Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.2% → 99.9%

Time: 11.2s

Alternatives: 17

Speedup: 9.3×

• 61.4% 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: 73.2% 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, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + (((a * a) * (4.0d0 + (a * (a + 4.0d0)))) + (b * (b * ((b * b) + (4.0d0 + (a * ((a * 2.0d0) + (-12.0d0))))))))
end function

Java

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

Python

def code(a, b):
	return -1.0 + (((a * a) * (4.0 + (a * (a + 4.0)))) + (b * (b * ((b * b) + (4.0 + (a * ((a * 2.0) + -12.0)))))))

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + (((a * a) * (4.0 + (a * (a + 4.0)))) + (b * (b * ((b * b) + (4.0 + (a * ((a * 2.0) + -12.0)))))));
end

Wolfram

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

Derivation

1. Initial program 69.8%

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

   1. associate--l+ N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

3. Simplified 69.8%

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

5. Taylor expanded in b around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \color{blue}{\left({b}^{2} \cdot \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)}\right), -1\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)}\right)\right), -1\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{b \cdot b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{-12}\right)\right)\right)\right), -1\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{{a}^{2}}{b}}{b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\color{blue}{a}, -12\right)\right)\right)\right), -1\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   10. *-lowering-*.f64 57.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

7. Simplified 57.0%

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

8. Taylor expanded in b around 0

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

    \[\leadsto -1 + \left(\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + b \cdot \left(b \cdot \left(b \cdot b + \left(4 + a \cdot \left(a \cdot 2 + -12\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 2: 97.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + \left(-1 + 4 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54)
   (+ (* (* a a) (+ 4.0 (* a (+ a 4.0)))) (+ -1.0 (* 4.0 (* b b))))
   (+ -1.0 (* (* b b) (+ 4.0 (+ (* b b) (* (* a a) 2.0)))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + (-1.0 + (4.0 * (b * b)));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = ((a * a) * (4.0d0 + (a * (a + 4.0d0)))) + ((-1.0d0) + (4.0d0 * (b * b)))
    else
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + ((b * b) + ((a * a) * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + (-1.0 + (4.0 * (b * b)));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + (-1.0 + (4.0 * (b * b)))
	else:
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + 4.0)))) + Float64(-1.0 + Float64(4.0 * Float64(b * b))));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(Float64(b * b) + Float64(Float64(a * a) * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + (-1.0 + (4.0 * (b * b)));
	else
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 79.0%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \color{blue}{\left({b}^{2} \cdot \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)}\right), -1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)}\right)\right), -1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{b \cdot b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{-12}\right)\right)\right)\right), -1\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{{a}^{2}}{b}}{b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\color{blue}{a}, -12\right)\right)\right)\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      10. *-lowering-*.f64 54.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 54.5%

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

   8. Taylor expanded in b around 0

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

   10. Simplified 99.9%

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

   11. Taylor expanded in a around 0

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \color{blue}{\left({b}^{2} \cdot \left(4 + {b}^{2}\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(4 + {b}^{2}\right)}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{4} + {b}^{2}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{4} + {b}^{2}\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + \color{blue}{4}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), \color{blue}{4}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right)\right)\right) \]

       7. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right)\right)\right) \]

   13. Simplified 99.6%

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

   14. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left({a}^{2} \cdot \left(4 + a \cdot \left(4 + a\right)\right)\right), \color{blue}{\left(4 \cdot {b}^{2} - 1\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left(4 + a \cdot \left(4 + a\right)\right)\right), \left(\color{blue}{4 \cdot {b}^{2}} - 1\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), \left(4 + a \cdot \left(4 + a\right)\right)\right), \left(\color{blue}{4} \cdot {b}^{2} - 1\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(4 + a \cdot \left(4 + a\right)\right)\right), \left(\color{blue}{4} \cdot {b}^{2} - 1\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \left(a \cdot \left(4 + a\right)\right)\right)\right), \left(4 \cdot \color{blue}{{b}^{2}} - 1\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(4 + a\right)\right)\right)\right), \left(4 \cdot {b}^{\color{blue}{2}} - 1\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \left(4 \cdot {b}^{2} - 1\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \left(4 \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \left(4 \cdot {b}^{2} + -1\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \left(-1 + \color{blue}{4 \cdot {b}^{2}}\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(4 \cdot {b}^{2}\right)}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \left({b}^{2} \cdot \color{blue}{4}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{4}\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(b \cdot b\right), 4\right)\right)\right) \]

       17. *-lowering-*.f64 98.4%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right)\right) \]

   16. Simplified 98.4%

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

   if 4.0000000000000003e54 < (*.f64 b b)

   1. Initial program 59.8%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 59.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(4 \cdot {b}^{2}\right)}, -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left({b}^{2}\right)\right), -1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(b \cdot b\right)\right), -1\right)\right) \]

      3. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(b, b\right)\right), -1\right)\right) \]

   7. Simplified 99.9%

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

   8. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left(4 + \left(2 \cdot {a}^{2} + {b}^{2}\right)\right)\right), \color{blue}{-1}\right) \]

   10. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + \left(-1 + 4 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \mathbf{if}\;a \leq -1.02 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-144}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1650:\\ \;\;\;\;-1 + 4 \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a (+ a 4.0)))))
   (if (<= a -1.02e+14)
     t_0
     (if (<= a -8e-144)
       (* (* b b) (* b b))
       (if (<= a 1650.0) (+ -1.0 (* 4.0 (* b b))) t_0)))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * (a + 4.0));
	double tmp;
	if (a <= -1.02e+14) {
		tmp = t_0;
	} else if (a <= -8e-144) {
		tmp = (b * b) * (b * b);
	} else if (a <= 1650.0) {
		tmp = -1.0 + (4.0 * (b * b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (a * (a + 4.0d0))
    if (a <= (-1.02d+14)) then
        tmp = t_0
    else if (a <= (-8d-144)) then
        tmp = (b * b) * (b * b)
    else if (a <= 1650.0d0) then
        tmp = (-1.0d0) + (4.0d0 * (b * b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) * (a * (a + 4.0));
	double tmp;
	if (a <= -1.02e+14) {
		tmp = t_0;
	} else if (a <= -8e-144) {
		tmp = (b * b) * (b * b);
	} else if (a <= 1650.0) {
		tmp = -1.0 + (4.0 * (b * b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * (a + 4.0))
	tmp = 0
	if a <= -1.02e+14:
		tmp = t_0
	elif a <= -8e-144:
		tmp = (b * b) * (b * b)
	elif a <= 1650.0:
		tmp = -1.0 + (4.0 * (b * b))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * Float64(a + 4.0)))
	tmp = 0.0
	if (a <= -1.02e+14)
		tmp = t_0;
	elseif (a <= -8e-144)
		tmp = Float64(Float64(b * b) * Float64(b * b));
	elseif (a <= 1650.0)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(b * b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * (a * (a + 4.0));
	tmp = 0.0;
	if (a <= -1.02e+14)
		tmp = t_0;
	elseif (a <= -8e-144)
		tmp = (b * b) * (b * b);
	elseif (a <= 1650.0)
		tmp = -1.0 + (4.0 * (b * b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.02e+14], t$95$0, If[LessEqual[a, -8e-144], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1650.0], N[(-1.0 + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.02e14 or 1650 < a

   1. Initial program 43.6%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 43.6%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{4}\right), \color{blue}{\left(1 + 4 \cdot \frac{1}{a}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{\left(2 \cdot 2\right)}\right), \left(1 + 4 \cdot \frac{1}{a}\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2} \cdot {a}^{2}\right), \left(\color{blue}{1} + 4 \cdot \frac{1}{a}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2}\right)\right), \left(\color{blue}{1} + 4 \cdot \frac{1}{a}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), \left({a}^{2}\right)\right), \left(1 + 4 \cdot \frac{1}{a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2}\right)\right), \left(1 + 4 \cdot \frac{1}{a}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a\right)\right), \left(1 + 4 \cdot \frac{1}{a}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right), \left(1 + 4 \cdot \frac{1}{a}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(4 \cdot \frac{1}{a}\right)}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(1, \left(\frac{4 \cdot 1}{\color{blue}{a}}\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(1, \left(\frac{4}{a}\right)\right)\right) \]

      12. /-lowering-/.f64 93.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(4, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 93.7%

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

   8. Taylor expanded in a around 0

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(a \cdot \left(4 + a\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{a} \cdot \left(4 + a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{a} \cdot \left(4 + a\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \color{blue}{\left(4 + a\right)}\right)\right) \]

      8. +-lowering-+.f64 93.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \color{blue}{a}\right)\right)\right) \]

   10. Simplified 93.7%

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

   if -1.02e14 < a < -7.9999999999999996e-144

   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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 72.5%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \color{blue}{\left(b \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{b} \cdot b\right)\right) \]

      4. *-lowering-*.f64 72.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   9. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]

   if -7.9999999999999996e-144 < a < 1650

   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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{-1}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), -1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), -1\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), -1\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), -1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), -1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), -1\right) \]

      13. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), -1\right) \]

   7. Simplified 98.2%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(4 \cdot {b}^{2}\right)}, -1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot 4\right), -1\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), 4\right), -1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 4\right), -1\right) \]

      4. *-lowering-*.f64 76.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right) \]

   10. Simplified 76.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+14}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-144}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1650:\\ \;\;\;\;-1 + 4 \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(4 + a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1000.0)
   (+ -1.0 (* a (* a (+ 4.0 (* a (+ a 4.0))))))
   (+ -1.0 (* (* b b) (+ 4.0 (+ (* b b) (* (* a a) 2.0)))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000.0) {
		tmp = -1.0 + (a * (a * (4.0 + (a * (a + 4.0)))));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1000.0d0) then
        tmp = (-1.0d0) + (a * (a * (4.0d0 + (a * (a + 4.0d0)))))
    else
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + ((b * b) + ((a * a) * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000.0) {
		tmp = -1.0 + (a * (a * (4.0 + (a * (a + 4.0)))));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1000.0:
		tmp = -1.0 + (a * (a * (4.0 + (a * (a + 4.0)))))
	else:
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1000.0)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a + 4.0))))));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(Float64(b * b) + Float64(Float64(a * a) * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1000.0)
		tmp = -1.0 + (a * (a * (4.0 + (a * (a + 4.0)))));
	else
		tmp = -1.0 + ((b * b) * (4.0 + ((b * b) + ((a * a) * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1000.0], N[(-1.0 + N[(a * N[(a * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e3

   1. Initial program 79.9%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.9%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \color{blue}{\left({b}^{2} \cdot \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)}\right), -1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)}\right)\right), -1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{b \cdot b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{-12}\right)\right)\right)\right), -1\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{{a}^{2}}{b}}{b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\color{blue}{a}, -12\right)\right)\right)\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      10. *-lowering-*.f64 54.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 54.7%

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

   8. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({a}^{2} \cdot \left(4 + 4 \cdot a\right) + {a}^{4}\right)}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(4 \cdot {a}^{2} + \left(4 \cdot a\right) \cdot {a}^{2}\right) + {\color{blue}{a}}^{4}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(4 \cdot {a}^{2} + \color{blue}{\left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{4}\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\color{blue}{\left(4 \cdot a\right) \cdot {a}^{2}} + {a}^{4}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

      9. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \color{blue}{\left(4 \cdot a + {a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \left(4 \cdot a + a \cdot \color{blue}{a}\right)\right)\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \left(a \cdot \color{blue}{\left(4 + a\right)}\right)\right)\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot \color{blue}{\left(4 + a \cdot \left(4 + a\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(4 + a \cdot \left(4 + a\right)\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{4} + a \cdot \left(4 + a\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{4} + a \cdot \left(4 + a\right)\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \color{blue}{\left(a \cdot \left(4 + a\right)\right)}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{\left(4 + a\right)}\right)\right)\right)\right) \]

      19. +-lowering-+.f64 98.5%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \color{blue}{a}\right)\right)\right)\right)\right) \]

   10. Simplified 98.5%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(a \cdot \color{blue}{\left(a \cdot \left(4 + a \cdot \left(4 + a\right)\right)\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(a \cdot \left(4 + a \cdot \left(4 + a\right)\right)\right) \cdot \color{blue}{a}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(a \cdot \left(4 + a \cdot \left(4 + a\right)\right)\right), \color{blue}{a}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(4 + a \cdot \left(4 + a\right)\right)\right), a\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \left(a \cdot \left(4 + a\right)\right)\right)\right), a\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(4 + a\right)\right)\right)\right), a\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a + 4\right)\right)\right)\right), a\right)\right) \]

       8. +-lowering-+.f64 98.6%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, 4\right)\right)\right)\right), a\right)\right) \]

   12. Applied egg-rr 98.6%

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

   if 1e3 < (*.f64 b b)

   1. Initial program 59.5%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 59.5%

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

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(4 \cdot {b}^{2}\right)}, -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left({b}^{2}\right)\right), -1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(b \cdot b\right)\right), -1\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(b, b\right)\right), -1\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left(4 + \left(2 \cdot {a}^{2} + {b}^{2}\right)\right)\right), \color{blue}{-1}\right) \]

   10. Simplified 99.2%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(4 + a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return -1.0 + (((a * a) * (4.0 + (a * (a + 4.0)))) + ((b * b) * (4.0 + (b * b))));
}

Python

def code(a, b):
	return -1.0 + (((a * a) * (4.0 + (a * (a + 4.0)))) + ((b * b) * (4.0 + (b * b))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

   1. associate--l+ N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

3. Simplified 69.8%

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

5. Taylor expanded in b around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \color{blue}{\left({b}^{2} \cdot \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)}\right), -1\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)}\right)\right), -1\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{b \cdot b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{-12}\right)\right)\right)\right), -1\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{{a}^{2}}{b}}{b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\color{blue}{a}, -12\right)\right)\right)\right), -1\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   10. *-lowering-*.f64 57.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

7. Simplified 57.0%

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

8. Taylor expanded in b around 0

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

10. Simplified 99.9%

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

11. Taylor expanded in a around 0

    \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \color{blue}{\left({b}^{2} \cdot \left(4 + {b}^{2}\right)\right)}\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(4 + {b}^{2}\right)}\right)\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{4} + {b}^{2}\right)\right)\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{4} + {b}^{2}\right)\right)\right)\right) \]

    4. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + \color{blue}{4}\right)\right)\right)\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), \color{blue}{4}\right)\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right)\right)\right) \]

    7. *-lowering-*.f64 99.6%

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, a\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right)\right)\right) \]

13. Simplified 99.6%

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

14. Final simplification 99.6%

    \[\leadsto -1 + \left(\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\right) \]
15. Add Preprocessing

Alternative 6: 80.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -2.6 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-144}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 4000:\\ \;\;\;\;-1 + 4 \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -2.6e+15)
     t_0
     (if (<= a -8e-144)
       (* (* b b) (* b b))
       (if (<= a 4000.0) (+ -1.0 (* 4.0 (* b b))) t_0)))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -2.6e+15) {
		tmp = t_0;
	} else if (a <= -8e-144) {
		tmp = (b * b) * (b * b);
	} else if (a <= 4000.0) {
		tmp = -1.0 + (4.0 * (b * b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * (a * a))
    if (a <= (-2.6d+15)) then
        tmp = t_0
    else if (a <= (-8d-144)) then
        tmp = (b * b) * (b * b)
    else if (a <= 4000.0d0) then
        tmp = (-1.0d0) + (4.0d0 * (b * b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -2.6e+15) {
		tmp = t_0;
	} else if (a <= -8e-144) {
		tmp = (b * b) * (b * b);
	} else if (a <= 4000.0) {
		tmp = -1.0 + (4.0 * (b * b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if a <= -2.6e+15:
		tmp = t_0
	elif a <= -8e-144:
		tmp = (b * b) * (b * b)
	elif a <= 4000.0:
		tmp = -1.0 + (4.0 * (b * b))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (a <= -2.6e+15)
		tmp = t_0;
	elseif (a <= -8e-144)
		tmp = Float64(Float64(b * b) * Float64(b * b));
	elseif (a <= 4000.0)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(b * b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * a));
	tmp = 0.0;
	if (a <= -2.6e+15)
		tmp = t_0;
	elseif (a <= -8e-144)
		tmp = (b * b) * (b * b);
	elseif (a <= 4000.0)
		tmp = -1.0 + (4.0 * (b * b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.6e+15], t$95$0, If[LessEqual[a, -8e-144], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4000.0], N[(-1.0 + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.6e15 or 4e3 < a

   1. Initial program 43.6%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 43.6%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left({\color{blue}{a}}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({\color{blue}{a}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \color{blue}{a}\right)\right) \]

      7. *-lowering-*.f64 93.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right) \]

   7. Simplified 93.4%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. associate-*r* N/A

         \[\leadsto \left(a \cdot \left(a \cdot a\right)\right) \cdot a \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), \color{blue}{a}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot a\right)\right), a\right) \]

      5. *-lowering-*.f64 93.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), a\right) \]

   9. Applied egg-rr 93.5%

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

   if -2.6e15 < a < -7.9999999999999996e-144

   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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 72.5%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 72.5%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \color{blue}{\left(b \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{b} \cdot b\right)\right) \]

      4. *-lowering-*.f64 72.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   9. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]

   if -7.9999999999999996e-144 < a < 4e3

   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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{-1}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), -1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), -1\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), -1\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), -1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), -1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), -1\right) \]

      13. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), -1\right) \]

   7. Simplified 98.2%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(4 \cdot {b}^{2}\right)}, -1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot 4\right), -1\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), 4\right), -1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 4\right), -1\right) \]

      4. *-lowering-*.f64 76.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right) \]

   10. Simplified 76.4%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+15}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-144}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 4000:\\ \;\;\;\;-1 + 4 \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(4 + a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54)
   (+ -1.0 (* a (* a (+ 4.0 (* a (+ a 4.0))))))
   (+ -1.0 (* (* b b) (+ 4.0 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = -1.0 + (a * (a * (4.0 + (a * (a + 4.0)))));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = (-1.0d0) + (a * (a * (4.0d0 + (a * (a + 4.0d0)))))
    else
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = -1.0 + (a * (a * (4.0 + (a * (a + 4.0)))));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = -1.0 + (a * (a * (4.0 + (a * (a + 4.0)))))
	else:
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a + 4.0))))));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = -1.0 + (a * (a * (4.0 + (a * (a + 4.0)))));
	else
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(-1.0 + N[(a * N[(a * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 79.0%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \color{blue}{\left({b}^{2} \cdot \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)}\right), -1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)}\right)\right), -1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{b \cdot b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{-12}\right)\right)\right)\right), -1\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{{a}^{2}}{b}}{b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\color{blue}{a}, -12\right)\right)\right)\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      10. *-lowering-*.f64 54.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 54.5%

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

   8. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({a}^{2} \cdot \left(4 + 4 \cdot a\right) + {a}^{4}\right)}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(4 \cdot {a}^{2} + \left(4 \cdot a\right) \cdot {a}^{2}\right) + {\color{blue}{a}}^{4}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(4 \cdot {a}^{2} + \color{blue}{\left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{4}\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\color{blue}{\left(4 \cdot a\right) \cdot {a}^{2}} + {a}^{4}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

      9. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \color{blue}{\left(4 \cdot a + {a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \left(4 \cdot a + a \cdot \color{blue}{a}\right)\right)\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \left(a \cdot \color{blue}{\left(4 + a\right)}\right)\right)\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot \color{blue}{\left(4 + a \cdot \left(4 + a\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(4 + a \cdot \left(4 + a\right)\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{4} + a \cdot \left(4 + a\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{4} + a \cdot \left(4 + a\right)\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \color{blue}{\left(a \cdot \left(4 + a\right)\right)}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{\left(4 + a\right)}\right)\right)\right)\right) \]

      19. +-lowering-+.f64 97.8%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \color{blue}{a}\right)\right)\right)\right)\right) \]

   10. Simplified 97.8%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(a \cdot \color{blue}{\left(a \cdot \left(4 + a \cdot \left(4 + a\right)\right)\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(a \cdot \left(4 + a \cdot \left(4 + a\right)\right)\right) \cdot \color{blue}{a}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(a \cdot \left(4 + a \cdot \left(4 + a\right)\right)\right), \color{blue}{a}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(4 + a \cdot \left(4 + a\right)\right)\right), a\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \left(a \cdot \left(4 + a\right)\right)\right)\right), a\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(4 + a\right)\right)\right)\right), a\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \left(a + 4\right)\right)\right)\right), a\right)\right) \]

       8. +-lowering-+.f64 97.9%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, 4\right)\right)\right)\right), a\right)\right) \]

   12. Applied egg-rr 97.9%

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

   if 4.0000000000000003e54 < (*.f64 b b)

   1. Initial program 59.8%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 59.8%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{-1}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), -1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), -1\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), -1\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), -1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), -1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), -1\right) \]

      13. *-lowering-*.f64 96.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), -1\right) \]

   7. Simplified 96.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(4 + a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54)
   (+ -1.0 (* (* a a) (+ 4.0 (* a (+ a 4.0)))))
   (+ -1.0 (* (* b b) (+ 4.0 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 + (a * (a + 4.0d0))))
    else
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))))
	else:
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + 4.0)))));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	else
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 79.0%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \color{blue}{\left({b}^{2} \cdot \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)}\right), -1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)}\right)\right), -1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{b \cdot b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{-12}\right)\right)\right)\right), -1\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{{a}^{2}}{b}}{b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\color{blue}{a}, -12\right)\right)\right)\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      10. *-lowering-*.f64 54.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 54.5%

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

   8. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({a}^{2} \cdot \left(4 + 4 \cdot a\right) + {a}^{4}\right)}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(4 \cdot {a}^{2} + \left(4 \cdot a\right) \cdot {a}^{2}\right) + {\color{blue}{a}}^{4}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(4 \cdot {a}^{2} + \color{blue}{\left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{4}\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\color{blue}{\left(4 \cdot a\right) \cdot {a}^{2}} + {a}^{4}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

      9. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \color{blue}{\left(4 \cdot a + {a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \left(4 \cdot a + a \cdot \color{blue}{a}\right)\right)\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \left(a \cdot \color{blue}{\left(4 + a\right)}\right)\right)\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot \color{blue}{\left(4 + a \cdot \left(4 + a\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(4 + a \cdot \left(4 + a\right)\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{4} + a \cdot \left(4 + a\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{4} + a \cdot \left(4 + a\right)\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \color{blue}{\left(a \cdot \left(4 + a\right)\right)}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{\left(4 + a\right)}\right)\right)\right)\right) \]

      19. +-lowering-+.f64 97.8%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \color{blue}{a}\right)\right)\right)\right)\right) \]

   10. Simplified 97.8%

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

   if 4.0000000000000003e54 < (*.f64 b b)

   1. Initial program 59.8%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 59.8%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{-1}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), -1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), -1\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), -1\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), -1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), -1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), -1\right) \]

      13. *-lowering-*.f64 96.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), -1\right) \]

   7. Simplified 96.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54)
   (+ -1.0 (* (* a a) (+ (* a a) 4.0)))
   (+ -1.0 (* (* b b) (+ 4.0 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = -1.0 + ((a * a) * ((a * a) + 4.0));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = (-1.0d0) + ((a * a) * ((a * a) + 4.0d0))
    else
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = -1.0 + ((a * a) * ((a * a) + 4.0));
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = -1.0 + ((a * a) * ((a * a) + 4.0))
	else:
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(Float64(a * a) + 4.0)));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = -1.0 + ((a * a) * ((a * a) + 4.0));
	else
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 79.0%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \color{blue}{\left({b}^{2} \cdot \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)}\right), -1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)}\right)\right), -1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{b \cdot b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{-12}\right)\right)\right)\right), -1\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{{a}^{2}}{b}}{b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\color{blue}{a}, -12\right)\right)\right)\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      10. *-lowering-*.f64 54.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 54.5%

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

   8. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({a}^{2} \cdot \left(4 + 4 \cdot a\right) + {a}^{4}\right)}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(4 \cdot {a}^{2} + \left(4 \cdot a\right) \cdot {a}^{2}\right) + {\color{blue}{a}}^{4}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(4 \cdot {a}^{2} + \color{blue}{\left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{4}\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\color{blue}{\left(4 \cdot a\right) \cdot {a}^{2}} + {a}^{4}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

      9. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \color{blue}{\left(4 \cdot a + {a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \left(4 \cdot a + a \cdot \color{blue}{a}\right)\right)\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \left(a \cdot \color{blue}{\left(4 + a\right)}\right)\right)\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot \color{blue}{\left(4 + a \cdot \left(4 + a\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(4 + a \cdot \left(4 + a\right)\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{4} + a \cdot \left(4 + a\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{4} + a \cdot \left(4 + a\right)\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \color{blue}{\left(a \cdot \left(4 + a\right)\right)}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{\left(4 + a\right)}\right)\right)\right)\right) \]

      19. +-lowering-+.f64 97.8%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \color{blue}{a}\right)\right)\right)\right)\right) \]

   10. Simplified 97.8%

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

   11. Taylor expanded in a around inf

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \color{blue}{\left({a}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 96.2%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   13. Simplified 96.2%

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

   if 4.0000000000000003e54 < (*.f64 b b)

   1. Initial program 59.8%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 59.8%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{-1}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), -1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), -1\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), -1\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), -1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), -1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), -1\right) \]

      13. *-lowering-*.f64 96.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), -1\right) \]

   7. Simplified 96.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54)
   (+ -1.0 (* (* a a) (+ (* a a) 4.0)))
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = -1.0 + ((a * a) * ((a * a) + 4.0));
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = (-1.0d0) + ((a * a) * ((a * a) + 4.0d0))
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = -1.0 + ((a * a) * ((a * a) + 4.0));
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = -1.0 + ((a * a) * ((a * a) + 4.0))
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(Float64(a * a) + 4.0)));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = -1.0 + ((a * a) * ((a * a) + 4.0));
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 79.0%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \color{blue}{\left({b}^{2} \cdot \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)}\right), -1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 + \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)}\right)\right), -1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{{a}^{2}}{b \cdot b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{-12}\right)\right)\right)\right), -1\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{{a}^{2}}{b}}{b}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \color{blue}{\mathsf{*.f64}\left(a, -12\right)}\right)\right)\right), -1\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\color{blue}{a}, -12\right)\right)\right)\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

      10. *-lowering-*.f64 54.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, 4\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, -12\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 54.5%

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

   8. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({a}^{2} \cdot \left(4 + 4 \cdot a\right) + {a}^{4}\right)}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(4 \cdot {a}^{2} + \left(4 \cdot a\right) \cdot {a}^{2}\right) + {\color{blue}{a}}^{4}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(4 \cdot {a}^{2} + \color{blue}{\left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{4}\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\color{blue}{\left(4 \cdot a\right) \cdot {a}^{2}} + {a}^{4}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

      9. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + \left(\left(4 \cdot a\right) \cdot {a}^{2} + {a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \color{blue}{\left(4 \cdot a + {a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \left(4 \cdot a + a \cdot \color{blue}{a}\right)\right)\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot 4 + {a}^{2} \cdot \left(a \cdot \color{blue}{\left(4 + a\right)}\right)\right)\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({a}^{2} \cdot \color{blue}{\left(4 + a \cdot \left(4 + a\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(4 + a \cdot \left(4 + a\right)\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{4} + a \cdot \left(4 + a\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{4} + a \cdot \left(4 + a\right)\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \color{blue}{\left(a \cdot \left(4 + a\right)\right)}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{\left(4 + a\right)}\right)\right)\right)\right) \]

      19. +-lowering-+.f64 97.8%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(4, \color{blue}{a}\right)\right)\right)\right)\right) \]

   10. Simplified 97.8%

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

   11. Taylor expanded in a around inf

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \color{blue}{\left({a}^{2}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 96.2%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   13. Simplified 96.2%

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

   if 4.0000000000000003e54 < (*.f64 b b)

   1. Initial program 59.8%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 59.8%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 96.5%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \color{blue}{\left(b \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{b} \cdot b\right)\right) \]

      4. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   9. Applied egg-rr 96.6%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.1% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54) (+ -1.0 (* a (* a 4.0))) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = (-1.0d0) + (a * (a * 4.0d0))
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = -1.0 + (a * (a * 4.0))
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64(-1.0 + Float64(a * Float64(a * 4.0)));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = -1.0 + (a * (a * 4.0));
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(-1.0 + N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 79.0%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1\right), \color{blue}{\left(a \cdot \left(-12 \cdot {b}^{2} + a \cdot \left(4 + 2 \cdot {b}^{2}\right)\right)\right)}\right) \]

   7. Simplified 66.7%

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

   8. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

         \[\leadsto 4 \cdot {a}^{2} + -1 \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {a}^{2}\right), \color{blue}{-1}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(a \cdot a\right)\right), -1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot a\right) \cdot a\right), -1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot \left(4 \cdot a\right)\right), -1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(4 \cdot a\right)\right), -1\right) \]

      8. *-lowering-*.f64 65.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(4, a\right)\right), -1\right) \]

   10. Simplified 65.0%

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

   if 4.0000000000000003e54 < (*.f64 b b)

   1. Initial program 59.8%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 59.8%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 96.5%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \color{blue}{\left(b \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{b} \cdot b\right)\right) \]

      4. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   9. Applied egg-rr 96.6%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54) (* a (* a (* a a))) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = a * (a * (a * a));
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = a * (a * (a * a))
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = a * (a * (a * a));
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = a * (a * (a * a))
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = a * (a * (a * a));
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 79.0%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left({\color{blue}{a}}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({\color{blue}{a}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \color{blue}{a}\right)\right) \]

      7. *-lowering-*.f64 58.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right) \]

   7. Simplified 58.7%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. associate-*r* N/A

         \[\leadsto \left(a \cdot \left(a \cdot a\right)\right) \cdot a \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), \color{blue}{a}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot a\right)\right), a\right) \]

      5. *-lowering-*.f64 58.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), a\right) \]

   9. Applied egg-rr 58.7%

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

   if 4.0000000000000003e54 < (*.f64 b b)

   1. Initial program 59.8%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 59.8%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 96.5%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \color{blue}{\left(b \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{b} \cdot b\right)\right) \]

      4. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   9. Applied egg-rr 96.6%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54) (* a (* a (* a a))) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = a * (a * (a * a));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = a * (a * (a * a))
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = a * (a * (a * a));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = a * (a * (a * a))
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = a * (a * (a * a));
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 79.0%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left({\color{blue}{a}}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({\color{blue}{a}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \color{blue}{a}\right)\right) \]

      7. *-lowering-*.f64 58.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right) \]

   7. Simplified 58.7%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. associate-*r* N/A

         \[\leadsto \left(a \cdot \left(a \cdot a\right)\right) \cdot a \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), \color{blue}{a}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot a\right)\right), a\right) \]

      5. *-lowering-*.f64 58.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), a\right) \]

   9. Applied egg-rr 58.7%

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

   if 4.0000000000000003e54 < (*.f64 b b)

   1. Initial program 59.8%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 59.8%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 96.5%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+54}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+54) (* (* a a) (* a a)) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+54) then
        tmp = (a * a) * (a * a)
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+54) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+54:
		tmp = (a * a) * (a * a)
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+54)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+54)
		tmp = (a * a) * (a * a);
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+54], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.0000000000000003e54

   1. Initial program 79.0%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.0%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left({\color{blue}{a}}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({\color{blue}{a}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \color{blue}{a}\right)\right) \]

      7. *-lowering-*.f64 58.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right) \]

   7. Simplified 58.7%

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

   if 4.0000000000000003e54 < (*.f64 b b)

   1. Initial program 59.8%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 59.8%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 96.5%

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

Alternative 15: 68.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.0005:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0005) -1.0 (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0005) {
		tmp = -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.0005d0) then
        tmp = -1.0d0
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0005) {
		tmp = -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.0005:
		tmp = -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0005)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.0005)
		tmp = -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0005], -1.0, N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000001e-4

   1. Initial program 79.5%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.5%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{-1}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), -1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), -1\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), -1\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), -1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), -1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), -1\right) \]

      13. *-lowering-*.f64 41.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), -1\right) \]

   7. Simplified 41.8%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1} \]
   9. Step-by-step derivation

   10. Simplified 40.8%

       \[\leadsto \color{blue}{-1} \]

   if 5.0000000000000001e-4 < (*.f64 b b)

   1. Initial program 60.1%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 60.1%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 92.9%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 92.9%

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

Alternative 16: 51.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.235:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 0.235) -1.0 (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.235) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.235d0) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.235) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.235:
		tmp = -1.0
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.235)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.235)
		tmp = -1.0;
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.235], -1.0, N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.23499999999999999

   1. Initial program 79.5%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 79.5%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{-1}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), -1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), -1\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), -1\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), -1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), -1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), -1\right) \]

      13. *-lowering-*.f64 41.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), -1\right) \]

   7. Simplified 41.8%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1} \]
   9. Step-by-step derivation

   10. Simplified 40.8%

       \[\leadsto \color{blue}{-1} \]

   if 0.23499999999999999 < (*.f64 b b)

   1. Initial program 60.1%

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 60.1%

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

   5. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{-1}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), -1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), -1\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), -1\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), -1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), -1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), -1\right) \]

      13. *-lowering-*.f64 93.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), -1\right) \]

   7. Simplified 93.0%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(4 \cdot {b}^{2}\right)}, -1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot 4\right), -1\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), 4\right), -1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), 4\right), -1\right) \]

      4. *-lowering-*.f64 55.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right), -1\right) \]

   10. Simplified 55.3%

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

   11. Taylor expanded in b around inf

       \[\leadsto \color{blue}{4 \cdot {b}^{2}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {b}^{2} \cdot \color{blue}{4} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{4}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), 4\right) \]

       4. *-lowering-*.f64 55.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right) \]

   13. Simplified 55.3%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot 4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.235:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 24.8% accurate, 130.0× 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 69.8%

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

   1. associate--l+ N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot a + b \cdot b\right)}^{2}\right), \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a + b \cdot b\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

3. Simplified 69.8%

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

5. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot {b}^{2} + {b}^{4}\right), \color{blue}{-1}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), -1\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), -1\right) \]

   6. pow-sqr N/A

      \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), -1\right) \]

   7. distribute-rgt-out N/A

      \[\leadsto \mathsf{+.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), -1\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), -1\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), -1\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), -1\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), -1\right) \]

   13. *-lowering-*.f64 67.4%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), -1\right) \]

7. Simplified 67.4%

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

8. Taylor expanded in b around 0

   \[\leadsto \color{blue}{-1} \]
9. Step-by-step derivation

10. Simplified 20.7%

    \[\leadsto \color{blue}{-1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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))