Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.2% → 98.0%

Time: 14.8s

Alternatives: 11

Speedup: 9.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% 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: 98.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 + a \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot \left(-12 + a \cdot 2\right) + \left(b \cdot b + 4\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1000000000.0)
   (+ (* (* a a) (+ (* a a) (+ 4.0 (* a 4.0)))) -1.0)
   (+ (* (* b b) (+ (* a (+ -12.0 (* a 2.0))) (+ (* b b) 4.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000000.0) {
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	} else {
		tmp = ((b * b) * ((a * (-12.0 + (a * 2.0))) + ((b * b) + 4.0))) + -1.0;
	}
	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) <= 1000000000.0d0) then
        tmp = ((a * a) * ((a * a) + (4.0d0 + (a * 4.0d0)))) + (-1.0d0)
    else
        tmp = ((b * b) * ((a * ((-12.0d0) + (a * 2.0d0))) + ((b * b) + 4.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000000.0) {
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	} else {
		tmp = ((b * b) * ((a * (-12.0 + (a * 2.0))) + ((b * b) + 4.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1000000000.0:
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0
	else:
		tmp = ((b * b) * ((a * (-12.0 + (a * 2.0))) + ((b * b) + 4.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1000000000.0)
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(a * a) + Float64(4.0 + Float64(a * 4.0)))) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(a * Float64(-12.0 + Float64(a * 2.0))) + Float64(Float64(b * b) + 4.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1000000000.0)
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	else
		tmp = ((b * b) * ((a * (-12.0 + (a * 2.0))) + ((b * b) + 4.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1000000000.0], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(4.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(a * N[(-12.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e9

   1. Initial program 81.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-lowering-*.f64 99.6%

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

   5. Simplified 99.6%

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

   if 1e9 < (*.f64 b b)

   1. Initial program 71.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

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

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

   5. Simplified 93.7%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in b around 0

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

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

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

       2. unpow2 N/A

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

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

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

       4. +-commutative N/A

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

       5. associate-+r+ N/A

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

       6. associate-+l+ N/A

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

       7. +-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*r* N/A

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

       11. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 97.6%

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

   11. Simplified 97.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 + a \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot \left(-12 + a \cdot 2\right) + \left(b \cdot b + 4\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;b \leq 1.18 \cdot 10^{-237}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-64}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= b 1.18e-237)
     -1.0
     (if (<= b 5.6e-136)
       t_0
       (if (<= b 1.7e-64) -1.0 (if (<= b 7.6e+20) t_0 (* b (* b (* b b)))))))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (b <= 1.18e-237) {
		tmp = -1.0;
	} else if (b <= 5.6e-136) {
		tmp = t_0;
	} else if (b <= 1.7e-64) {
		tmp = -1.0;
	} else if (b <= 7.6e+20) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = a * (a * (a * a))
    if (b <= 1.18d-237) then
        tmp = -1.0d0
    else if (b <= 5.6d-136) then
        tmp = t_0
    else if (b <= 1.7d-64) then
        tmp = -1.0d0
    else if (b <= 7.6d+20) then
        tmp = t_0
    else
        tmp = b * (b * (b * b))
    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 (b <= 1.18e-237) {
		tmp = -1.0;
	} else if (b <= 5.6e-136) {
		tmp = t_0;
	} else if (b <= 1.7e-64) {
		tmp = -1.0;
	} else if (b <= 7.6e+20) {
		tmp = t_0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if b <= 1.18e-237:
		tmp = -1.0
	elif b <= 5.6e-136:
		tmp = t_0
	elif b <= 1.7e-64:
		tmp = -1.0
	elif b <= 7.6e+20:
		tmp = t_0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (b <= 1.18e-237)
		tmp = -1.0;
	elseif (b <= 5.6e-136)
		tmp = t_0;
	elseif (b <= 1.7e-64)
		tmp = -1.0;
	elseif (b <= 7.6e+20)
		tmp = t_0;
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * a));
	tmp = 0.0;
	if (b <= 1.18e-237)
		tmp = -1.0;
	elseif (b <= 5.6e-136)
		tmp = t_0;
	elseif (b <= 1.7e-64)
		tmp = -1.0;
	elseif (b <= 7.6e+20)
		tmp = t_0;
	else
		tmp = b * (b * (b * b));
	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[b, 1.18e-237], -1.0, If[LessEqual[b, 5.6e-136], t$95$0, If[LessEqual[b, 1.7e-64], -1.0, If[LessEqual[b, 7.6e+20], t$95$0, N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.18e-237 or 5.6000000000000002e-136 < b < 1.70000000000000006e-64

   1. Initial program 77.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. cube-unmult N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 66.9%

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

   5. Simplified 66.9%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 32.8%

      \[\leadsto \color{blue}{-1} \]

   if 1.18e-237 < b < 5.6000000000000002e-136 or 1.70000000000000006e-64 < b < 7.6e20

   1. Initial program 74.2%

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

      1. unpow2 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. clear-num N/A

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

      10. flip-+ N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 74.2%

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

   4. Applied egg-rr 74.2%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto {a}^{3} \cdot \color{blue}{a} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{{a}^{3}} \]

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

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right) \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 75.7%

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

   10. Simplified 75.7%

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

   if 7.6e20 < b

   1. Initial program 76.5%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   4. 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 96.8%

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

   5. Simplified 96.8%

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

Alternative 3: 98.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 + a \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(4 + \left(a \cdot a\right) \cdot 2\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1000000000.0)
   (+ (* (* a a) (+ (* a a) (+ 4.0 (* a 4.0)))) -1.0)
   (+ (* (* b b) (+ (* b b) (+ 4.0 (* (* a a) 2.0)))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000000.0) {
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	} else {
		tmp = ((b * b) * ((b * b) + (4.0 + ((a * a) * 2.0)))) + -1.0;
	}
	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) <= 1000000000.0d0) then
        tmp = ((a * a) * ((a * a) + (4.0d0 + (a * 4.0d0)))) + (-1.0d0)
    else
        tmp = ((b * b) * ((b * b) + (4.0d0 + ((a * a) * 2.0d0)))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000000.0) {
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	} else {
		tmp = ((b * b) * ((b * b) + (4.0 + ((a * a) * 2.0)))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1000000000.0:
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0
	else:
		tmp = ((b * b) * ((b * b) + (4.0 + ((a * a) * 2.0)))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1000000000.0)
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(a * a) + Float64(4.0 + Float64(a * 4.0)))) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(b * b) + Float64(4.0 + Float64(Float64(a * a) * 2.0)))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1000000000.0)
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	else
		tmp = ((b * b) * ((b * b) + (4.0 + ((a * a) * 2.0)))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1000000000.0], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(4.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + N[(4.0 + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e9

   1. Initial program 81.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-lowering-*.f64 99.6%

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

   5. Simplified 99.6%

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

   if 1e9 < (*.f64 b b)

   1. Initial program 71.9%

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

      1. unpow2 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. clear-num N/A

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

      10. flip-+ N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 71.9%

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

   4. Applied egg-rr 71.9%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+r+ N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-+r+ N/A

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

      13. +-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. metadata-eval N/A

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

      18. distribute-rgt1-in N/A

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

      19. metadata-eval N/A

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

      20. cancel-sign-sub-inv N/A

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

   10. Simplified 97.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 + a \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(4 + \left(a \cdot a\right) \cdot 2\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot a + b \cdot b\\ \left(t\_0 \cdot t\_0 + b \cdot \left(b \cdot 4\right)\right) + -1 \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* a a) (* b b)))) (+ (+ (* t_0 t_0) (* b (* b 4.0))) -1.0)))

C

double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return ((t_0 * t_0) + (b * (b * 4.0))) + -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    t_0 = (a * a) + (b * b)
    code = ((t_0 * t_0) + (b * (b * 4.0d0))) + (-1.0d0)
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return ((t_0 * t_0) + (b * (b * 4.0))) + -1.0;
}

Python

def code(a, b):
	t_0 = (a * a) + (b * b)
	return ((t_0 * t_0) + (b * (b * 4.0))) + -1.0

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) + Float64(b * b))
	return Float64(Float64(Float64(t_0 * t_0) + Float64(b * Float64(b * 4.0))) + -1.0)
end

MATLAB

function tmp = code(a, b)
	t_0 = (a * a) + (b * b);
	tmp = ((t_0 * t_0) + (b * (b * 4.0))) + -1.0;
end

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. unpow2 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   9. clear-num N/A

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

   10. flip-+ N/A

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

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

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

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

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

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

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

   14. *-lowering-*.f64 76.8%

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

4. Applied egg-rr 76.8%

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

5. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 98.4%

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

7. Simplified 98.4%

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

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

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

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

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

   3. associate-/r/ N/A

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

   4. /-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 98.5%

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

9. Applied egg-rr 98.5%

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

10. Final simplification 98.5%

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

Alternative 5: 98.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 + a \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1000000000.0)
   (+ (* (* a a) (+ (* a a) (+ 4.0 (* a 4.0)))) -1.0)
   (+ (* (* b b) (+ (* b b) (* (* a a) 2.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000000.0) {
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	} else {
		tmp = ((b * b) * ((b * b) + ((a * a) * 2.0))) + -1.0;
	}
	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) <= 1000000000.0d0) then
        tmp = ((a * a) * ((a * a) + (4.0d0 + (a * 4.0d0)))) + (-1.0d0)
    else
        tmp = ((b * b) * ((b * b) + ((a * a) * 2.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1000000000.0) {
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	} else {
		tmp = ((b * b) * ((b * b) + ((a * a) * 2.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1000000000.0:
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0
	else:
		tmp = ((b * b) * ((b * b) + ((a * a) * 2.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1000000000.0)
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(a * a) + Float64(4.0 + Float64(a * 4.0)))) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(b * b) + Float64(Float64(a * a) * 2.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1000000000.0)
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	else
		tmp = ((b * b) * ((b * b) + ((a * a) * 2.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1000000000.0], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(4.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e9

   1. Initial program 81.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-lowering-*.f64 99.6%

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

   5. Simplified 99.6%

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

   if 1e9 < (*.f64 b b)

   1. Initial program 71.9%

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

      1. unpow2 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. clear-num N/A

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

      10. flip-+ N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 71.9%

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

   4. Applied egg-rr 71.9%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(4, \color{blue}{\left({a}^{3}\right)}\right)\right), 1\right) \]
   6. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 87.9%

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

   7. Simplified 87.9%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

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

      12. cancel-sign-sub-inv N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. cancel-sign-sub-inv N/A

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

      17. metadata-eval N/A

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

      18. distribute-rgt1-in N/A

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

      19. metadata-eval N/A

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

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

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

      21. unpow2 N/A

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

      22. *-lowering-*.f64 97.4%

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

   10. Simplified 97.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 1000000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 + a \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 + a \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + a \cdot -12\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+14)
   (+ (* (* a a) (+ (* a a) (+ 4.0 (* a 4.0)))) -1.0)
   (+ (* (* b b) (+ (* b b) (* a -12.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+14) {
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	} else {
		tmp = ((b * b) * ((b * b) + (a * -12.0))) + -1.0;
	}
	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+14) then
        tmp = ((a * a) * ((a * a) + (4.0d0 + (a * 4.0d0)))) + (-1.0d0)
    else
        tmp = ((b * b) * ((b * b) + (a * (-12.0d0)))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+14) {
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	} else {
		tmp = ((b * b) * ((b * b) + (a * -12.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+14:
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0
	else:
		tmp = ((b * b) * ((b * b) + (a * -12.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+14)
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(a * a) + Float64(4.0 + Float64(a * 4.0)))) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(b * b) + Float64(a * -12.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+14)
		tmp = ((a * a) * ((a * a) + (4.0 + (a * 4.0)))) + -1.0;
	else
		tmp = ((b * b) * ((b * b) + (a * -12.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+14], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(4.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + N[(a * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e14

   1. Initial program 81.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-lowering-*.f64 99.6%

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

   5. Simplified 99.6%

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

   if 4e14 < (*.f64 b b)

   1. Initial program 72.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. +-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 91.7%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 91.6%

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

   8. Simplified 91.6%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + \left(4 + a \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + a \cdot -12\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+35}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 150000:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+35)
   (* (* a a) (* a a))
   (if (<= a 150000.0)
     (+ (* b (* b (+ (* b b) 4.0))) -1.0)
     (* a (* (* a a) (+ a 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+35) {
		tmp = (a * a) * (a * a);
	} else if (a <= 150000.0) {
		tmp = (b * (b * ((b * b) + 4.0))) + -1.0;
	} else {
		tmp = a * ((a * a) * (a + 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.5d+35)) then
        tmp = (a * a) * (a * a)
    else if (a <= 150000.0d0) then
        tmp = (b * (b * ((b * b) + 4.0d0))) + (-1.0d0)
    else
        tmp = a * ((a * a) * (a + 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+35) {
		tmp = (a * a) * (a * a);
	} else if (a <= 150000.0) {
		tmp = (b * (b * ((b * b) + 4.0))) + -1.0;
	} else {
		tmp = a * ((a * a) * (a + 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.5e+35:
		tmp = (a * a) * (a * a)
	elif a <= 150000.0:
		tmp = (b * (b * ((b * b) + 4.0))) + -1.0
	else:
		tmp = a * ((a * a) * (a + 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+35)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 150000.0)
		tmp = Float64(Float64(b * Float64(b * Float64(Float64(b * b) + 4.0))) + -1.0);
	else
		tmp = Float64(a * Float64(Float64(a * a) * Float64(a + 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.5e+35)
		tmp = (a * a) * (a * a);
	elseif (a <= 150000.0)
		tmp = (b * (b * ((b * b) + 4.0))) + -1.0;
	else
		tmp = a * ((a * a) * (a + 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e+35], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 150000.0], N[(N[(b * N[(b * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(N[(a * a), $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.49999999999999995e35

   1. Initial program 29.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   4. 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 96.5%

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

   5. Simplified 96.5%

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

   if -1.49999999999999995e35 < a < 1.5e5

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. cube-unmult N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 96.5%

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

   5. Simplified 96.5%

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

   if 1.5e5 < a

   1. Initial program 64.1%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   4. 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 94.7%

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

   5. Simplified 94.7%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 94.7%

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

   7. Applied egg-rr 94.7%

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

   8. Taylor expanded in a around 0

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-lowering-+.f64 94.8%

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

   10. Simplified 94.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+35}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 150000:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -4.5 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.41:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -4.5e-12) t_0 (if (<= a 0.41) -1.0 t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -4.5e-12) {
		tmp = t_0;
	} else if (a <= 0.41) {
		tmp = -1.0;
	} 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 <= (-4.5d-12)) then
        tmp = t_0
    else if (a <= 0.41d0) then
        tmp = -1.0d0
    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 <= -4.5e-12) {
		tmp = t_0;
	} else if (a <= 0.41) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if a <= -4.5e-12:
		tmp = t_0
	elif a <= 0.41:
		tmp = -1.0
	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 <= -4.5e-12)
		tmp = t_0;
	elseif (a <= 0.41)
		tmp = -1.0;
	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 <= -4.5e-12)
		tmp = t_0;
	elseif (a <= 0.41)
		tmp = -1.0;
	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, -4.5e-12], t$95$0, If[LessEqual[a, 0.41], -1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.49999999999999981e-12 or 0.409999999999999976 < a

   1. Initial program 53.4%

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

      1. unpow2 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. clear-num N/A

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

      10. flip-+ N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 53.4%

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

   4. Applied egg-rr 53.4%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto {a}^{3} \cdot \color{blue}{a} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{{a}^{3}} \]

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

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right) \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 84.3%

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

   10. Simplified 84.3%

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

   if -4.49999999999999981e-12 < a < 0.409999999999999976

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. cube-unmult N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 46.1%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 93.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+31) (+ (* (* a a) (* a a)) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+31) {
		tmp = ((a * a) * (a * a)) + -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) <= 5d+31) then
        tmp = ((a * a) * (a * a)) + (-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) <= 5e+31) {
		tmp = ((a * a) * (a * a)) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+31:
		tmp = ((a * a) * (a * a)) + -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+31)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) + -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) <= 5e+31)
		tmp = ((a * a) * (a * a)) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+31], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.00000000000000027e31

   1. Initial program 80.7%

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

   3. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{\left(2 \cdot 2\right)}\right), 1\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({a}^{2} \cdot {a}^{2}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2}\right)\right), 1\right) \]

      4. unpow2 N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2}\right)\right), 1\right) \]

      6. unpow2 N/A

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

      7. *-lowering-*.f64 95.1%

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

   5. Simplified 95.1%

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

   if 5.00000000000000027e31 < (*.f64 b b)

   1. Initial program 72.3%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   4. 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%

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

   5. 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. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+31) (+ (* (* a a) 4.0) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+31) {
		tmp = ((a * a) * 4.0) + -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) <= 5d+31) then
        tmp = ((a * a) * 4.0d0) + (-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) <= 5e+31) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+31:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+31)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -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) <= 5e+31)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+31], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.00000000000000027e31

   1. Initial program 80.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

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

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

   5. Simplified 73.5%

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

   6. Taylor expanded in b around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 71.2%

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

   8. Simplified 71.2%

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

   if 5.00000000000000027e31 < (*.f64 b b)

   1. Initial program 72.3%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   4. 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%

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

   5. 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. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 25.2% 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 76.8%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. pow-plus N/A

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

   4. cube-unmult N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-out N/A

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

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

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

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

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 68.0%

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

5. Simplified 68.0%

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

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 23.6%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(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))