Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.5% → 99.9%

Time: 10.3s

Alternatives: 12

Speedup: 5.5×

• 61.3% 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: 73.5% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot a\right)}\right)}^{-1}\\ t_1 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ t_2 := 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\\ \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + t\_2 \leq 4 \cdot 10^{+278}:\\ \;\;\;\;\left(t\_2 + \frac{t\_1}{\frac{1}{t\_1}}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{t\_1}{t\_0 \cdot t\_0} + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (pow (sqrt (fma b b (* a a))) -1.0))
        (t_1 (fma a a (* b b)))
        (t_2 (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0))))))
   (if (<= (+ (pow (+ (* a a) (* b b)) 2.0) t_2) 4e+278)
     (+ (+ t_2 (/ t_1 (/ 1.0 t_1))) -1.0)
     (+ -1.0 (+ (/ t_1 (* t_0 t_0)) (* 4.0 (* (* b b) 3.0)))))))

C

double code(double a, double b) {
	double t_0 = pow(sqrt(fma(b, b, (a * a))), -1.0);
	double t_1 = fma(a, a, (b * b));
	double t_2 = 4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)));
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + t_2) <= 4e+278) {
		tmp = (t_2 + (t_1 / (1.0 / t_1))) + -1.0;
	} else {
		tmp = -1.0 + ((t_1 / (t_0 * t_0)) + (4.0 * ((b * b) * 3.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = sqrt(fma(b, b, Float64(a * a))) ^ -1.0
	t_1 = fma(a, a, Float64(b * b))
	t_2 = Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0))))
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + t_2) <= 4e+278)
		tmp = Float64(Float64(t_2 + Float64(t_1 / Float64(1.0 / t_1))) + -1.0);
	else
		tmp = Float64(-1.0 + Float64(Float64(t_1 / Float64(t_0 * t_0)) + Float64(4.0 * Float64(Float64(b * b) * 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[Power[N[Sqrt[N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision], 4e+278], N[(N[(t$95$2 + N[(t$95$1 / N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(N[(t$95$1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(b * b), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $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))))) < 3.99999999999999985e278

   1. Initial program 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. clear-num N/A

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

   4. Applied rewrites 99.8%

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

   if 3.99999999999999985e278 < (+.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 53.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. clear-num N/A

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. unpow1 N/A

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

      4. sqr-pow N/A

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

      5. unpow-prod-down N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. unpow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-pow.f64 N/A

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

   9. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq 4 \cdot 10^{+278}:\\ \;\;\;\;\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + \frac{\mathsf{fma}\left(a, a, b \cdot b\right)}{\frac{1}{\mathsf{fma}\left(a, a, b \cdot b\right)}}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\mathsf{fma}\left(a, a, b \cdot b\right)}{{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot a\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot a\right)}\right)}^{-1}} + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ t_1 := \frac{t\_0}{\frac{1}{t\_0}}\\ t_2 := 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\\ \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + t\_2 \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\left(t\_2 + t\_1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(t\_1 + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b)))
        (t_1 (/ t_0 (/ 1.0 t_0)))
        (t_2 (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0))))))
   (if (<= (+ (pow (+ (* a a) (* b b)) 2.0) t_2) 2e+62)
     (+ (+ t_2 t_1) -1.0)
     (+ -1.0 (+ t_1 (* 4.0 (* (* b b) 3.0)))))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double t_1 = t_0 / (1.0 / t_0);
	double t_2 = 4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)));
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + t_2) <= 2e+62) {
		tmp = (t_2 + t_1) + -1.0;
	} else {
		tmp = -1.0 + (t_1 + (4.0 * ((b * b) * 3.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	t_1 = Float64(t_0 / Float64(1.0 / t_0))
	t_2 = Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0))))
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + t_2) <= 2e+62)
		tmp = Float64(Float64(t_2 + t_1) + -1.0);
	else
		tmp = Float64(-1.0 + Float64(t_1 + Float64(4.0 * Float64(Float64(b * b) * 3.0))));
	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[(t$95$0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision], 2e+62], N[(N[(t$95$2 + t$95$1), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(t$95$1 + N[(4.0 * N[(N[(b * b), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $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 99.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. clear-num N/A

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

   4. Applied rewrites 99.9%

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

   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 63.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. clear-num N/A

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.9%

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

Reproduce

herbie shell --seed 2024228 
(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))