Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.9% → 99.9%

Time: 9.1s

Alternatives: 12

Speedup: 8.0×

• 60.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot a + b \cdot b\\ \mathbf{if}\;{t\_0}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot t\_0 + \left(4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right) + b \cdot \left(b \cdot \left(a + 3\right)\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(1 + \frac{\frac{4 + b \cdot \left(b \cdot 2\right)}{a} - 4}{a}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* a a) (* b b))))
   (if (<=
        (+
         (pow t_0 2.0)
         (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
        INFINITY)
     (+
      (* t_0 t_0)
      (+ (* 4.0 (+ (* a (* a (- 1.0 a))) (* b (* b (+ a 3.0))))) -1.0))
     (+
      (*
       (* a (* a (* a a)))
       (+ 1.0 (/ (- (/ (+ 4.0 (* b (* b 2.0))) a) 4.0) a)))
      -1.0))))

C

double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	double tmp;
	if ((pow(t_0, 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= ((double) INFINITY)) {
		tmp = (t_0 * t_0) + ((4.0 * ((a * (a * (1.0 - a))) + (b * (b * (a + 3.0))))) + -1.0);
	} else {
		tmp = ((a * (a * (a * a))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	double tmp;
	if ((Math.pow(t_0, 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * t_0) + ((4.0 * ((a * (a * (1.0 - a))) + (b * (b * (a + 3.0))))) + -1.0);
	} else {
		tmp = ((a * (a * (a * a))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) + (b * b)
	tmp = 0
	if (math.pow(t_0, 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= math.inf:
		tmp = (t_0 * t_0) + ((4.0 * ((a * (a * (1.0 - a))) + (b * (b * (a + 3.0))))) + -1.0)
	else:
		tmp = ((a * (a * (a * a))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) + Float64(b * b))
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0))))) <= Inf)
		tmp = Float64(Float64(t_0 * t_0) + Float64(Float64(4.0 * Float64(Float64(a * Float64(a * Float64(1.0 - a))) + Float64(b * Float64(b * Float64(a + 3.0))))) + -1.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(1.0 + Float64(Float64(Float64(Float64(4.0 + Float64(b * Float64(b * 2.0))) / a) - 4.0) / a))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) + (b * b);
	tmp = 0.0;
	if (((t_0 ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= Inf)
		tmp = (t_0 * t_0) + ((4.0 * ((a * (a * (1.0 - a))) + (b * (b * (a + 3.0))))) + -1.0);
	else
		tmp = ((a * (a * (a * a))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(4.0 * N[(N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(N[(4.0 + N[(b * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] - 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < +inf.0

   1. Initial program 99.8%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 99.9%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around -inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

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

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

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

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 97.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(1 + \frac{\frac{4 + b \cdot \left(b \cdot 2\right)}{a} - 4}{a}\right) + -1\\ \mathbf{if}\;a \leq -7500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+33}:\\ \;\;\;\;\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (* a (* a (* a a)))
           (+ 1.0 (/ (- (/ (+ 4.0 (* b (* b 2.0))) a) 4.0) a)))
          -1.0)))
   (if (<= a -7500000000000.0)
     t_0
     (if (<= a 4.2e+33)
       (+
        (+
         (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0))))
         (* b (* b (* b b))))
        -1.0)
       t_0))))

C

double code(double a, double b) {
	double t_0 = ((a * (a * (a * a))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0;
	double tmp;
	if (a <= -7500000000000.0) {
		tmp = t_0;
	} else if (a <= 4.2e+33) {
		tmp = ((4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))) + (b * (b * (b * b)))) + -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))) * (1.0d0 + ((((4.0d0 + (b * (b * 2.0d0))) / a) - 4.0d0) / a))) + (-1.0d0)
    if (a <= (-7500000000000.0d0)) then
        tmp = t_0
    else if (a <= 4.2d+33) then
        tmp = ((4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (a + 3.0d0)))) + (b * (b * (b * b)))) + (-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))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0;
	double tmp;
	if (a <= -7500000000000.0) {
		tmp = t_0;
	} else if (a <= 4.2e+33) {
		tmp = ((4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))) + (b * (b * (b * b)))) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((a * (a * (a * a))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0
	tmp = 0
	if a <= -7500000000000.0:
		tmp = t_0
	elif a <= 4.2e+33:
		tmp = ((4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))) + (b * (b * (b * b)))) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(1.0 + Float64(Float64(Float64(Float64(4.0 + Float64(b * Float64(b * 2.0))) / a) - 4.0) / a))) + -1.0)
	tmp = 0.0
	if (a <= -7500000000000.0)
		tmp = t_0;
	elseif (a <= 4.2e+33)
		tmp = Float64(Float64(Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))) + Float64(b * Float64(b * Float64(b * b)))) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((a * (a * (a * a))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0;
	tmp = 0.0;
	if (a <= -7500000000000.0)
		tmp = t_0;
	elseif (a <= 4.2e+33)
		tmp = ((4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))) + (b * (b * (b * b)))) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(N[(4.0 + N[(b * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] - 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[a, -7500000000000.0], t$95$0, If[LessEqual[a, 4.2e+33], N[(N[(N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5e12 or 4.2000000000000001e33 < a

   1. Initial program 44.6%

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

   3. Taylor expanded in a around -inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   if -7.5e12 < a < 4.2000000000000001e33

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7500000000000:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(1 + \frac{\frac{4 + b \cdot \left(b \cdot 2\right)}{a} - 4}{a}\right) + -1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+33}:\\ \;\;\;\;\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(1 + \frac{\frac{4 + b \cdot \left(b \cdot 2\right)}{a} - 4}{a}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(1 + \frac{\frac{4 + b \cdot \left(b \cdot 2\right)}{a} - 4}{a}\right) + -1\\ \mathbf{if}\;a \leq -8 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+33}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (* a (* a (* a a)))
           (+ 1.0 (/ (- (/ (+ 4.0 (* b (* b 2.0))) a) 4.0) a)))
          -1.0)))
   (if (<= a -8e+15)
     t_0
     (if (<= a 4.2e+33) (+ (* b (* b (+ (* b b) 12.0))) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = ((a * (a * (a * a))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0;
	double tmp;
	if (a <= -8e+15) {
		tmp = t_0;
	} else if (a <= 4.2e+33) {
		tmp = (b * (b * ((b * b) + 12.0))) + -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))) * (1.0d0 + ((((4.0d0 + (b * (b * 2.0d0))) / a) - 4.0d0) / a))) + (-1.0d0)
    if (a <= (-8d+15)) then
        tmp = t_0
    else if (a <= 4.2d+33) then
        tmp = (b * (b * ((b * b) + 12.0d0))) + (-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))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0;
	double tmp;
	if (a <= -8e+15) {
		tmp = t_0;
	} else if (a <= 4.2e+33) {
		tmp = (b * (b * ((b * b) + 12.0))) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((a * (a * (a * a))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0
	tmp = 0
	if a <= -8e+15:
		tmp = t_0
	elif a <= 4.2e+33:
		tmp = (b * (b * ((b * b) + 12.0))) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(1.0 + Float64(Float64(Float64(Float64(4.0 + Float64(b * Float64(b * 2.0))) / a) - 4.0) / a))) + -1.0)
	tmp = 0.0
	if (a <= -8e+15)
		tmp = t_0;
	elseif (a <= 4.2e+33)
		tmp = Float64(Float64(b * Float64(b * Float64(Float64(b * b) + 12.0))) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((a * (a * (a * a))) * (1.0 + ((((4.0 + (b * (b * 2.0))) / a) - 4.0) / a))) + -1.0;
	tmp = 0.0;
	if (a <= -8e+15)
		tmp = t_0;
	elseif (a <= 4.2e+33)
		tmp = (b * (b * ((b * b) + 12.0))) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(N[(4.0 + N[(b * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] - 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[a, -8e+15], t$95$0, If[LessEqual[a, 4.2e+33], N[(N[(b * N[(b * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -8e15 or 4.2000000000000001e33 < a

   1. Initial program 45.3%

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

   3. Taylor expanded in a around -inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   if -8e15 < a < 4.2000000000000001e33

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

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

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

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

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

      19. +-commutative N/A

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

      20. +-lowering-+.f64 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+15}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(1 + \frac{\frac{4 + b \cdot \left(b \cdot 2\right)}{a} - 4}{a}\right) + -1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+33}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(1 + \frac{\frac{4 + b \cdot \left(b \cdot 2\right)}{a} - 4}{a}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ t_1 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))) (t_1 (* b (* b (* b b)))))
   (if (<= a -2.4e+59)
     t_0
     (if (<= a -1.65e-50)
       t_1
       (if (<= a 2.95e-145) -1.0 (if (<= a 9e+33) t_1 t_0))))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double t_1 = b * (b * (b * b));
	double tmp;
	if (a <= -2.4e+59) {
		tmp = t_0;
	} else if (a <= -1.65e-50) {
		tmp = t_1;
	} else if (a <= 2.95e-145) {
		tmp = -1.0;
	} else if (a <= 9e+33) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = a * (a * (a * a))
    t_1 = b * (b * (b * b))
    if (a <= (-2.4d+59)) then
        tmp = t_0
    else if (a <= (-1.65d-50)) then
        tmp = t_1
    else if (a <= 2.95d-145) then
        tmp = -1.0d0
    else if (a <= 9d+33) then
        tmp = t_1
    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 t_1 = b * (b * (b * b));
	double tmp;
	if (a <= -2.4e+59) {
		tmp = t_0;
	} else if (a <= -1.65e-50) {
		tmp = t_1;
	} else if (a <= 2.95e-145) {
		tmp = -1.0;
	} else if (a <= 9e+33) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	t_1 = b * (b * (b * b))
	tmp = 0
	if a <= -2.4e+59:
		tmp = t_0
	elif a <= -1.65e-50:
		tmp = t_1
	elif a <= 2.95e-145:
		tmp = -1.0
	elif a <= 9e+33:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	t_1 = Float64(b * Float64(b * Float64(b * b)))
	tmp = 0.0
	if (a <= -2.4e+59)
		tmp = t_0;
	elseif (a <= -1.65e-50)
		tmp = t_1;
	elseif (a <= 2.95e-145)
		tmp = -1.0;
	elseif (a <= 9e+33)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * a));
	t_1 = b * (b * (b * b));
	tmp = 0.0;
	if (a <= -2.4e+59)
		tmp = t_0;
	elseif (a <= -1.65e-50)
		tmp = t_1;
	elseif (a <= 2.95e-145)
		tmp = -1.0;
	elseif (a <= 9e+33)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.4e+59], t$95$0, If[LessEqual[a, -1.65e-50], t$95$1, If[LessEqual[a, 2.95e-145], -1.0, If[LessEqual[a, 9e+33], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.4000000000000002e59 or 9.0000000000000001e33 < a

   1. Initial program 46.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 46.3%

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

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 96.2%

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

   7. Simplified 96.2%

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

   if -2.4000000000000002e59 < a < -1.6499999999999999e-50 or 2.9499999999999999e-145 < a < 9.0000000000000001e33

   1. Initial program 85.4%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 85.5%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right) + b \cdot \left(b \cdot \left(a + 3\right)\right)\right) + -1\right)} \]
   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 68.6%

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

   7. Simplified 68.6%

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

   if -1.6499999999999999e-50 < a < 2.9499999999999999e-145

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. --lowering--.f64 64.4%

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

   5. Simplified 64.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 64.4%

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

Alternative 5: 67.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ t_1 := \left(b \cdot b\right) \cdot 12\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-10}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))) (t_1 (* (* b b) 12.0)))
   (if (<= a -2.4e+59)
     t_0
     (if (<= a -2.8e-9)
       t_1
       (if (<= a 3.8e-10) -1.0 (if (<= a 2.5e+30) t_1 t_0))))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double t_1 = (b * b) * 12.0;
	double tmp;
	if (a <= -2.4e+59) {
		tmp = t_0;
	} else if (a <= -2.8e-9) {
		tmp = t_1;
	} else if (a <= 3.8e-10) {
		tmp = -1.0;
	} else if (a <= 2.5e+30) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = a * (a * (a * a))
    t_1 = (b * b) * 12.0d0
    if (a <= (-2.4d+59)) then
        tmp = t_0
    else if (a <= (-2.8d-9)) then
        tmp = t_1
    else if (a <= 3.8d-10) then
        tmp = -1.0d0
    else if (a <= 2.5d+30) then
        tmp = t_1
    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 t_1 = (b * b) * 12.0;
	double tmp;
	if (a <= -2.4e+59) {
		tmp = t_0;
	} else if (a <= -2.8e-9) {
		tmp = t_1;
	} else if (a <= 3.8e-10) {
		tmp = -1.0;
	} else if (a <= 2.5e+30) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	t_1 = (b * b) * 12.0
	tmp = 0
	if a <= -2.4e+59:
		tmp = t_0
	elif a <= -2.8e-9:
		tmp = t_1
	elif a <= 3.8e-10:
		tmp = -1.0
	elif a <= 2.5e+30:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	t_1 = Float64(Float64(b * b) * 12.0)
	tmp = 0.0
	if (a <= -2.4e+59)
		tmp = t_0;
	elseif (a <= -2.8e-9)
		tmp = t_1;
	elseif (a <= 3.8e-10)
		tmp = -1.0;
	elseif (a <= 2.5e+30)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * a));
	t_1 = (b * b) * 12.0;
	tmp = 0.0;
	if (a <= -2.4e+59)
		tmp = t_0;
	elseif (a <= -2.8e-9)
		tmp = t_1;
	elseif (a <= 3.8e-10)
		tmp = -1.0;
	elseif (a <= 2.5e+30)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]}, If[LessEqual[a, -2.4e+59], t$95$0, If[LessEqual[a, -2.8e-9], t$95$1, If[LessEqual[a, 3.8e-10], -1.0, If[LessEqual[a, 2.5e+30], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.4000000000000002e59 or 2.4999999999999999e30 < a

   1. Initial program 46.7%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 46.7%

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

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 95.5%

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

   7. Simplified 95.5%

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

   if -2.4000000000000002e59 < a < -2.79999999999999984e-9 or 3.7999999999999998e-10 < a < 2.4999999999999999e30

   1. Initial program 64.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

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

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

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

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

      19. +-commutative N/A

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

      20. +-lowering-+.f64 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in b around 0

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

   11. Simplified 67.2%

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

   12. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(12, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(12, \left(b \cdot \color{blue}{b}\right)\right) \]

       3. *-lowering-*.f64 66.7%

          \[\leadsto \mathsf{*.f64}\left(12, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   14. Simplified 66.7%

       \[\leadsto \color{blue}{12 \cdot \left(b \cdot b\right)} \]

   if -2.79999999999999984e-9 < a < 3.7999999999999998e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. --lowering--.f64 59.1%

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

   5. Simplified 59.1%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 59.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-9}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-10}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+30}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + -4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 4 \cdot \left(a + 3\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4.5e-5)
   (+ (* (* a a) (+ 4.0 (* a (+ a -4.0)))) -1.0)
   (+ (* b (* b (+ (* b b) (* 4.0 (+ a 3.0))))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.5e-5) {
		tmp = ((a * a) * (4.0 + (a * (a + -4.0)))) + -1.0;
	} else {
		tmp = (b * (b * ((b * b) + (4.0 * (a + 3.0))))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4.5d-5) then
        tmp = ((a * a) * (4.0d0 + (a * (a + (-4.0d0))))) + (-1.0d0)
    else
        tmp = (b * (b * ((b * b) + (4.0d0 * (a + 3.0d0))))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.5e-5) {
		tmp = ((a * a) * (4.0 + (a * (a + -4.0)))) + -1.0;
	} else {
		tmp = (b * (b * ((b * b) + (4.0 * (a + 3.0))))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4.5e-5:
		tmp = ((a * a) * (4.0 + (a * (a + -4.0)))) + -1.0
	else:
		tmp = (b * (b * ((b * b) + (4.0 * (a + 3.0))))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4.5e-5)
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + -4.0)))) + -1.0);
	else
		tmp = Float64(Float64(b * Float64(b * Float64(Float64(b * b) + Float64(4.0 * Float64(a + 3.0))))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4.5e-5)
		tmp = ((a * a) * (4.0 + (a * (a + -4.0)))) + -1.0;
	else
		tmp = (b * (b * ((b * b) + (4.0 * (a + 3.0))))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4.5e-5], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(b * N[(N[(b * b), $MachinePrecision] + N[(4.0 * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.50000000000000028e-5

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. --lowering--.f64 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-lowering-+.f64 99.6%

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

   8. Simplified 99.6%

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

   if 4.50000000000000028e-5 < (*.f64 b b)

   1. Initial program 60.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

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

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

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

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

      19. +-commutative N/A

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

      20. +-lowering-+.f64 91.8%

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

   5. Simplified 91.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + -4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 4 \cdot \left(a + 3\right)\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+33}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -2.4e+59)
     t_0
     (if (<= a 4.2e+33) (+ (* b (* b (+ (* b b) 12.0))) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -2.4e+59) {
		tmp = t_0;
	} else if (a <= 4.2e+33) {
		tmp = (b * (b * ((b * b) + 12.0))) + -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 <= (-2.4d+59)) then
        tmp = t_0
    else if (a <= 4.2d+33) then
        tmp = (b * (b * ((b * b) + 12.0d0))) + (-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 <= -2.4e+59) {
		tmp = t_0;
	} else if (a <= 4.2e+33) {
		tmp = (b * (b * ((b * b) + 12.0))) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if a <= -2.4e+59:
		tmp = t_0
	elif a <= 4.2e+33:
		tmp = (b * (b * ((b * b) + 12.0))) + -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 <= -2.4e+59)
		tmp = t_0;
	elseif (a <= 4.2e+33)
		tmp = Float64(Float64(b * Float64(b * Float64(Float64(b * b) + 12.0))) + -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 <= -2.4e+59)
		tmp = t_0;
	elseif (a <= 4.2e+33)
		tmp = (b * (b * ((b * b) + 12.0))) + -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, -2.4e+59], t$95$0, If[LessEqual[a, 4.2e+33], N[(N[(b * N[(b * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4000000000000002e59 or 4.2000000000000001e33 < a

   1. Initial program 46.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 46.3%

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

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 96.2%

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

   7. Simplified 96.2%

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

   if -2.4000000000000002e59 < a < 4.2000000000000001e33

   1. Initial program 94.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

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

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

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

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

      19. +-commutative N/A

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

      20. +-lowering-+.f64 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 97.8%

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

   8. Simplified 97.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+33}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+34}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -2.4e+59)
     t_0
     (if (<= a 1.7e+34) (+ (* b (* b (* b b))) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -2.4e+59) {
		tmp = t_0;
	} else if (a <= 1.7e+34) {
		tmp = (b * (b * (b * b))) + -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 <= (-2.4d+59)) then
        tmp = t_0
    else if (a <= 1.7d+34) then
        tmp = (b * (b * (b * b))) + (-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 <= -2.4e+59) {
		tmp = t_0;
	} else if (a <= 1.7e+34) {
		tmp = (b * (b * (b * b))) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if a <= -2.4e+59:
		tmp = t_0
	elif a <= 1.7e+34:
		tmp = (b * (b * (b * b))) + -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 <= -2.4e+59)
		tmp = t_0;
	elseif (a <= 1.7e+34)
		tmp = Float64(Float64(b * Float64(b * Float64(b * b))) + -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 <= -2.4e+59)
		tmp = t_0;
	elseif (a <= 1.7e+34)
		tmp = (b * (b * (b * b))) + -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, -2.4e+59], t$95$0, If[LessEqual[a, 1.7e+34], N[(N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4000000000000002e59 or 1.7e34 < a

   1. Initial program 46.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 46.3%

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

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 96.2%

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

   7. Simplified 96.2%

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

   if -2.4000000000000002e59 < a < 1.7e34

   1. Initial program 94.6%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 95.5%

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

   5. Simplified 95.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+34}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+29}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -2.4e+59) t_0 (if (<= a 9e+29) (+ (* b (* b 12.0)) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -2.4e+59) {
		tmp = t_0;
	} else if (a <= 9e+29) {
		tmp = (b * (b * 12.0)) + -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 <= (-2.4d+59)) then
        tmp = t_0
    else if (a <= 9d+29) then
        tmp = (b * (b * 12.0d0)) + (-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 <= -2.4e+59) {
		tmp = t_0;
	} else if (a <= 9e+29) {
		tmp = (b * (b * 12.0)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if a <= -2.4e+59:
		tmp = t_0
	elif a <= 9e+29:
		tmp = (b * (b * 12.0)) + -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 <= -2.4e+59)
		tmp = t_0;
	elseif (a <= 9e+29)
		tmp = Float64(Float64(b * Float64(b * 12.0)) + -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 <= -2.4e+59)
		tmp = t_0;
	elseif (a <= 9e+29)
		tmp = (b * (b * 12.0)) + -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, -2.4e+59], t$95$0, If[LessEqual[a, 9e+29], N[(N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4000000000000002e59 or 9.0000000000000005e29 < a

   1. Initial program 46.7%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 46.7%

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

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 95.5%

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

   7. Simplified 95.5%

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

   if -2.4000000000000002e59 < a < 9.0000000000000005e29

   1. Initial program 94.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

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

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

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

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

      19. +-commutative N/A

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

      20. +-lowering-+.f64 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in b around 0

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

   11. Simplified 75.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+29}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10000:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 10000.0)
   (+ (* a (* a (* a a))) -1.0)
   (* b (* b (+ (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 10000.0) {
		tmp = (a * (a * (a * a))) + -1.0;
	} else {
		tmp = b * (b * ((b * b) + 12.0));
	}
	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) <= 10000.0d0) then
        tmp = (a * (a * (a * a))) + (-1.0d0)
    else
        tmp = b * (b * ((b * b) + 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 10000.0) {
		tmp = (a * (a * (a * a))) + -1.0;
	} else {
		tmp = b * (b * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 10000.0:
		tmp = (a * (a * (a * a))) + -1.0
	else:
		tmp = b * (b * ((b * b) + 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 10000.0)
		tmp = Float64(Float64(a * Float64(a * Float64(a * a))) + -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(Float64(b * b) + 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 10000.0)
		tmp = (a * (a * (a * a))) + -1.0;
	else
		tmp = b * (b * ((b * b) + 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 10000.0], N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e4

   1. Initial program 81.3%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 97.4%

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

   5. Simplified 97.4%

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

   if 1e4 < (*.f64 b b)

   1. Initial program 59.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

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

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

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

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

      19. +-commutative N/A

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

      20. +-lowering-+.f64 92.5%

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

   5. Simplified 92.5%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-lft1-in N/A

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

       4. associate-*l* N/A

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

       5. fma-define N/A

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

       6. metadata-eval N/A

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

       7. pow-sqr N/A

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

       8. associate-*r* N/A

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

       9. lft-mult-inverse N/A

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

       10. *-lft-identity N/A

           \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{\left(2 \cdot 2\right)}\right) \]

       12. pow-sqr N/A

           \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{2} \cdot {b}^{2}\right) \]

       13. fma-define N/A

           \[\leadsto 12 \cdot {b}^{2} + \color{blue}{{b}^{2} \cdot {b}^{2}} \]

       14. distribute-rgt-in N/A

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

       15. unpow2 N/A

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

       16. associate-*l* N/A

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

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

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

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

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

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

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

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

       21. *-lowering-*.f64 89.9%

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   11. Simplified 89.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10000:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.082:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.082) -1.0 (* (* b b) 12.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.082) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 12.0;
	}
	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) <= 0.082d0) then
        tmp = -1.0d0
    else
        tmp = (b * b) * 12.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.082) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * 12.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.082:
		tmp = -1.0
	else:
		tmp = (b * b) * 12.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.082)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * 12.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.082)
		tmp = -1.0;
	else
		tmp = (b * b) * 12.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.082], -1.0, N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0820000000000000034

   1. Initial program 81.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. --lowering--.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 48.2%

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

   if 0.0820000000000000034 < (*.f64 b b)

   1. Initial program 60.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

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

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

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

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

      19. +-commutative N/A

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

      20. +-lowering-+.f64 91.8%

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

   5. Simplified 91.8%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 89.6%

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

   8. Simplified 89.6%

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

   9. Taylor expanded in b around 0

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

   11. Simplified 53.5%

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

   12. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(12, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(12, \left(b \cdot \color{blue}{b}\right)\right) \]

       3. *-lowering-*.f64 53.5%

          \[\leadsto \mathsf{*.f64}\left(12, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   14. Simplified 53.5%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.082:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.6% accurate, 128.0× 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 71.4%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

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

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. --lowering--.f64 73.5%

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

5. Simplified 73.5%

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

6. Taylor expanded in a around 0

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

8. Simplified 26.3%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))