Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.1% → 99.9%

Time: 12.4s

Alternatives: 17

Speedup: 3.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ t_1 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(1 - a, \left(a \cdot a\right) \cdot 4, \mathsf{fma}\left(b, 4 \cdot \left(b \cdot \left(a + 3\right)\right), t\_1 \cdot t\_1\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))) (t_1 (fma a a (* b b))))
   (if (<=
        (+
         (pow (+ (* a a) (* b b)) 2.0)
         (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
        2e+62)
     (+
      (fma
       (- 1.0 a)
       (* (* a a) 4.0)
       (fma b (* 4.0 (* b (+ a 3.0))) (* t_1 t_1)))
      -1.0)
     (fma t_0 t_0 (fma (* b b) 12.0 -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double t_1 = fma(a, a, (b * b));
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 2e+62) {
		tmp = fma((1.0 - a), ((a * a) * 4.0), fma(b, (4.0 * (b * (a + 3.0))), (t_1 * t_1))) + -1.0;
	} else {
		tmp = fma(t_0, t_0, fma((b * b), 12.0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	t_1 = fma(a, a, Float64(b * b))
	tmp = 0.0
	if (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(a + 3.0))))) <= 2e+62)
		tmp = Float64(fma(Float64(1.0 - a), Float64(Float64(a * a) * 4.0), fma(b, Float64(4.0 * Float64(b * Float64(a + 3.0))), Float64(t_1 * t_1))) + -1.0);
	else
		tmp = fma(t_0, t_0, fma(Float64(b * b), 12.0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+62], N[(N[(N[(1.0 - a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[(b * N[(4.0 * N[(b * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 2.00000000000000007e62

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   if 2.00000000000000007e62 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 67.3%

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

   4. Applied egg-rr 68.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(4, \color{blue}{3 \cdot {b}^{2}}, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

   7. Simplified 99.9%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(4, \color{blue}{b \cdot \left(b \cdot 3\right)}, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. remove-double-div 99.9

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

      10. lift-fma.f64 N/A

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(1 - a, \left(a \cdot a\right) \cdot 4, \mathsf{fma}\left(b, 4 \cdot \left(b \cdot \left(a + 3\right)\right), \mathsf{fma}\left(a, a, b \cdot b\right) \cdot \mathsf{fma}\left(a, a, b \cdot b\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \left(1 - a\right), b \cdot \left(b \cdot \left(a + 3\right)\right)\right), 4, \mathsf{fma}\left(t\_0, t\_0, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1, \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))) (t_1 (fma b b (* a a))))
   (if (<=
        (+
         (pow (+ (* a a) (* b b)) 2.0)
         (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
        2e+62)
     (fma (fma a (* a (- 1.0 a)) (* b (* b (+ a 3.0)))) 4.0 (fma t_0 t_0 -1.0))
     (fma t_1 t_1 (fma (* b b) 12.0 -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double t_1 = fma(b, b, (a * a));
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 2e+62) {
		tmp = fma(fma(a, (a * (1.0 - a)), (b * (b * (a + 3.0)))), 4.0, fma(t_0, t_0, -1.0));
	} else {
		tmp = fma(t_1, t_1, fma((b * b), 12.0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	t_1 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (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(a + 3.0))))) <= 2e+62)
		tmp = fma(fma(a, Float64(a * Float64(1.0 - a)), Float64(b * Float64(b * Float64(a + 3.0)))), 4.0, fma(t_0, t_0, -1.0));
	else
		tmp = fma(t_1, t_1, fma(Float64(b * b), 12.0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+62], N[(N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$1 + N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 2.00000000000000007e62

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   if 2.00000000000000007e62 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 67.3%

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

   4. Applied egg-rr 68.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(4, \color{blue}{3 \cdot {b}^{2}}, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

   7. Simplified 99.9%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(4, \color{blue}{b \cdot \left(b \cdot 3\right)}, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. remove-double-div 99.9

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

      10. lift-fma.f64 N/A

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \left(1 - a\right), b \cdot \left(b \cdot \left(a + 3\right)\right)\right), 4, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \left(1 - a\right), b \cdot \left(b \cdot \left(a + 3\right)\right)\right), 4, \mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<=
        (+
         (pow (+ (* a a) (* b b)) 2.0)
         (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
        2e+38)
     (fma
      (fma a (* a (- 1.0 a)) (* b (* b (+ a 3.0))))
      4.0
      (fma (* a a) (* a a) -1.0))
     (fma t_0 t_0 (fma (* b b) 12.0 -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 2e+38) {
		tmp = fma(fma(a, (a * (1.0 - a)), (b * (b * (a + 3.0)))), 4.0, fma((a * a), (a * a), -1.0));
	} else {
		tmp = fma(t_0, t_0, fma((b * b), 12.0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (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(a + 3.0))))) <= 2e+38)
		tmp = fma(fma(a, Float64(a * Float64(1.0 - a)), Float64(b * Float64(b * Float64(a + 3.0)))), 4.0, fma(Float64(a * a), Float64(a * a), -1.0));
	else
		tmp = fma(t_0, t_0, fma(Float64(b * b), 12.0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+38], N[(N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 1.99999999999999995e38

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 1.99999999999999995e38 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 68.3%

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

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(4, \color{blue}{3 \cdot {b}^{2}}, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   7. Simplified 99.7%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(4, \color{blue}{b \cdot \left(b \cdot 3\right)}, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. remove-double-div 99.7

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

      10. lift-fma.f64 N/A

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \left(1 - a\right), b \cdot \left(b \cdot \left(a + 3\right)\right)\right), 4, \mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot a\right), a, \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(1 - a\right), b \cdot \left(b \cdot \left(a + 3\right)\right)\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<=
        (+
         (pow (+ (* a a) (* b b)) 2.0)
         (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
        2e+38)
     (fma
      (* a (* a a))
      a
      (fma 4.0 (fma a (* a (- 1.0 a)) (* b (* b (+ a 3.0)))) -1.0))
     (fma t_0 t_0 (fma (* b b) 12.0 -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 2e+38) {
		tmp = fma((a * (a * a)), a, fma(4.0, fma(a, (a * (1.0 - a)), (b * (b * (a + 3.0)))), -1.0));
	} else {
		tmp = fma(t_0, t_0, fma((b * b), 12.0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (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(a + 3.0))))) <= 2e+38)
		tmp = fma(Float64(a * Float64(a * a)), a, fma(4.0, fma(a, Float64(a * Float64(1.0 - a)), Float64(b * Float64(b * Float64(a + 3.0)))), -1.0));
	else
		tmp = fma(t_0, t_0, fma(Float64(b * b), 12.0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+38], N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * a + N[(4.0 * N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 1.99999999999999995e38

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 1.99999999999999995e38 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 68.3%

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

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(4, \color{blue}{3 \cdot {b}^{2}}, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   7. Simplified 99.7%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(4, \color{blue}{b \cdot \left(b \cdot 3\right)}, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. remove-double-div 99.7

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

      10. lift-fma.f64 N/A

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot a\right), a, \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \left(1 - a\right), b \cdot \left(b \cdot \left(a + 3\right)\right)\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right) + 4 \cdot \left(a \cdot \mathsf{fma}\left(a, -a, a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<=
        (+
         (pow (+ (* a a) (* b b)) 2.0)
         (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
        2e+38)
     (+ (+ (* a (* a (* a a))) (* 4.0 (* a (fma a (- a) a)))) -1.0)
     (fma t_0 t_0 (fma (* b b) 12.0 -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) <= 2e+38) {
		tmp = ((a * (a * (a * a))) + (4.0 * (a * fma(a, -a, a)))) + -1.0;
	} else {
		tmp = fma(t_0, t_0, fma((b * b), 12.0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (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(a + 3.0))))) <= 2e+38)
		tmp = Float64(Float64(Float64(a * Float64(a * Float64(a * a))) + Float64(4.0 * Float64(a * fma(a, Float64(-a), a)))) + -1.0);
	else
		tmp = fma(t_0, t_0, fma(Float64(b * b), 12.0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+38], N[(N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(a * N[(a * (-a) + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 1.99999999999999995e38

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. unpow2 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. *-rgt-identity N/A

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

      12. mul-1-neg N/A

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

      13. unpow2 N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. mul-1-neg N/A

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

      16. distribute-lft-in N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. mul-1-neg N/A

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

      23. lower-neg.f64 99.6

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

   8. Simplified 99.6%

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

   if 1.99999999999999995e38 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 68.3%

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

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(4, \color{blue}{3 \cdot {b}^{2}}, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   7. Simplified 99.7%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(4, \color{blue}{b \cdot \left(b \cdot 3\right)}, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. remove-double-div 99.7

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

      10. lift-fma.f64 N/A

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right) + 4 \cdot \left(a \cdot \mathsf{fma}\left(a, -a, a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, 12 \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq 7.7 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.45)
   (fma (* a a) (* a a) (* 12.0 (* b (* b b))))
   (if (<= a 7.7e-31)
     (+ (+ (* (* b b) (* b b)) (* 4.0 (* (* b b) 3.0))) -1.0)
     (+ (+ (* b b) (* a (* a (* a a)))) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.45) {
		tmp = fma((a * a), (a * a), (12.0 * (b * (b * b))));
	} else if (a <= 7.7e-31) {
		tmp = (((b * b) * (b * b)) + (4.0 * ((b * b) * 3.0))) + -1.0;
	} else {
		tmp = ((b * b) + (a * (a * (a * a)))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.45)
		tmp = fma(Float64(a * a), Float64(a * a), Float64(12.0 * Float64(b * Float64(b * b))));
	elseif (a <= 7.7e-31)
		tmp = Float64(Float64(Float64(Float64(b * b) * Float64(b * b)) + Float64(4.0 * Float64(Float64(b * b) * 3.0))) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) + Float64(a * Float64(a * Float64(a * a)))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.45], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(12.0 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7.7e-31], N[(N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(b * b), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.44999999999999996

   1. Initial program 65.1%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.1

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

   5. Simplified 61.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 93.5

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

   8. Simplified 93.5%

      \[\leadsto \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right) + 4 \cdot \color{blue}{\left(3 \cdot \left(b \cdot b\right)\right)}\right) - 1 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

   10. Applied egg-rr 93.5%

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

   11. Taylor expanded in b around inf

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \color{blue}{12 \cdot {b}^{3}}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \color{blue}{{b}^{3} \cdot 12}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \color{blue}{{b}^{3} \cdot 12}\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot 12\right) \]

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot 12\right) \]

       7. lower-*.f64 80.9

          \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot 12\right) \]

   13. Simplified 80.9%

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \color{blue}{\left(b \cdot \left(b \cdot b\right)\right) \cdot 12}\right) \]

   if -1.44999999999999996 < a < 7.70000000000000012e-31

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.6

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 99.1

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

   11. Simplified 99.1%

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

   if 7.70000000000000012e-31 < a

   1. Initial program 41.5%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 40.0

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

   5. Simplified 40.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 95.1

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

   8. Simplified 95.1%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 95.1

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

   11. Simplified 95.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, 12 \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq 7.7 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+118}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+118)
   (+ (+ (* a (* a (* a a))) (* 4.0 (* (* b b) 3.0))) -1.0)
   (* (* b b) (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+118) {
		tmp = ((a * (a * (a * a))) + (4.0 * ((b * b) * 3.0))) + -1.0;
	} else {
		tmp = (b * 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) <= 1d+118) then
        tmp = ((a * (a * (a * a))) + (4.0d0 * ((b * b) * 3.0d0))) + (-1.0d0)
    else
        tmp = (b * b) * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+118) {
		tmp = ((a * (a * (a * a))) + (4.0 * ((b * b) * 3.0))) + -1.0;
	} else {
		tmp = (b * b) * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+118:
		tmp = ((a * (a * (a * a))) + (4.0 * ((b * b) * 3.0))) + -1.0
	else:
		tmp = (b * b) * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+118)
		tmp = Float64(Float64(Float64(a * Float64(a * Float64(a * a))) + Float64(4.0 * Float64(Float64(b * b) * 3.0))) + -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+118)
		tmp = ((a * (a * (a * a))) + (4.0 * ((b * b) * 3.0))) + -1.0;
	else
		tmp = (b * b) * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+118], N[(N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(b * b), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999967e117

   1. Initial program 86.0%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 78.1

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

   5. Simplified 78.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 88.9

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

   8. Simplified 88.9%

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

   if 9.99999999999999967e117 < (*.f64 b b)

   1. Initial program 62.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 48.3

         \[\leadsto \color{blue}{{b}^{9}} \]

   5. Simplified 48.3%

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\color{blue}{\left(3 \cdot 3\right)}} \]

      2. pow-pow N/A

         \[\leadsto \color{blue}{{\left({b}^{3}\right)}^{3}} \]

      3. pow3 N/A

         \[\leadsto {\color{blue}{\left(\left(b \cdot b\right) \cdot b\right)}}^{3} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(\color{blue}{\left(b \cdot b\right)} \cdot b\right)}^{3} \]

      5. unpow-prod-down N/A

         \[\leadsto \color{blue}{{\left(b \cdot b\right)}^{3} \cdot {b}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(b \cdot b\right)}^{3} \cdot {b}^{3}} \]

      7. lift-*.f64 N/A

         \[\leadsto {\color{blue}{\left(b \cdot b\right)}}^{3} \cdot {b}^{3} \]

      8. unpow-prod-down N/A

         \[\leadsto \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)} \cdot {b}^{3} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)} \cdot {b}^{3} \]

      10. cube-unmult N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{3}\right) \cdot {b}^{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {b}^{3}\right) \cdot {b}^{3} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{3}\right) \cdot {b}^{3} \]

      13. cube-unmult N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right) \cdot {b}^{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \cdot {b}^{3} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right) \cdot {b}^{3} \]

      16. cube-unmult N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 48.3

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

   7. Applied egg-rr 48.3%

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

   8. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 48.3

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

   10. Simplified 48.3%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+118}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+118)
   (fma (* a a) (* a a) (fma (* b b) 12.0 -1.0))
   (* (* b b) (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+118) {
		tmp = fma((a * a), (a * a), fma((b * b), 12.0, -1.0));
	} else {
		tmp = (b * b) * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+118)
		tmp = fma(Float64(a * a), Float64(a * a), fma(Float64(b * b), 12.0, -1.0));
	else
		tmp = Float64(Float64(b * b) * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+118], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999967e117

   1. Initial program 86.0%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 78.1

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

   5. Simplified 78.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 88.9

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

   8. Simplified 88.9%

      \[\leadsto \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right) + 4 \cdot \color{blue}{\left(3 \cdot \left(b \cdot b\right)\right)}\right) - 1 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

   10. Applied egg-rr 88.9%

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

   if 9.99999999999999967e117 < (*.f64 b b)

   1. Initial program 62.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 48.3

         \[\leadsto \color{blue}{{b}^{9}} \]

   5. Simplified 48.3%

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\color{blue}{\left(3 \cdot 3\right)}} \]

      2. pow-pow N/A

         \[\leadsto \color{blue}{{\left({b}^{3}\right)}^{3}} \]

      3. pow3 N/A

         \[\leadsto {\color{blue}{\left(\left(b \cdot b\right) \cdot b\right)}}^{3} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(\color{blue}{\left(b \cdot b\right)} \cdot b\right)}^{3} \]

      5. unpow-prod-down N/A

         \[\leadsto \color{blue}{{\left(b \cdot b\right)}^{3} \cdot {b}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(b \cdot b\right)}^{3} \cdot {b}^{3}} \]

      7. lift-*.f64 N/A

         \[\leadsto {\color{blue}{\left(b \cdot b\right)}}^{3} \cdot {b}^{3} \]

      8. unpow-prod-down N/A

         \[\leadsto \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)} \cdot {b}^{3} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)} \cdot {b}^{3} \]

      10. cube-unmult N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{3}\right) \cdot {b}^{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {b}^{3}\right) \cdot {b}^{3} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{3}\right) \cdot {b}^{3} \]

      13. cube-unmult N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right) \cdot {b}^{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \cdot {b}^{3} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right) \cdot {b}^{3} \]

      16. cube-unmult N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 48.3

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

   7. Applied egg-rr 48.3%

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

   8. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 48.3

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

   10. Simplified 48.3%

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

Alternative 9: 72.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+118}:\\ \;\;\;\;\left(b \cdot b + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+118)
   (+ (+ (* b b) (* a (* a (* a a)))) -1.0)
   (* (* b b) (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+118) {
		tmp = ((b * b) + (a * (a * (a * a)))) + -1.0;
	} else {
		tmp = (b * 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) <= 1d+118) then
        tmp = ((b * b) + (a * (a * (a * a)))) + (-1.0d0)
    else
        tmp = (b * b) * (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+118) {
		tmp = ((b * b) + (a * (a * (a * a)))) + -1.0;
	} else {
		tmp = (b * b) * (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+118:
		tmp = ((b * b) + (a * (a * (a * a)))) + -1.0
	else:
		tmp = (b * b) * (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+118)
		tmp = Float64(Float64(Float64(b * b) + Float64(a * Float64(a * Float64(a * a)))) + -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+118)
		tmp = ((b * b) + (a * (a * (a * a)))) + -1.0;
	else
		tmp = (b * b) * (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+118], N[(N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999967e117

   1. Initial program 86.0%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 78.1

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

   5. Simplified 78.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 88.9

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

   8. Simplified 88.9%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 88.8

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

   11. Simplified 88.8%

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

   if 9.99999999999999967e117 < (*.f64 b b)

   1. Initial program 62.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 48.3

         \[\leadsto \color{blue}{{b}^{9}} \]

   5. Simplified 48.3%

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\color{blue}{\left(3 \cdot 3\right)}} \]

      2. pow-pow N/A

         \[\leadsto \color{blue}{{\left({b}^{3}\right)}^{3}} \]

      3. pow3 N/A

         \[\leadsto {\color{blue}{\left(\left(b \cdot b\right) \cdot b\right)}}^{3} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(\color{blue}{\left(b \cdot b\right)} \cdot b\right)}^{3} \]

      5. unpow-prod-down N/A

         \[\leadsto \color{blue}{{\left(b \cdot b\right)}^{3} \cdot {b}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(b \cdot b\right)}^{3} \cdot {b}^{3}} \]

      7. lift-*.f64 N/A

         \[\leadsto {\color{blue}{\left(b \cdot b\right)}}^{3} \cdot {b}^{3} \]

      8. unpow-prod-down N/A

         \[\leadsto \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)} \cdot {b}^{3} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)} \cdot {b}^{3} \]

      10. cube-unmult N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{3}\right) \cdot {b}^{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {b}^{3}\right) \cdot {b}^{3} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{3}\right) \cdot {b}^{3} \]

      13. cube-unmult N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right) \cdot {b}^{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \cdot {b}^{3} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right) \cdot {b}^{3} \]

      16. cube-unmult N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 48.3

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

   7. Applied egg-rr 48.3%

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

   8. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 48.3

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

   10. Simplified 48.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+118}:\\ \;\;\;\;\left(b \cdot b + a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+118) (fma (* a a) (* a a) -1.0) (* (* b b) (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+118) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = (b * b) * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+118)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+118], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999967e117

   1. Initial program 86.0%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 78.1

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

   5. Simplified 78.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 88.9

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

   8. Simplified 88.9%

      \[\leadsto \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right) + 4 \cdot \color{blue}{\left(3 \cdot \left(b \cdot b\right)\right)}\right) - 1 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

   10. Applied egg-rr 88.9%

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

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \color{blue}{-1}\right) \]
   12. Step-by-step derivation

   13. Simplified 88.3%

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

   if 9.99999999999999967e117 < (*.f64 b b)

   1. Initial program 62.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 48.3

         \[\leadsto \color{blue}{{b}^{9}} \]

   5. Simplified 48.3%

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

      1. metadata-eval N/A

         \[\leadsto {b}^{\color{blue}{\left(3 \cdot 3\right)}} \]

      2. pow-pow N/A

         \[\leadsto \color{blue}{{\left({b}^{3}\right)}^{3}} \]

      3. pow3 N/A

         \[\leadsto {\color{blue}{\left(\left(b \cdot b\right) \cdot b\right)}}^{3} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(\color{blue}{\left(b \cdot b\right)} \cdot b\right)}^{3} \]

      5. unpow-prod-down N/A

         \[\leadsto \color{blue}{{\left(b \cdot b\right)}^{3} \cdot {b}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(b \cdot b\right)}^{3} \cdot {b}^{3}} \]

      7. lift-*.f64 N/A

         \[\leadsto {\color{blue}{\left(b \cdot b\right)}}^{3} \cdot {b}^{3} \]

      8. unpow-prod-down N/A

         \[\leadsto \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)} \cdot {b}^{3} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)} \cdot {b}^{3} \]

      10. cube-unmult N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{3}\right) \cdot {b}^{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {b}^{3}\right) \cdot {b}^{3} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{3}\right) \cdot {b}^{3} \]

      13. cube-unmult N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right) \cdot {b}^{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \cdot {b}^{3} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right) \cdot {b}^{3} \]

      16. cube-unmult N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 48.3

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

   7. Applied egg-rr 48.3%

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

   8. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 48.3

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

   10. Simplified 48.3%

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

Alternative 11: 71.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(b \cdot b\right)\right) \cdot 16\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 3e+193) (fma (* a a) (* a a) -1.0) (* (* b (* b b)) 16.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3e+193) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = (b * (b * b)) * 16.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 3e+193)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(b * Float64(b * b)) * 16.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 3e+193], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * 16.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3e193

   1. Initial program 83.1%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 73.0

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

   5. Simplified 73.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 87.0

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

   8. Simplified 87.0%

      \[\leadsto \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right) + 4 \cdot \color{blue}{\left(3 \cdot \left(b \cdot b\right)\right)}\right) - 1 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

   10. Applied egg-rr 87.0%

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

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{fma}\left(a \cdot a, a \cdot a, \color{blue}{-1}\right) \]
   12. Step-by-step derivation

   13. Simplified 86.4%

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

   if 3e193 < (*.f64 b b)

   1. Initial program 64.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 64.0

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

   5. Simplified 64.0%

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

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 64.0

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

   8. Simplified 64.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{16 \cdot {b}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{b}^{3} \cdot 16} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{{b}^{3} \cdot 16} \]

       3. cube-mult N/A

          \[\leadsto \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot 16 \]

       4. unpow2 N/A

          \[\leadsto \left(b \cdot \color{blue}{{b}^{2}}\right) \cdot 16 \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(b \cdot {b}^{2}\right)} \cdot 16 \]

       6. unpow2 N/A

          \[\leadsto \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot 16 \]

       7. lower-*.f64 49.4

          \[\leadsto \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot 16 \]

   11. Simplified 49.4%

       \[\leadsto \color{blue}{\left(b \cdot \left(b \cdot b\right)\right) \cdot 16} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 52000:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot \left(b \cdot b\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 52000.0) (fma 12.0 (* b (* b b)) -1.0) (* 4.0 (* b (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 52000.0) {
		tmp = fma(12.0, (b * (b * b)), -1.0);
	} else {
		tmp = 4.0 * (b * (a * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 52000.0)
		tmp = fma(12.0, Float64(b * Float64(b * b)), -1.0);
	else
		tmp = Float64(4.0 * Float64(b * Float64(a * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 52000.0], N[(12.0 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(4.0 * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 52000

   1. Initial program 87.5%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 73.1

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

   5. Simplified 73.1%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{12 \cdot {b}^{3} - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{12 \cdot {b}^{3} + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 12 \cdot {b}^{3} + \color{blue}{-1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{3}, -1\right)} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 48.1

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

   8. Simplified 48.1%

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

   if 52000 < a

   1. Initial program 36.2%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 34.6

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

   5. Simplified 34.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 19.4

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

   8. Simplified 19.4%

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

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 37.2

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

   11. Simplified 37.2%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 52000:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot \left(b \cdot b\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 24.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 23000000:\\ \;\;\;\;\left(b \cdot \left(b \cdot b\right)\right) \cdot 16\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 23000000.0) (* (* b (* b b)) 16.0) (* 4.0 (* b (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 23000000.0) {
		tmp = (b * (b * b)) * 16.0;
	} else {
		tmp = 4.0 * (b * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 23000000.0d0) then
        tmp = (b * (b * b)) * 16.0d0
    else
        tmp = 4.0d0 * (b * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 23000000.0) {
		tmp = (b * (b * b)) * 16.0;
	} else {
		tmp = 4.0 * (b * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 23000000.0:
		tmp = (b * (b * b)) * 16.0
	else:
		tmp = 4.0 * (b * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 23000000.0)
		tmp = Float64(Float64(b * Float64(b * b)) * 16.0);
	else
		tmp = Float64(4.0 * Float64(b * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 23000000.0)
		tmp = (b * (b * b)) * 16.0;
	else
		tmp = 4.0 * (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 23000000.0], N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * 16.0), $MachinePrecision], N[(4.0 * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.3e7

   1. Initial program 87.5%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.0

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

   5. Simplified 80.0%

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

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.0

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

   8. Simplified 80.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{16 \cdot {b}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{b}^{3} \cdot 16} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{{b}^{3} \cdot 16} \]

       3. cube-mult N/A

          \[\leadsto \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot 16 \]

       4. unpow2 N/A

          \[\leadsto \left(b \cdot \color{blue}{{b}^{2}}\right) \cdot 16 \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(b \cdot {b}^{2}\right)} \cdot 16 \]

       6. unpow2 N/A

          \[\leadsto \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot 16 \]

       7. lower-*.f64 20.3

          \[\leadsto \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot 16 \]

   11. Simplified 20.3%

       \[\leadsto \color{blue}{\left(b \cdot \left(b \cdot b\right)\right) \cdot 16} \]

   if 2.3e7 < a

   1. Initial program 36.2%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 34.6

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

   5. Simplified 34.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 19.4

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

   8. Simplified 19.4%

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

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 37.2

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

   11. Simplified 37.2%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 23000000:\\ \;\;\;\;\left(b \cdot \left(b \cdot b\right)\right) \cdot 16\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 24.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 23000000:\\ \;\;\;\;12 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 23000000.0) (* 12.0 (* b (* b b))) (* 4.0 (* b (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 23000000.0) {
		tmp = 12.0 * (b * (b * b));
	} else {
		tmp = 4.0 * (b * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 23000000.0d0) then
        tmp = 12.0d0 * (b * (b * b))
    else
        tmp = 4.0d0 * (b * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 23000000.0) {
		tmp = 12.0 * (b * (b * b));
	} else {
		tmp = 4.0 * (b * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 23000000.0:
		tmp = 12.0 * (b * (b * b))
	else:
		tmp = 4.0 * (b * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 23000000.0)
		tmp = Float64(12.0 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(4.0 * Float64(b * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 23000000.0)
		tmp = 12.0 * (b * (b * b));
	else
		tmp = 4.0 * (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 23000000.0], N[(12.0 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.3e7

   1. Initial program 87.5%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 73.1

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

   5. Simplified 73.1%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 20.8

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

   8. Simplified 20.8%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{12 \cdot {b}^{3}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{12 \cdot {b}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 20.3

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

   11. Simplified 20.3%

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

   if 2.3e7 < a

   1. Initial program 36.2%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 34.6

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

   5. Simplified 34.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 19.4

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

   8. Simplified 19.4%

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

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 37.2

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

   11. Simplified 37.2%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 23000000:\\ \;\;\;\;12 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.7 \cdot 10^{-31}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 7.7e-31) (* b (* b 12.0)) (* 4.0 (* b (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 7.7e-31) {
		tmp = b * (b * 12.0);
	} else {
		tmp = 4.0 * (b * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 7.7d-31) then
        tmp = b * (b * 12.0d0)
    else
        tmp = 4.0d0 * (b * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 7.7e-31) {
		tmp = b * (b * 12.0);
	} else {
		tmp = 4.0 * (b * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 7.7e-31:
		tmp = b * (b * 12.0)
	else:
		tmp = 4.0 * (b * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 7.7e-31)
		tmp = Float64(b * Float64(b * 12.0));
	else
		tmp = Float64(4.0 * Float64(b * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 7.7e-31)
		tmp = b * (b * 12.0);
	else
		tmp = 4.0 * (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 7.7e-31], N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.70000000000000012e-31

   1. Initial program 87.1%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 72.4

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

   5. Simplified 72.4%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 20.7

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

   8. Simplified 20.7%

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

   9. Taylor expanded in a around 0

      \[\leadsto 4 \cdot \color{blue}{\left(3 \cdot {b}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 32.3

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

   11. Simplified 32.3%

       \[\leadsto 4 \cdot \color{blue}{\left(b \cdot \left(b \cdot 3\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*l* N/A

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

       9. metadata-eval N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 32.3

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

   13. Applied egg-rr 32.3%

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

   if 7.70000000000000012e-31 < a

   1. Initial program 41.5%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 40.0

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

   5. Simplified 40.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 19.7

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

   8. Simplified 19.7%

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

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 36.0

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

   11. Simplified 36.0%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.7 \cdot 10^{-31}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.9% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(b \cdot 12\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* b (* b 12.0)))

C

double code(double a, double b) {
	return b * (b * 12.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * (b * 12.0d0)
end function

Java

public static double code(double a, double b) {
	return b * (b * 12.0);
}

Python

def code(a, b):
	return b * (b * 12.0)

Julia

function code(a, b)
	return Float64(b * Float64(b * 12.0))
end

MATLAB

function tmp = code(a, b)
	tmp = b * (b * 12.0);
end

Wolfram

code[a_, b_] := N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

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

3. Taylor expanded in b around 0

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. cube-mult N/A

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

   7. lower-*.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 64.8

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

5. Simplified 64.8%

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

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{4 \cdot \left({b}^{3} \cdot \left(3 + a\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{4 \cdot \left({b}^{3} \cdot \left(3 + a\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto 4 \cdot \color{blue}{\left({b}^{3} \cdot \left(3 + a\right)\right)} \]

   3. cube-mult N/A

      \[\leadsto 4 \cdot \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot \left(3 + a\right)\right) \]

   4. unpow2 N/A

      \[\leadsto 4 \cdot \left(\left(b \cdot \color{blue}{{b}^{2}}\right) \cdot \left(3 + a\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto 4 \cdot \left(\color{blue}{\left(b \cdot {b}^{2}\right)} \cdot \left(3 + a\right)\right) \]

   6. unpow2 N/A

      \[\leadsto 4 \cdot \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \left(3 + a\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto 4 \cdot \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \left(3 + a\right)\right) \]

   8. lower-+.f64 20.5

      \[\leadsto 4 \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(3 + a\right)}\right) \]

8. Simplified 20.5%

   \[\leadsto \color{blue}{4 \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(3 + a\right)\right)} \]

9. Taylor expanded in a around 0

   \[\leadsto 4 \cdot \color{blue}{\left(3 \cdot {b}^{2}\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto 4 \cdot \color{blue}{\left({b}^{2} \cdot 3\right)} \]

    2. unpow2 N/A

       \[\leadsto 4 \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot 3\right) \]

    3. associate-*l* N/A

       \[\leadsto 4 \cdot \color{blue}{\left(b \cdot \left(b \cdot 3\right)\right)} \]

    4. *-commutative N/A

       \[\leadsto 4 \cdot \left(b \cdot \color{blue}{\left(3 \cdot b\right)}\right) \]

    5. lower-*.f64 N/A

       \[\leadsto 4 \cdot \color{blue}{\left(b \cdot \left(3 \cdot b\right)\right)} \]

    6. *-commutative N/A

       \[\leadsto 4 \cdot \left(b \cdot \color{blue}{\left(b \cdot 3\right)}\right) \]

    7. lower-*.f64 29.7

       \[\leadsto 4 \cdot \left(b \cdot \color{blue}{\left(b \cdot 3\right)}\right) \]

11. Simplified 29.7%

    \[\leadsto 4 \cdot \color{blue}{\left(b \cdot \left(b \cdot 3\right)\right)} \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto 4 \cdot \left(b \cdot \color{blue}{\left(b \cdot 3\right)}\right) \]

    2. lift-*.f64 N/A

       \[\leadsto 4 \cdot \color{blue}{\left(b \cdot \left(b \cdot 3\right)\right)} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{\left(b \cdot \left(b \cdot 3\right)\right) \cdot 4} \]

    4. lift-*.f64 N/A

       \[\leadsto \color{blue}{\left(b \cdot \left(b \cdot 3\right)\right)} \cdot 4 \]

    5. lift-*.f64 N/A

       \[\leadsto \left(b \cdot \color{blue}{\left(b \cdot 3\right)}\right) \cdot 4 \]

    6. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(b \cdot b\right) \cdot 3\right)} \cdot 4 \]

    7. lift-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(b \cdot b\right)} \cdot 3\right) \cdot 4 \]

    8. associate-*l* N/A

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(3 \cdot 4\right)} \]

    9. metadata-eval N/A

       \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{12} \]

    10. lift-*.f64 N/A

        \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot 12 \]

    11. associate-*l* N/A

        \[\leadsto \color{blue}{b \cdot \left(b \cdot 12\right)} \]

    12. *-commutative N/A

        \[\leadsto \color{blue}{\left(b \cdot 12\right) \cdot b} \]

    13. lower-*.f64 N/A

        \[\leadsto \color{blue}{\left(b \cdot 12\right) \cdot b} \]

    14. lower-*.f64 29.7

        \[\leadsto \color{blue}{\left(b \cdot 12\right)} \cdot b \]

13. Applied egg-rr 29.7%

    \[\leadsto \color{blue}{\left(b \cdot 12\right) \cdot b} \]

14. Final simplification 29.7%

    \[\leadsto b \cdot \left(b \cdot 12\right) \]
15. Add Preprocessing

Alternative 17: 27.8% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* b b))

C

double code(double a, double b) {
	return b * b;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * b
end function

Java

public static double code(double a, double b) {
	return b * b;
}

Python

def code(a, b):
	return b * b

Julia

function code(a, b)
	return Float64(b * b)
end

MATLAB

function tmp = code(a, b)
	tmp = b * b;
end

Wolfram

code[a_, b_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{{b}^{9}} \]
4. Step-by-step derivation

   1. lower-pow.f64 21.5

      \[\leadsto \color{blue}{{b}^{9}} \]

5. Simplified 21.5%

   \[\leadsto \color{blue}{{b}^{9}} \]

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{{b}^{2}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{b \cdot b} \]

   2. lower-*.f64 29.7

      \[\leadsto \color{blue}{b \cdot b} \]

8. Simplified 29.7%

   \[\leadsto \color{blue}{b \cdot b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))