Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.3% → 99.9%

Time: 12.1s

Alternatives: 20

Speedup: 5.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 2 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(a, a + 4, 4\right), a, \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), \mathsf{fma}\left(b, b, 4\right)\right), b \cdot b, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, -12\right), \mathsf{fma}\left(b, b, 4\right)\right), b, a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
      2e+191)
   (fma
    (* a (fma a (+ a 4.0) 4.0))
    a
    (fma (fma a (fma 2.0 a -12.0) (fma b b 4.0)) (* b b) -1.0))
   (fma (* b (fma a (fma a 2.0 -12.0) (fma b b 4.0))) b (* a (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 2e+191) {
		tmp = fma((a * fma(a, (a + 4.0), 4.0)), a, fma(fma(a, fma(2.0, a, -12.0), fma(b, b, 4.0)), (b * b), -1.0));
	} else {
		tmp = fma((b * fma(a, fma(a, 2.0, -12.0), fma(b, b, 4.0))), b, (a * (a * (a * a))));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= 2e+191)
		tmp = fma(Float64(a * fma(a, Float64(a + 4.0), 4.0)), a, fma(fma(a, fma(2.0, a, -12.0), fma(b, b, 4.0)), Float64(b * b), -1.0));
	else
		tmp = fma(Float64(b * fma(a, fma(a, 2.0, -12.0), fma(b, b, 4.0))), b, Float64(a * Float64(a * Float64(a * a))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+191], N[(N[(a * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * a + N[(N[(a * N[(2.0 * a + -12.0), $MachinePrecision] + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(a * N[(a * 2.0 + -12.0), $MachinePrecision] + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < 2.00000000000000015e191

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.9%

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

   if 2.00000000000000015e191 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 61.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 99.4%

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

   12. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 2 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(a, a + 4, 4\right), a, \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), \mathsf{fma}\left(b, b, 4\right)\right), b \cdot b, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, -12\right), \mathsf{fma}\left(b, b, 4\right)\right), b, a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 10000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), 4\right), \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, -12\right), \mathsf{fma}\left(b, b, 4\right)\right), b, a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
      10000000.0)
   (fma
    (* b b)
    (fma a (fma 2.0 a -12.0) 4.0)
    (fma (* a a) (fma a (+ a 4.0) 4.0) -1.0))
   (fma (* b (fma a (fma a 2.0 -12.0) (fma b b 4.0))) b (* a (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 10000000.0) {
		tmp = fma((b * b), fma(a, fma(2.0, a, -12.0), 4.0), fma((a * a), fma(a, (a + 4.0), 4.0), -1.0));
	} else {
		tmp = fma((b * fma(a, fma(a, 2.0, -12.0), fma(b, b, 4.0))), b, (a * (a * (a * a))));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= 10000000.0)
		tmp = fma(Float64(b * b), fma(a, fma(2.0, a, -12.0), 4.0), fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), -1.0));
	else
		tmp = fma(Float64(b * fma(a, fma(a, 2.0, -12.0), fma(b, b, 4.0))), b, Float64(a * Float64(a * Float64(a * a))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 10000000.0], N[(N[(b * b), $MachinePrecision] * N[(a * N[(2.0 * a + -12.0), $MachinePrecision] + 4.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(a * N[(a * 2.0 + -12.0), $MachinePrecision] + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < 1e7

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.0%

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

   if 1e7 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 65.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 99.2%

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

   12. Applied rewrites 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 10000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), 4\right), \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, -12\right), \mathsf{fma}\left(b, b, 4\right)\right), b, a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))