Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.3% → 99.0%

Time: 10.1s

Alternatives: 12

Speedup: 9.3×

• 60.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

   1. associate--l+ N/A

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. sub-neg N/A

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

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

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

3. Simplified 74.4%

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

5. Taylor expanded in a around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 2: 94.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{if}\;a \leq -170000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 42000000:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a (+ a 4.0))))))
   (if (<= a -170000000000.0)
     t_0
     (if (<= a 42000000.0) (+ (* b (* b (+ (* b b) 4.0))) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * (a + 4.0)));
	double tmp;
	if (a <= -170000000000.0) {
		tmp = t_0;
	} else if (a <= 42000000.0) {
		tmp = (b * (b * ((b * b) + 4.0))) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * (a * (a + 4.0d0)))
    if (a <= (-170000000000.0d0)) then
        tmp = t_0
    else if (a <= 42000000.0d0) then
        tmp = (b * (b * ((b * b) + 4.0d0))) + (-1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * (a * (a + 4.0)));
	double tmp;
	if (a <= -170000000000.0) {
		tmp = t_0;
	} else if (a <= 42000000.0) {
		tmp = (b * (b * ((b * b) + 4.0))) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * (a + 4.0)))
	tmp = 0
	if a <= -170000000000.0:
		tmp = t_0
	elif a <= 42000000.0:
		tmp = (b * (b * ((b * b) + 4.0))) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * Float64(a + 4.0))))
	tmp = 0.0
	if (a <= -170000000000.0)
		tmp = t_0;
	elseif (a <= 42000000.0)
		tmp = Float64(Float64(b * Float64(b * Float64(Float64(b * b) + 4.0))) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * (a + 4.0)));
	tmp = 0.0;
	if (a <= -170000000000.0)
		tmp = t_0;
	elseif (a <= 42000000.0)
		tmp = (b * (b * ((b * b) + 4.0))) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -170000000000.0], t$95$0, If[LessEqual[a, 42000000.0], N[(N[(b * N[(b * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7e11 or 4.2e7 < a

   1. Initial program 44.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 94.2%

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

   5. Simplified 94.2%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 94.2%

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

   8. Simplified 94.2%

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

   9. Taylor expanded in a around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. associate-*l* N/A

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

       9. lft-mult-inverse N/A

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

       10. metadata-eval N/A

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

       11. *-lft-identity N/A

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

       12. associate-*r* N/A

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

       13. unpow2 N/A

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

       14. associate-*l* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

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

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

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

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

       20. +-commutative N/A

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

       21. +-lowering-+.f64 94.2%

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

   11. Simplified 94.2%

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

   if -1.7e11 < a < 4.2e7

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 98.8%

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

   5. Simplified 98.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -170000000000:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{elif}\;a \leq 42000000:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{if}\;a \leq -105000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 42000000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a (+ a 4.0))))))
   (if (<= a -105000000000.0)
     t_0
     (if (<= a 42000000.0) (+ (* (* b b) (+ (* b b) 4.0)) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * (a + 4.0)));
	double tmp;
	if (a <= -105000000000.0) {
		tmp = t_0;
	} else if (a <= 42000000.0) {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * (a * (a + 4.0d0)))
    if (a <= (-105000000000.0d0)) then
        tmp = t_0
    else if (a <= 42000000.0d0) then
        tmp = ((b * b) * ((b * b) + 4.0d0)) + (-1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * (a * (a + 4.0)));
	double tmp;
	if (a <= -105000000000.0) {
		tmp = t_0;
	} else if (a <= 42000000.0) {
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * (a + 4.0)))
	tmp = 0
	if a <= -105000000000.0:
		tmp = t_0
	elif a <= 42000000.0:
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * Float64(a + 4.0))))
	tmp = 0.0
	if (a <= -105000000000.0)
		tmp = t_0;
	elseif (a <= 42000000.0)
		tmp = Float64(Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * (a + 4.0)));
	tmp = 0.0;
	if (a <= -105000000000.0)
		tmp = t_0;
	elseif (a <= 42000000.0)
		tmp = ((b * b) * ((b * b) + 4.0)) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -105000000000.0], t$95$0, If[LessEqual[a, 42000000.0], N[(N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.05e11 or 4.2e7 < a

   1. Initial program 44.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 94.2%

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

   5. Simplified 94.2%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 94.2%

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

   8. Simplified 94.2%

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

   9. Taylor expanded in a around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. associate-*l* N/A

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

       9. lft-mult-inverse N/A

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

       10. metadata-eval N/A

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

       11. *-lft-identity N/A

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

       12. associate-*r* N/A

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

       13. unpow2 N/A

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

       14. associate-*l* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

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

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

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

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

       20. +-commutative N/A

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

       21. +-lowering-+.f64 94.2%

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

   11. Simplified 94.2%

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

   if -1.05e11 < a < 4.2e7

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -105000000000:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{elif}\;a \leq 42000000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-213}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 0.5:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-213)
   -1.0
   (if (<= (* b b) 0.5) (* a (* a (* a a))) (* b (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-213) {
		tmp = -1.0;
	} else if ((b * b) <= 0.5) {
		tmp = a * (a * (a * a));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d-213) then
        tmp = -1.0d0
    else if ((b * b) <= 0.5d0) then
        tmp = a * (a * (a * a))
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-213) {
		tmp = -1.0;
	} else if ((b * b) <= 0.5) {
		tmp = a * (a * (a * a));
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-213:
		tmp = -1.0
	elif (b * b) <= 0.5:
		tmp = a * (a * (a * a))
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-213)
		tmp = -1.0;
	elseif (Float64(b * b) <= 0.5)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-213)
		tmp = -1.0;
	elseif ((b * b) <= 0.5)
		tmp = a * (a * (a * a));
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-213], -1.0, If[LessEqual[N[(b * b), $MachinePrecision], 0.5], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 1.9999999999999999e-213

   1. Initial program 84.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 58.0%

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

   5. Simplified 58.0%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 58.0%

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

   if 1.9999999999999999e-213 < (*.f64 b b) < 0.5

   1. Initial program 82.4%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 82.3%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.8%

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

   7. Simplified 60.8%

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

   if 0.5 < (*.f64 b b)

   1. Initial program 65.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 65.3%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 90.2%

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

   7. Simplified 90.2%

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

Alternative 5: 97.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot \left(a + 4\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.5)
   (+ (* a (* a (+ 4.0 (* a (+ a 4.0))))) -1.0)
   (* (* b b) (+ (* b b) (* (* a a) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0;
	} else {
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.5d0) then
        tmp = (a * (a * (4.0d0 + (a * (a + 4.0d0))))) + (-1.0d0)
    else
        tmp = (b * b) * ((b * b) + ((a * a) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0;
	} else {
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.5:
		tmp = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0
	else:
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.5)
		tmp = Float64(Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a + 4.0))))) + -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + Float64(Float64(a * a) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.5)
		tmp = (a * (a * (4.0 + (a * (a + 4.0))))) + -1.0;
	else
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.5], N[(N[(a * N[(a * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.5

   1. Initial program 83.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 99.1%

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

   8. Simplified 99.1%

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

   if 0.5 < (*.f64 b b)

   1. Initial program 65.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 65.3%

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

   5. Taylor expanded in a around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in b around 0

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

   10. Simplified 99.7%

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

   11. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

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

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

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. *-commutative N/A

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

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

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

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

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

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

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

       13. unpow2 N/A

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

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

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

       15. +-commutative N/A

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

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

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

       17. unpow2 N/A

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

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

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

       19. unpow2 N/A

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

       20. associate-*r* N/A

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

       21. *-commutative N/A

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

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

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

       23. *-commutative N/A

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

       24. *-lowering-*.f64 91.0%

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

   13. Simplified 91.0%

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

   14. Taylor expanded in b around inf

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*r/ N/A

          \[\leadsto {b}^{2} \cdot {b}^{2} + \frac{2 \cdot {a}^{2}}{{b}^{2}} \cdot {\color{blue}{b}}^{4} \]

       6. associate-*l/ N/A

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

       7. associate-/l* N/A

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

       8. *-lft-identity N/A

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

       9. associate-*l/ N/A

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

       10. metadata-eval N/A

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

       11. pow-sqr N/A

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

       12. associate-*r* N/A

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

       13. lft-mult-inverse N/A

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

       14. *-lft-identity N/A

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

       15. distribute-rgt-in N/A

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

       16. +-commutative N/A

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

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

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

       18. unpow2 N/A

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

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

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

   16. Simplified 97.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot \left(a + 4\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.5)
   (+ (* a (* a (* a a))) -1.0)
   (* (* b b) (+ (* b b) (* (* a a) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = (a * (a * (a * a))) + -1.0;
	} else {
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.5d0) then
        tmp = (a * (a * (a * a))) + (-1.0d0)
    else
        tmp = (b * b) * ((b * b) + ((a * a) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = (a * (a * (a * a))) + -1.0;
	} else {
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.5:
		tmp = (a * (a * (a * a))) + -1.0
	else:
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.5)
		tmp = Float64(Float64(a * Float64(a * Float64(a * a))) + -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + Float64(Float64(a * a) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.5)
		tmp = (a * (a * (a * a))) + -1.0;
	else
		tmp = (b * b) * ((b * b) + ((a * a) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.5], N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.5

   1. Initial program 83.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 97.5%

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

   5. Simplified 97.5%

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

   if 0.5 < (*.f64 b b)

   1. Initial program 65.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 65.3%

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

   5. Taylor expanded in a around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in b around 0

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

   10. Simplified 99.7%

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

   11. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

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

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

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. *-commutative N/A

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

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

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

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

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

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

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

       13. unpow2 N/A

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

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

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

       15. +-commutative N/A

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

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

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

       17. unpow2 N/A

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

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

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

       19. unpow2 N/A

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

       20. associate-*r* N/A

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

       21. *-commutative N/A

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

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

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

       23. *-commutative N/A

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

       24. *-lowering-*.f64 91.0%

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

   13. Simplified 91.0%

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

   14. Taylor expanded in b around inf

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*r/ N/A

          \[\leadsto {b}^{2} \cdot {b}^{2} + \frac{2 \cdot {a}^{2}}{{b}^{2}} \cdot {\color{blue}{b}}^{4} \]

       6. associate-*l/ N/A

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

       7. associate-/l* N/A

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

       8. *-lft-identity N/A

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

       9. associate-*l/ N/A

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

       10. metadata-eval N/A

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

       11. pow-sqr N/A

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

       12. associate-*r* N/A

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

       13. lft-mult-inverse N/A

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

       14. *-lft-identity N/A

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

       15. distribute-rgt-in N/A

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

       16. +-commutative N/A

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

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

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

       18. unpow2 N/A

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

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

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

   16. Simplified 97.2%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{if}\;a \leq -80000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 180000000000:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a (+ a 4.0))))))
   (if (<= a -80000000000.0)
     t_0
     (if (<= a 180000000000.0) (+ (* b (* b (* b b))) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * (a + 4.0)));
	double tmp;
	if (a <= -80000000000.0) {
		tmp = t_0;
	} else if (a <= 180000000000.0) {
		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 + 4.0d0)))
    if (a <= (-80000000000.0d0)) then
        tmp = t_0
    else if (a <= 180000000000.0d0) 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 + 4.0)));
	double tmp;
	if (a <= -80000000000.0) {
		tmp = t_0;
	} else if (a <= 180000000000.0) {
		tmp = (b * (b * (b * b))) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * (a + 4.0)))
	tmp = 0
	if a <= -80000000000.0:
		tmp = t_0
	elif a <= 180000000000.0:
		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 * Float64(a + 4.0))))
	tmp = 0.0
	if (a <= -80000000000.0)
		tmp = t_0;
	elseif (a <= 180000000000.0)
		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 + 4.0)));
	tmp = 0.0;
	if (a <= -80000000000.0)
		tmp = t_0;
	elseif (a <= 180000000000.0)
		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 * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -80000000000.0], t$95$0, If[LessEqual[a, 180000000000.0], 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 < -8e10 or 1.8e11 < a

   1. Initial program 44.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 94.2%

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

   5. Simplified 94.2%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 94.2%

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

   8. Simplified 94.2%

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

   9. Taylor expanded in a around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. associate-*l* N/A

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

       9. lft-mult-inverse N/A

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

       10. metadata-eval N/A

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

       11. *-lft-identity N/A

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

       12. associate-*r* N/A

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

       13. unpow2 N/A

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

       14. associate-*l* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

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

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

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

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

       20. +-commutative N/A

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

       21. +-lowering-+.f64 94.2%

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

   11. Simplified 94.2%

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

   if -8e10 < a < 1.8e11

   1. Initial program 99.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 98.1%

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

   5. Simplified 98.1%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -80000000000:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{elif}\;a \leq 180000000000:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(a + 4\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -4 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.41:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -4e-31) t_0 (if (<= a 0.41) -1.0 t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -4e-31) {
		tmp = t_0;
	} else if (a <= 0.41) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * (a * a))
    if (a <= (-4d-31)) then
        tmp = t_0
    else if (a <= 0.41d0) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -4e-31) {
		tmp = t_0;
	} else if (a <= 0.41) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * (a * a))
	tmp = 0
	if a <= -4e-31:
		tmp = t_0
	elif a <= 0.41:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (a <= -4e-31)
		tmp = t_0;
	elseif (a <= 0.41)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * (a * a));
	tmp = 0.0;
	if (a <= -4e-31)
		tmp = t_0;
	elseif (a <= 0.41)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4e-31], t$95$0, If[LessEqual[a, 0.41], -1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4e-31 or 0.409999999999999976 < a

   1. Initial program 47.4%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 46.5%

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

   5. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 88.4%

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

   7. Simplified 88.4%

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

   if -4e-31 < a < 0.409999999999999976

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 49.3%

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

Alternative 9: 98.4% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

   1. associate--l+ N/A

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. sub-neg N/A

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

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

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

3. Simplified 74.4%

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

5. Taylor expanded in a around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in b around 0

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

10. Simplified 98.5%

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

Alternative 10: 93.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.5) (+ (* a (* a (* a a))) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = (a * (a * (a * a))) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.5d0) then
        tmp = (a * (a * (a * a))) + (-1.0d0)
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = (a * (a * (a * a))) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.5:
		tmp = (a * (a * (a * a))) + -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.5)
		tmp = Float64(Float64(a * Float64(a * Float64(a * a))) + -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.5)
		tmp = (a * (a * (a * a))) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.5], N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.5

   1. Initial program 83.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 97.5%

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

   5. Simplified 97.5%

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

   if 0.5 < (*.f64 b b)

   1. Initial program 65.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. sub-neg N/A

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

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

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

   3. Simplified 65.3%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 90.2%

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

   7. Simplified 90.2%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.5) (+ (* (* a a) 4.0) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 0.5d0) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else
        tmp = b * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.5:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = b * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.5)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.5)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = b * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.5

   1. Initial program 83.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. *-lowering-*.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 80.6%

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

   if 0.5 < (*.f64 b b)

   1. Initial program 65.3%

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

      1. associate--l+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(\color{blue}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \left(a \cdot a + b \cdot b\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(a \cdot a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot b\right)\right)\right), \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(1 + a\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(\left(\left(a \cdot a\right) \cdot \left(4 + a \cdot 4\right) + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right) + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {b}^{\left(2 \cdot \color{blue}{2}\right)} \]

      2. pow-sqr N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(b \cdot b\right) \cdot {\color{blue}{b}}^{2} \]

      4. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left(b \cdot {b}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {b}^{2}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      8. *-lowering-*.f64 90.2%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   7. Simplified 90.2%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.1% accurate, 130.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

double code(double a, double b) {
	return -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -1.0d0
end function

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0;
end

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.8%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)}, 1\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{4} + 4 \cdot {b}^{2}\right), 1\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{\left(2 \cdot 2\right)} + 4 \cdot {b}^{2}\right), 1\right) \]

   3. pow-sqr N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{2} \cdot {b}^{2} + 4 \cdot {b}^{2}\right), 1\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \mathsf{\_.f64}\left(\left({b}^{2} \cdot \left({b}^{2} + 4\right)\right), 1\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({b}^{2} + 4\right)\right), 1\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({b}^{2} + 4\right)\right), 1\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({b}^{2} + 4\right)\right), 1\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left({b}^{2}\right), 4\right)\right), 1\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\left(b \cdot b\right), 4\right)\right), 1\right) \]

   10. *-lowering-*.f64 70.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), 4\right)\right), 1\right) \]

5. Simplified 70.2%

   \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)} - 1 \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 26.2%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))