

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

   8. Applied rewrites 98.8%

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

   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 90.0%

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

   if -2.49999999999999992e117 < a < 6.7999999999999995e153

   1. Initial program 86.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 99.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 81.5%

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

Alternative 7: 93.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 100000000000.0)
   (fma (* (* a a) a) a -1.0)
   (fma (* (* b b) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 100000000000.0) {
		tmp = fma(((a * a) * a), a, -1.0);
	} else {
		tmp = fma(((b * b) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 100000000000.0)
		tmp = fma(Float64(Float64(a * a) * a), a, -1.0);
	else
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 100000000000.0], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e11

   1. Initial program 80.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in b around 0

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 98.7

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

   8. Applied rewrites 98.7%

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

   10. Applied rewrites 98.8%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 96.7%

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

   if 1e11 < (*.f64 b b)

   1. Initial program 54.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 91.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 91.0%

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

Alternative 8: 93.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 100000000000.0)
   (fma (* (* a a) a) a -1.0)
   (fma (* b b) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 100000000000.0) {
		tmp = fma(((a * a) * a), a, -1.0);
	} else {
		tmp = fma((b * b), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 100000000000.0)
		tmp = fma(Float64(Float64(a * a) * a), a, -1.0);
	else
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 100000000000.0], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e11

   1. Initial program 80.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in b around 0

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 98.7

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

   8. Applied rewrites 98.7%

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

   10. Applied rewrites 98.8%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 96.7%

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

   if 1e11 < (*.f64 b b)

   1. Initial program 54.7%

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

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 99.2%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 91.0%

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

Alternative 9: 93.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 100000000000.0)
   (fma (* a a) (* a a) -1.0)
   (fma (* b b) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 100000000000.0) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma((b * b), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 100000000000.0)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 100000000000.0], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e11

   1. Initial program 80.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in b around 0

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 96.7%

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

   if 1e11 < (*.f64 b b)

   1. Initial program 54.7%

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

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 99.2%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 91.0%

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

Alternative 10: 69.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+293) (fma (* a a) 4.0 -1.0) (fma 12.0 (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+293) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = fma(12.0, (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+293)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = fma(12.0, Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+293], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(12.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.9999999999999998e293

   1. Initial program 73.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in b around 0

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 80.9

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

   8. Applied rewrites 80.9%

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

   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 58.8%

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

   if 1.9999999999999998e293 < (*.f64 b b)

   1. Initial program 48.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-rgt-identity N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   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 96.4%

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

Alternative 11: 51.5% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. *-rgt-identity N/A

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

   12. *-inverses N/A

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

   13. associate-/l* N/A

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

   14. associate-*l/ N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

8. Applied rewrites 69.6%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 50.2%

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

Alternative 12: 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 66.7%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. *-rgt-identity N/A

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

   12. *-inverses N/A

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

   13. associate-/l* N/A

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

   14. associate-*l/ N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

8. Applied rewrites 69.6%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 21.8%

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

Reproduce

herbie shell --seed 2024257 
(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))