Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.2% → 99.9%

Time: 8.0s

Alternatives: 11

Speedup: 6.0×

• 61.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 73.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(3.0 + a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 + a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 * t$95$0 + N[(N[(N[(N[(1.0 - a), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(3.0 + a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 - N[(N[(4.0 - N[(N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{+48}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right) \cdot 4 + {\left(b \cdot b + a \cdot a\right)}^{2}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} \cdot \left(1 - \frac{4 - \frac{\mathsf{fma}\left(b \cdot b, 2, 4\right)}{a}}{a}\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.25e+48)
   (- (+ (* (* (fma (- a) a a) a) 4.0) (pow (+ (* b b) (* a a)) 2.0)) 1.0)
   (- (* (pow a 4.0) (- 1.0 (/ (- 4.0 (/ (fma (* b b) 2.0 4.0) a)) a))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.25e+48) {
		tmp = (((fma(-a, a, a) * a) * 4.0) + pow(((b * b) + (a * a)), 2.0)) - 1.0;
	} else {
		tmp = (pow(a, 4.0) * (1.0 - ((4.0 - (fma((b * b), 2.0, 4.0) / a)) / a))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.25e+48)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(-a), a, a) * a) * 4.0) + (Float64(Float64(b * b) + Float64(a * a)) ^ 2.0)) - 1.0);
	else
		tmp = Float64(Float64((a ^ 4.0) * Float64(1.0 - Float64(Float64(4.0 - Float64(fma(Float64(b * b), 2.0, 4.0) / a)) / a))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.25e+48], N[(N[(N[(N[(N[((-a) * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 - N[(N[(4.0 - N[(N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.24999999999999993e48

   1. Initial program 87.2%

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

   3. Taylor expanded in a around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. cancel-sign-sub N/A

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

      6. distribute-rgt-out-- N/A

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

      7. unsub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. sub-neg N/A

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

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

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-commutative N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft1-in N/A

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

   5. Applied rewrites 99.1%

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

   if 1.24999999999999993e48 < a

   1. Initial program 21.0%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{+48}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right) \cdot 4 + {\left(b \cdot b + a \cdot a\right)}^{2}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} \cdot \left(1 - \frac{4 - \frac{\mathsf{fma}\left(b \cdot b, 2, 4\right)}{a}}{a}\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right) \cdot 4 + {\left(b \cdot b + a \cdot a\right)}^{2}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2e+69)
   (- (+ (* (* (fma (- a) a a) a) 4.0) (pow (+ (* b b) (* a a)) 2.0)) 1.0)
   (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2e+69) {
		tmp = (((fma(-a, a, a) * a) * 4.0) + pow(((b * b) + (a * a)), 2.0)) - 1.0;
	} else {
		tmp = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2e+69)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(-a), a, a) * a) * 4.0) + (Float64(Float64(b * b) + Float64(a * a)) ^ 2.0)) - 1.0);
	else
		tmp = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2e+69], N[(N[(N[(N[(N[((-a) * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.0000000000000001e69

   1. Initial program 87.7%

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

   3. Taylor expanded in a around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. cancel-sign-sub N/A

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

      6. distribute-rgt-out-- N/A

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

      7. unsub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. sub-neg N/A

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

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

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-commutative N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft1-in N/A

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

   5. Applied rewrites 99.1%

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

   if 2.0000000000000001e69 < a

   1. Initial program 10.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 44.0%

      \[\leadsto \mathsf{fma}\left(\left(1 - a\right) \cdot 4, a \cdot a, \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b\right) \cdot b\right)\right) - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 84.0%

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

   10. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

   12. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right) \cdot 4 + {\left(b \cdot b + a \cdot a\right)}^{2}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000305:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, \mathsf{fma}\left(\left(\left(1 - a\right) \cdot 4\right) \cdot a, a, \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b\right) \cdot b\right)\right) - 1\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.000305)
   (-
    (fma
     (* a a)
     (* a a)
     (fma (* (* (- 1.0 a) 4.0) a) a (* (* (fma (fma 2.0 a 4.0) a 12.0) b) b)))
    1.0)
   (if (<= a 2.7e-5)
     (- (fma (* b b) 12.0 (pow b 4.0)) 1.0)
     (fma (* a a) (fma (- a 4.0) a 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.000305) {
		tmp = fma((a * a), (a * a), fma((((1.0 - a) * 4.0) * a), a, ((fma(fma(2.0, a, 4.0), a, 12.0) * b) * b))) - 1.0;
	} else if (a <= 2.7e-5) {
		tmp = fma((b * b), 12.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.000305)
		tmp = Float64(fma(Float64(a * a), Float64(a * a), fma(Float64(Float64(Float64(1.0 - a) * 4.0) * a), a, Float64(Float64(fma(fma(2.0, a, 4.0), a, 12.0) * b) * b))) - 1.0);
	elseif (a <= 2.7e-5)
		tmp = Float64(fma(Float64(b * b), 12.0, (b ^ 4.0)) - 1.0);
	else
		tmp = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.000305], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(N[(1.0 - a), $MachinePrecision] * 4.0), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 2.7e-5], N[(N[(N[(b * b), $MachinePrecision] * 12.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.04999999999999987e-4

   1. Initial program 58.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 98.4%

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

   if -3.04999999999999987e-4 < a < 2.6999999999999999e-5

   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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 2.6999999999999999e-5 < a

   1. Initial program 36.3%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 58.9%

      \[\leadsto \mathsf{fma}\left(\left(1 - a\right) \cdot 4, a \cdot a, \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b\right) \cdot b\right)\right) - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 87.1%

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

   10. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

   12. Applied rewrites 93.2%

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

Alternative 6: 96.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000305:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, \mathsf{fma}\left(\left(\left(1 - a\right) \cdot 4\right) \cdot a, a, \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b\right) \cdot b\right)\right) - 1\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.000305)
   (-
    (fma
     (* a a)
     (* a a)
     (fma (* (* (- 1.0 a) 4.0) a) a (* (* (fma (fma 2.0 a 4.0) a 12.0) b) b)))
    1.0)
   (if (<= a 2.7e-5)
     (fma (* (fma b b 12.0) b) b -1.0)
     (fma (* a a) (fma (- a 4.0) a 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.000305) {
		tmp = fma((a * a), (a * a), fma((((1.0 - a) * 4.0) * a), a, ((fma(fma(2.0, a, 4.0), a, 12.0) * b) * b))) - 1.0;
	} else if (a <= 2.7e-5) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.000305)
		tmp = Float64(fma(Float64(a * a), Float64(a * a), fma(Float64(Float64(Float64(1.0 - a) * 4.0) * a), a, Float64(Float64(fma(fma(2.0, a, 4.0), a, 12.0) * b) * b))) - 1.0);
	elseif (a <= 2.7e-5)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.000305], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(N[(1.0 - a), $MachinePrecision] * 4.0), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 2.7e-5], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.04999999999999987e-4

   1. Initial program 58.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 98.4%

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

   if -3.04999999999999987e-4 < a < 2.6999999999999999e-5

   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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. cancel-sign-sub N/A

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

      6. distribute-rgt-out-- N/A

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

      7. unsub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. sub-neg N/A

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

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

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-commutative N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft1-in N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 2.6999999999999999e-5 < a

   1. Initial program 36.3%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 58.9%

      \[\leadsto \mathsf{fma}\left(\left(1 - a\right) \cdot 4, a \cdot a, \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b\right) \cdot b\right)\right) - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 87.1%

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

   10. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

   12. Applied rewrites 93.2%

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

Alternative 7: 96.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000355:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a + \left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right) \cdot 4\right) - 1\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.000355)
   (-
    (+ (* (* (fma (* b b) 2.0 (* a a)) a) a) (* (* (fma (- a) a a) a) 4.0))
    1.0)
   (if (<= a 2.7e-5)
     (fma (* (fma b b 12.0) b) b -1.0)
     (fma (* a a) (fma (- a 4.0) a 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.000355) {
		tmp = (((fma((b * b), 2.0, (a * a)) * a) * a) + ((fma(-a, a, a) * a) * 4.0)) - 1.0;
	} else if (a <= 2.7e-5) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.000355)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a) + Float64(Float64(fma(Float64(-a), a, a) * a) * 4.0)) - 1.0);
	elseif (a <= 2.7e-5)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.000355], N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[((-a) * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 2.7e-5], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.5500000000000001e-4

   1. Initial program 58.9%

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

   3. Taylor expanded in a around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. cancel-sign-sub N/A

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

      6. distribute-rgt-out-- N/A

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

      7. unsub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. sub-neg N/A

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

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

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-commutative N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft1-in N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-lft-in N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

   if -3.5500000000000001e-4 < a < 2.6999999999999999e-5

   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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. cancel-sign-sub N/A

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

      6. distribute-rgt-out-- N/A

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

      7. unsub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. sub-neg N/A

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

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

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-commutative N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft1-in N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 2.6999999999999999e-5 < a

   1. Initial program 36.3%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 58.9%

      \[\leadsto \mathsf{fma}\left(\left(1 - a\right) \cdot 4, a \cdot a, \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b\right) \cdot b\right)\right) - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 87.1%

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

   10. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

   12. Applied rewrites 93.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000355:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a + \left(\mathsf{fma}\left(-a, a, a\right) \cdot a\right) \cdot 4\right) - 1\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a - 4, a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+46)
   (fma (* a a) (fma (- a 4.0) a 4.0) -1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+46) {
		tmp = fma((a * a), fma((a - 4.0), a, 4.0), -1.0);
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+46)
		tmp = fma(Float64(a * a), fma(Float64(a - 4.0), a, 4.0), -1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+46], N[(N[(a * a), $MachinePrecision] * N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000002e46

   1. Initial program 80.5%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 78.1%

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

   7. Applied rewrites 86.6%

      \[\leadsto \mathsf{fma}\left(\left(1 - a\right) \cdot 4, a \cdot a, \mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right) \cdot b\right) \cdot b\right)\right) - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

   12. Applied rewrites 97.6%

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

   if 5.0000000000000002e46 < (*.f64 b b)

   1. Initial program 62.8%

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

   3. Taylor expanded in a around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. cancel-sign-sub N/A

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

      6. distribute-rgt-out-- N/A

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

      7. unsub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. sub-neg N/A

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

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

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-commutative N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft1-in N/A

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 92.7

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

   8. Applied rewrites 92.7%

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

Alternative 9: 79.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+102}:\\ \;\;\;\;\left(\mathsf{fma}\left(-4, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.5e+102)
   (- (* (* (fma -4.0 a 4.0) a) a) 1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.5e+102) {
		tmp = ((fma(-4.0, a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.5e+102)
		tmp = Float64(Float64(Float64(fma(-4.0, a, 4.0) * a) * a) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.5e+102], N[(N[(N[(N[(-4.0 * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.50000000000000011e102

   1. Initial program 61.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 70.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 100.0%

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

   if -3.50000000000000011e102 < a

   1. Initial program 74.7%

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

   3. Taylor expanded in a around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. cancel-sign-sub N/A

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

      6. distribute-rgt-out-- N/A

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

      7. unsub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. sub-neg N/A

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

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

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-commutative N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft1-in N/A

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 73.7

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

   8. Applied rewrites 73.7%

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

Alternative 10: 68.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma (* (fma b b 12.0) b) b -1.0))

C

double code(double a, double b) {
	return fma((fma(b, b, 12.0) * b), b, -1.0);
}

Julia

function code(a, b)
	return fma(Float64(fma(b, b, 12.0) * b), b, -1.0)
end

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in a around inf

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. cancel-sign-sub N/A

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

   6. distribute-rgt-out-- N/A

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

   7. unsub-neg N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-in N/A

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

   10. sub-neg N/A

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

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

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

   12. unpow3 N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-commutative N/A

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

   17. sub-neg N/A

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

   18. +-commutative N/A

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

   19. distribute-lft1-in N/A

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

5. Applied rewrites 81.7%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 67.2

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

8. Applied rewrites 67.2%

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

Alternative 11: 50.7% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ 12 \cdot \left(b \cdot b\right) - 1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* 12.0 (* b b)) 1.0))

C

double code(double a, double b) {
	return (12.0 * (b * b)) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (12.0d0 * (b * b)) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (12.0 * (b * b)) - 1.0;
}

Python

def code(a, b):
	return (12.0 * (b * b)) - 1.0

Julia

function code(a, b)
	return Float64(Float64(12.0 * Float64(b * b)) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = (12.0 * (b * b)) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 73.5%

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

6. Taylor expanded in a around 0

   \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
7. Step-by-step derivation

8. Applied rewrites 49.1%

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

Reproduce

herbie shell --seed 2024288 
(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))