Bouland and Aaronson, Equation (25)

Percentage Accurate: 72.9% → 99.8%

Time: 10.4s

Alternatives: 13

Speedup: 9.3×

• 61.7% 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: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b + a \cdot a\\ \mathbf{if}\;b \cdot b \leq 10^{-58}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_0 + \left(\left(b \cdot b\right) \cdot 4 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* b b) (* a a))))
   (if (<= (* b b) 1e-58)
     (+ (* (* a a) (+ 4.0 (* a (+ a 4.0)))) -1.0)
     (+ (* t_0 t_0) (+ (* (* b b) 4.0) -1.0)))))

C

double code(double a, double b) {
	double t_0 = (b * b) + (a * a);
	double tmp;
	if ((b * b) <= 1e-58) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = (t_0 * t_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) :: t_0
    real(8) :: tmp
    t_0 = (b * b) + (a * a)
    if ((b * b) <= 1d-58) then
        tmp = ((a * a) * (4.0d0 + (a * (a + 4.0d0)))) + (-1.0d0)
    else
        tmp = (t_0 * t_0) + (((b * b) * 4.0d0) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (b * b) + (a * a);
	double tmp;
	if ((b * b) <= 1e-58) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = (t_0 * t_0) + (((b * b) * 4.0) + -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b * b) + (a * a)
	tmp = 0
	if (b * b) <= 1e-58:
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0
	else:
		tmp = (t_0 * t_0) + (((b * b) * 4.0) + -1.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) + Float64(a * a))
	tmp = 0.0
	if (Float64(b * b) <= 1e-58)
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + 4.0)))) + -1.0);
	else
		tmp = Float64(Float64(t_0 * t_0) + Float64(Float64(Float64(b * b) * 4.0) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b * b) + (a * a);
	tmp = 0.0;
	if ((b * b) <= 1e-58)
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	else
		tmp = (t_0 * t_0) + (((b * b) * 4.0) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 1e-58], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e-58

   1. Initial program 81.7%

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

   3. Taylor expanded in 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-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. +-lowering-+.f64 99.9%

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

   8. Simplified 99.9%

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

   if 1e-58 < (*.f64 b b)

   1. Initial program 70.2%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 70.3%

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

   5. Taylor expanded in a around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-58}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b + a \cdot a\right) \cdot \left(b \cdot b + a \cdot a\right) + \left(\left(b \cdot b\right) \cdot 4 + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;b \cdot b \leq 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{-46}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= (* b b) 1e-308)
     t_0
     (if (<= (* b b) 2e-46)
       -1.0
       (if (<= (* b b) 5e+46) t_0 (* b (* b (* b b))))))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if ((b * b) <= 1e-308) {
		tmp = t_0;
	} else if ((b * b) <= 2e-46) {
		tmp = -1.0;
	} else if ((b * b) <= 5e+46) {
		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 * b) <= 1d-308) then
        tmp = t_0
    else if ((b * b) <= 2d-46) then
        tmp = -1.0d0
    else if ((b * b) <= 5d+46) 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 * b) <= 1e-308) {
		tmp = t_0;
	} else if ((b * b) <= 2e-46) {
		tmp = -1.0;
	} else if ((b * b) <= 5e+46) {
		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 * b) <= 1e-308:
		tmp = t_0
	elif (b * b) <= 2e-46:
		tmp = -1.0
	elif (b * b) <= 5e+46:
		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 (Float64(b * b) <= 1e-308)
		tmp = t_0;
	elseif (Float64(b * b) <= 2e-46)
		tmp = -1.0;
	elseif (Float64(b * b) <= 5e+46)
		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 * b) <= 1e-308)
		tmp = t_0;
	elseif ((b * b) <= 2e-46)
		tmp = -1.0;
	elseif ((b * b) <= 5e+46)
		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[N[(b * b), $MachinePrecision], 1e-308], t$95$0, If[LessEqual[N[(b * b), $MachinePrecision], 2e-46], -1.0, If[LessEqual[N[(b * b), $MachinePrecision], 5e+46], t$95$0, N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 9.9999999999999991e-309 or 2.00000000000000005e-46 < (*.f64 b b) < 5.0000000000000002e46

   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. Step-by-step derivation

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 80.7%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 59.5%

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

   7. Simplified 59.5%

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

   if 9.9999999999999991e-309 < (*.f64 b b) < 2.00000000000000005e-46

   1. Initial program 88.1%

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

   3. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({a}^{4}\right)}, 1\right) \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 60.2%

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

   if 5.0000000000000002e46 < (*.f64 b b)

   1. Initial program 66.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 66.3%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 92.7%

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

   7. Simplified 92.7%

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

Alternative 3: 67.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -8.5 \cdot 10^{+54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq -2.32 \cdot 10^{-57}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{elif}\;a \leq 0.42:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -8.5e+54)
     t_0
     (if (<= a -2.32e-57) (* (* b b) 4.0) (if (<= a 0.42) -1.0 t_0)))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -8.5e+54) {
		tmp = t_0;
	} else if (a <= -2.32e-57) {
		tmp = (b * b) * 4.0;
	} else if (a <= 0.42) {
		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 <= (-8.5d+54)) then
        tmp = t_0
    else if (a <= (-2.32d-57)) then
        tmp = (b * b) * 4.0d0
    else if (a <= 0.42d0) 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 <= -8.5e+54) {
		tmp = t_0;
	} else if (a <= -2.32e-57) {
		tmp = (b * b) * 4.0;
	} else if (a <= 0.42) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if a <= -8.5e+54:
		tmp = t_0
	elif a <= -2.32e-57:
		tmp = (b * b) * 4.0
	elif a <= 0.42:
		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 <= -8.5e+54)
		tmp = t_0;
	elseif (a <= -2.32e-57)
		tmp = Float64(Float64(b * b) * 4.0);
	elseif (a <= 0.42)
		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 <= -8.5e+54)
		tmp = t_0;
	elseif (a <= -2.32e-57)
		tmp = (b * b) * 4.0;
	elseif (a <= 0.42)
		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, -8.5e+54], t$95$0, If[LessEqual[a, -2.32e-57], N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[a, 0.42], -1.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.4999999999999995e54 or 0.419999999999999984 < a

   1. Initial program 46.9%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 46.9%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 93.5%

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

   7. Simplified 93.5%

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

   if -8.4999999999999995e54 < a < -2.31999999999999992e-57

   1. Initial program 99.7%

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

   3. Taylor expanded in a around 0

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

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 72.4%

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

   5. Simplified 72.4%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 51.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{4 \cdot {b}^{2}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 46.2%

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

   11. Simplified 46.2%

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

   if -2.31999999999999992e-57 < a < 0.419999999999999984

   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 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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 50.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+54}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq -2.32 \cdot 10^{-57}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{elif}\;a \leq 0.42:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+46}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-46)
   (+ (* (* a a) 4.0) -1.0)
   (if (<= (* b b) 5e+46) (* a (* a (* a a))) (* b (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-46) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else if ((b * b) <= 5e+46) {
		tmp = a * (a * (a * a));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d-46) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else if ((b * b) <= 5d+46) then
        tmp = a * (a * (a * a))
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-46) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else if ((b * b) <= 5e+46) {
		tmp = a * (a * (a * a));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-46:
		tmp = ((a * a) * 4.0) + -1.0
	elif (b * b) <= 5e+46:
		tmp = a * (a * (a * a))
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-46)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	elseif (Float64(b * b) <= 5e+46)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-46)
		tmp = ((a * a) * 4.0) + -1.0;
	elseif ((b * b) <= 5e+46)
		tmp = a * (a * (a * a));
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-46], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 5e+46], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 2.00000000000000005e-46

   1. Initial program 82.2%

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

   3. 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-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 72.2%

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

   if 2.00000000000000005e-46 < (*.f64 b b) < 5.0000000000000002e46

   1. Initial program 93.4%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 93.5%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 76.0%

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

   7. Simplified 76.0%

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

   if 5.0000000000000002e46 < (*.f64 b b)

   1. Initial program 66.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 66.3%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+46}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.1 \cdot 10^{-42}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 1.5 \cdot 10^{+238}:\\ \;\;\;\;4 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4.1e-42)
   -1.0
   (if (<= (* b b) 1.5e+238) (* 4.0 (* a (* a a))) (* (* b b) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.1e-42) {
		tmp = -1.0;
	} else if ((b * b) <= 1.5e+238) {
		tmp = 4.0 * (a * (a * a));
	} else {
		tmp = (b * b) * 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 ((b * b) <= 4.1d-42) then
        tmp = -1.0d0
    else if ((b * b) <= 1.5d+238) then
        tmp = 4.0d0 * (a * (a * a))
    else
        tmp = (b * b) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.1e-42) {
		tmp = -1.0;
	} else if ((b * b) <= 1.5e+238) {
		tmp = 4.0 * (a * (a * a));
	} else {
		tmp = (b * b) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4.1e-42:
		tmp = -1.0
	elif (b * b) <= 1.5e+238:
		tmp = 4.0 * (a * (a * a))
	else:
		tmp = (b * b) * 4.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4.1e-42)
		tmp = -1.0;
	elseif (Float64(b * b) <= 1.5e+238)
		tmp = Float64(4.0 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(Float64(b * b) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4.1e-42)
		tmp = -1.0;
	elseif ((b * b) <= 1.5e+238)
		tmp = 4.0 * (a * (a * a));
	else
		tmp = (b * b) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4.1e-42], -1.0, If[LessEqual[N[(b * b), $MachinePrecision], 1.5e+238], N[(4.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 4.1000000000000001e-42

   1. Initial program 82.2%

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

   3. 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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 48.6%

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

   if 4.1000000000000001e-42 < (*.f64 b b) < 1.5e238

   1. Initial program 69.2%

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

   3. 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-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 56.2%

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

   5. Simplified 56.2%

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

   6. Taylor expanded in a around 0

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

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

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

      2. *-lowering-*.f64 34.6%

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

   8. Simplified 34.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{4 \cdot {a}^{3}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(4, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 33.8%

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

   11. Simplified 33.8%

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

   if 1.5e238 < (*.f64 b b)

   1. Initial program 69.9%

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

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 82.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{4 \cdot {b}^{2}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 82.9%

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

   11. Simplified 82.9%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.1 \cdot 10^{-42}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 1.5 \cdot 10^{+238}:\\ \;\;\;\;4 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+46)
   (+ (* (* a a) (+ 4.0 (* a (+ a 4.0)))) -1.0)
   (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+46) {
		tmp = ((a * a) * (4.0 + (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+46) then
        tmp = ((a * a) * (4.0d0 + (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+46) {
		tmp = ((a * a) * (4.0 + (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+46:
		tmp = ((a * a) * (4.0 + (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+46)
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(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+46)
		tmp = ((a * a) * (4.0 + (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+46], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $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) < 5.0000000000000002e46

   1. Initial program 83.4%

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. +-lowering-+.f64 97.6%

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

   8. Simplified 97.6%

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

   if 5.0000000000000002e46 < (*.f64 b b)

   1. Initial program 66.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 66.3%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1520:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + 4\right)\right)\\ \mathbf{elif}\;a \leq 2.1:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a + 4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1520.0)
   (* a (* (* a a) (+ a 4.0)))
   (if (<= a 2.1) (+ (* (* b b) 4.0) -1.0) (* (+ a 4.0) (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1520.0) {
		tmp = a * ((a * a) * (a + 4.0));
	} else if (a <= 2.1) {
		tmp = ((b * b) * 4.0) + -1.0;
	} else {
		tmp = (a + 4.0) * (a * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1520.0d0)) then
        tmp = a * ((a * a) * (a + 4.0d0))
    else if (a <= 2.1d0) then
        tmp = ((b * b) * 4.0d0) + (-1.0d0)
    else
        tmp = (a + 4.0d0) * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1520.0) {
		tmp = a * ((a * a) * (a + 4.0));
	} else if (a <= 2.1) {
		tmp = ((b * b) * 4.0) + -1.0;
	} else {
		tmp = (a + 4.0) * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1520.0:
		tmp = a * ((a * a) * (a + 4.0))
	elif a <= 2.1:
		tmp = ((b * b) * 4.0) + -1.0
	else:
		tmp = (a + 4.0) * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1520.0)
		tmp = Float64(a * Float64(Float64(a * a) * Float64(a + 4.0)));
	elseif (a <= 2.1)
		tmp = Float64(Float64(Float64(b * b) * 4.0) + -1.0);
	else
		tmp = Float64(Float64(a + 4.0) * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1520.0)
		tmp = a * ((a * a) * (a + 4.0));
	elseif (a <= 2.1)
		tmp = ((b * b) * 4.0) + -1.0;
	else
		tmp = (a + 4.0) * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1520.0], N[(a * N[(N[(a * a), $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.1], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a + 4.0), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1520

   1. Initial program 30.5%

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

   3. Taylor expanded in b around 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-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 87.2%

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

   5. Simplified 87.2%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. +-lowering-+.f64 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r/ N/A

          \[\leadsto {a}^{4} \cdot \frac{4 \cdot 1}{a} + {a}^{\color{blue}{4}} \cdot 1 \]

       4. metadata-eval N/A

          \[\leadsto {a}^{4} \cdot \frac{4}{a} + {a}^{4} \cdot 1 \]

       5. associate-*r/ N/A

          \[\leadsto \frac{{a}^{4} \cdot 4}{a} + \color{blue}{{a}^{4}} \cdot 1 \]

       6. associate-*l/ N/A

          \[\leadsto \frac{{a}^{4}}{a} \cdot 4 + \color{blue}{{a}^{4}} \cdot 1 \]

       7. metadata-eval N/A

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

       8. pow-sqr N/A

          \[\leadsto \frac{{a}^{2} \cdot {a}^{2}}{a} \cdot 4 + {a}^{4} \cdot 1 \]

       9. associate-*r/ N/A

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

       10. *-rgt-identity N/A

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

       11. associate-*r/ N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. rgt-mult-inverse N/A

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

       15. *-rgt-identity N/A

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

       16. pow-plus N/A

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

       17. metadata-eval N/A

           \[\leadsto {a}^{3} \cdot 4 + {a}^{4} \cdot 1 \]

       18. *-rgt-identity N/A

           \[\leadsto {a}^{3} \cdot 4 + {a}^{\color{blue}{4}} \]

       19. metadata-eval N/A

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

       20. pow-plus N/A

           \[\leadsto {a}^{3} \cdot 4 + {a}^{3} \cdot \color{blue}{a} \]

   11. Simplified 86.4%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

          \[\leadsto \left(a \cdot \left(a + 4\right)\right) \cdot \left(a \cdot a\right) \]

       4. associate-*r* N/A

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

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

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

       6. *-commutative N/A

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

       7. +-commutative N/A

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

       8. associate-*r* N/A

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

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

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

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

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

       11. +-commutative N/A

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

       12. +-lowering-+.f64 86.5%

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

   13. Applied egg-rr 86.5%

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

   if -1520 < a < 2.10000000000000009

   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

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 78.0%

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

   if 2.10000000000000009 < a

   1. Initial program 69.8%

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 93.1%

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

   5. Simplified 93.1%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. +-lowering-+.f64 93.1%

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

   8. Simplified 93.1%

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

   9. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r/ N/A

          \[\leadsto {a}^{4} \cdot \frac{4 \cdot 1}{a} + {a}^{\color{blue}{4}} \cdot 1 \]

       4. metadata-eval N/A

          \[\leadsto {a}^{4} \cdot \frac{4}{a} + {a}^{4} \cdot 1 \]

       5. associate-*r/ N/A

          \[\leadsto \frac{{a}^{4} \cdot 4}{a} + \color{blue}{{a}^{4}} \cdot 1 \]

       6. associate-*l/ N/A

          \[\leadsto \frac{{a}^{4}}{a} \cdot 4 + \color{blue}{{a}^{4}} \cdot 1 \]

       7. metadata-eval N/A

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

       8. pow-sqr N/A

          \[\leadsto \frac{{a}^{2} \cdot {a}^{2}}{a} \cdot 4 + {a}^{4} \cdot 1 \]

       9. associate-*r/ N/A

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

       10. *-rgt-identity N/A

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

       11. associate-*r/ N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. rgt-mult-inverse N/A

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

       15. *-rgt-identity N/A

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

       16. pow-plus N/A

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

       17. metadata-eval N/A

           \[\leadsto {a}^{3} \cdot 4 + {a}^{4} \cdot 1 \]

       18. *-rgt-identity N/A

           \[\leadsto {a}^{3} \cdot 4 + {a}^{\color{blue}{4}} \]

       19. metadata-eval N/A

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

       20. pow-plus N/A

           \[\leadsto {a}^{3} \cdot 4 + {a}^{3} \cdot \color{blue}{a} \]

   11. Simplified 91.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1520:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + 4\right)\right)\\ \mathbf{elif}\;a \leq 2.1:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a + 4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -360:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + 4\right)\right)\\ \mathbf{elif}\;a \leq 2.4:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -360.0)
   (* a (* (* a a) (+ a 4.0)))
   (if (<= a 2.4) (+ (* (* b b) 4.0) -1.0) (* (* a a) (* a (+ a 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -360.0) {
		tmp = a * ((a * a) * (a + 4.0));
	} else if (a <= 2.4) {
		tmp = ((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 <= (-360.0d0)) then
        tmp = a * ((a * a) * (a + 4.0d0))
    else if (a <= 2.4d0) then
        tmp = ((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 <= -360.0) {
		tmp = a * ((a * a) * (a + 4.0));
	} else if (a <= 2.4) {
		tmp = ((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 <= -360.0:
		tmp = a * ((a * a) * (a + 4.0))
	elif a <= 2.4:
		tmp = ((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 <= -360.0)
		tmp = Float64(a * Float64(Float64(a * a) * Float64(a + 4.0)));
	elseif (a <= 2.4)
		tmp = Float64(Float64(Float64(b * b) * 4.0) + -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * Float64(a + 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -360.0)
		tmp = a * ((a * a) * (a + 4.0));
	elseif (a <= 2.4)
		tmp = ((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, -360.0], N[(a * N[(N[(a * a), $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.4], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -360

   1. Initial program 30.5%

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

   3. Taylor expanded in b around 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-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 87.2%

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

   5. Simplified 87.2%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. +-lowering-+.f64 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r/ N/A

          \[\leadsto {a}^{4} \cdot \frac{4 \cdot 1}{a} + {a}^{\color{blue}{4}} \cdot 1 \]

       4. metadata-eval N/A

          \[\leadsto {a}^{4} \cdot \frac{4}{a} + {a}^{4} \cdot 1 \]

       5. associate-*r/ N/A

          \[\leadsto \frac{{a}^{4} \cdot 4}{a} + \color{blue}{{a}^{4}} \cdot 1 \]

       6. associate-*l/ N/A

          \[\leadsto \frac{{a}^{4}}{a} \cdot 4 + \color{blue}{{a}^{4}} \cdot 1 \]

       7. metadata-eval N/A

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

       8. pow-sqr N/A

          \[\leadsto \frac{{a}^{2} \cdot {a}^{2}}{a} \cdot 4 + {a}^{4} \cdot 1 \]

       9. associate-*r/ N/A

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

       10. *-rgt-identity N/A

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

       11. associate-*r/ N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. rgt-mult-inverse N/A

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

       15. *-rgt-identity N/A

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

       16. pow-plus N/A

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

       17. metadata-eval N/A

           \[\leadsto {a}^{3} \cdot 4 + {a}^{4} \cdot 1 \]

       18. *-rgt-identity N/A

           \[\leadsto {a}^{3} \cdot 4 + {a}^{\color{blue}{4}} \]

       19. metadata-eval N/A

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

       20. pow-plus N/A

           \[\leadsto {a}^{3} \cdot 4 + {a}^{3} \cdot \color{blue}{a} \]

   11. Simplified 86.4%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

          \[\leadsto \left(a \cdot \left(a + 4\right)\right) \cdot \left(a \cdot a\right) \]

       4. associate-*r* N/A

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

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

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

       6. *-commutative N/A

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

       7. +-commutative N/A

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

       8. associate-*r* N/A

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

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

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

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

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

       11. +-commutative N/A

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

       12. +-lowering-+.f64 86.5%

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

   13. Applied egg-rr 86.5%

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

   if -360 < a < 2.39999999999999991

   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

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 78.0%

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

   if 2.39999999999999991 < a

   1. Initial program 69.8%

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 93.1%

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

   5. Simplified 93.1%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. fma-define N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lft-mult-inverse N/A

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

      12. *-lft-identity N/A

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

      13. unpow2 N/A

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

      14. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(4, {a}^{\color{blue}{3}}, {a}^{4}\right) \]

      15. fma-define N/A

          \[\leadsto 4 \cdot {a}^{3} + \color{blue}{{a}^{4}} \]

      16. cube-mult N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. metadata-eval N/A

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

      20. pow-sqr N/A

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

      21. distribute-rgt-in N/A

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

      22. unpow2 N/A

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

   8. Simplified 91.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -360:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + 4\right)\right)\\ \mathbf{elif}\;a \leq 2.4:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \mathbf{if}\;a \leq -2600:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.4:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a (+ a 4.0)))))
   (if (<= a -2600.0) t_0 (if (<= a 2.4) (+ (* (* b b) 4.0) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * (a + 4.0));
	double tmp;
	if (a <= -2600.0) {
		tmp = t_0;
	} else if (a <= 2.4) {
		tmp = ((b * b) * 4.0) + -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 + 4.0d0))
    if (a <= (-2600.0d0)) then
        tmp = t_0
    else if (a <= 2.4d0) then
        tmp = ((b * b) * 4.0d0) + (-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 + 4.0));
	double tmp;
	if (a <= -2600.0) {
		tmp = t_0;
	} else if (a <= 2.4) {
		tmp = ((b * b) * 4.0) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * (a + 4.0))
	tmp = 0
	if a <= -2600.0:
		tmp = t_0
	elif a <= 2.4:
		tmp = ((b * b) * 4.0) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * Float64(a + 4.0)))
	tmp = 0.0
	if (a <= -2600.0)
		tmp = t_0;
	elseif (a <= 2.4)
		tmp = Float64(Float64(Float64(b * b) * 4.0) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * (a * (a + 4.0));
	tmp = 0.0;
	if (a <= -2600.0)
		tmp = t_0;
	elseif (a <= 2.4)
		tmp = ((b * b) * 4.0) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2600.0], t$95$0, If[LessEqual[a, 2.4], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2600 or 2.39999999999999991 < a

   1. Initial program 51.8%

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 90.4%

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

   5. Simplified 90.4%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. fma-define N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lft-mult-inverse N/A

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

      12. *-lft-identity N/A

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

      13. unpow2 N/A

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

      14. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(4, {a}^{\color{blue}{3}}, {a}^{4}\right) \]

      15. fma-define N/A

          \[\leadsto 4 \cdot {a}^{3} + \color{blue}{{a}^{4}} \]

      16. cube-mult N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. metadata-eval N/A

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

      20. pow-sqr N/A

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

      21. distribute-rgt-in N/A

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

      22. unpow2 N/A

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

   8. Simplified 89.3%

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

   if -2600 < a < 2.39999999999999991

   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

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 78.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2600:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \mathbf{elif}\;a \leq 2.4:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -8.5 \cdot 10^{+54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.95:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -8.5e+54) t_0 (if (<= a 1.95) (+ (* (* b b) 4.0) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -8.5e+54) {
		tmp = t_0;
	} else if (a <= 1.95) {
		tmp = ((b * b) * 4.0) + -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 <= (-8.5d+54)) then
        tmp = t_0
    else if (a <= 1.95d0) then
        tmp = ((b * b) * 4.0d0) + (-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 <= -8.5e+54) {
		tmp = t_0;
	} else if (a <= 1.95) {
		tmp = ((b * b) * 4.0) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if a <= -8.5e+54:
		tmp = t_0
	elif a <= 1.95:
		tmp = ((b * b) * 4.0) + -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 <= -8.5e+54)
		tmp = t_0;
	elseif (a <= 1.95)
		tmp = Float64(Float64(Float64(b * b) * 4.0) + -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 <= -8.5e+54)
		tmp = t_0;
	elseif (a <= 1.95)
		tmp = ((b * b) * 4.0) + -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, -8.5e+54], t$95$0, If[LessEqual[a, 1.95], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.4999999999999995e54 or 1.94999999999999996 < a

   1. Initial program 46.9%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 46.9%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 93.5%

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

   7. Simplified 93.5%

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

   if -8.4999999999999995e54 < a < 1.94999999999999996

   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

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 74.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+54}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 1.95:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.9% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{+24}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.35e+24) (+ (* a (* a (* a a))) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.35e+24) {
		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 <= 2.35d+24) 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 <= 2.35e+24) {
		tmp = (a * (a * (a * a))) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.35e+24:
		tmp = (a * (a * (a * a))) + -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.35e+24)
		tmp = Float64(Float64(a * Float64(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 <= 2.35e+24)
		tmp = (a * (a * (a * a))) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.35e+24], N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.35e24

   1. Initial program 77.9%

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

   3. Taylor expanded in a around 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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 78.2%

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

   5. Simplified 78.2%

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

   if 2.35e24 < b

   1. Initial program 68.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 68.3%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(\left(\left(a \cdot a\right) \cdot \left(4 + a \cdot 4\right) + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right) + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {b}^{\left(2 \cdot \color{blue}{2}\right)} \]

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(b \cdot b\right) \cdot {\color{blue}{b}}^{2} \]

      4. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left(b \cdot {b}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 92.1%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 92.1%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{+24}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.5% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.235:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 0.235) -1.0 (* (* b b) 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.235) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 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 ((b * b) <= 0.235d0) then
        tmp = -1.0d0
    else
        tmp = (b * b) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.235) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.235:
		tmp = -1.0
	else:
		tmp = (b * b) * 4.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.235)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.235)
		tmp = -1.0;
	else
		tmp = (b * b) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.235], -1.0, N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.23499999999999999

   1. Initial program 83.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 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. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(a \cdot a\right) \cdot {a}^{2}\right), 1\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot \left(a \cdot {a}^{2}\right)\right), 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), 1\right) \]

      12. *-lowering-*.f64 95.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), 1\right) \]

   5. Simplified 95.7%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)} - 1 \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 47.0%

      \[\leadsto \color{blue}{-1} \]

   if 0.23499999999999999 < (*.f64 b b)

   1. Initial program 68.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), 1\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), 1\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), 1\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), 1\right) \]

      10. *-lowering-*.f64 87.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), 1\right) \]

   5. Simplified 87.1%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)} - 1 \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \color{blue}{4}\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 50.3%

      \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{4} - 1 \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{4 \cdot {b}^{2}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(4, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(4, \left(b \cdot \color{blue}{b}\right)\right) \]

       3. *-lowering-*.f64 50.3%

          \[\leadsto \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   11. Simplified 50.3%

       \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.235:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 24.3% 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 75.7%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
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. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(a \cdot a\right) \cdot {a}^{2}\right), 1\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot \left(a \cdot {a}^{2}\right)\right), 1\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

   6. cube-mult N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot {a}^{3}\right), 1\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{3}\right)\right), 1\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right), 1\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right), 1\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right), 1\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right), 1\right) \]

   12. *-lowering-*.f64 68.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), 1\right) \]

5. Simplified 68.2%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)} - 1 \]

6. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 24.2%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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))