Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.6% → 100.0%

Time: 8.8s

Alternatives: 14

Speedup: 5.3×

• 60.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision], Infinity], N[(N[(N[(N[(t$95$0 * a), $MachinePrecision] * a + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(t$95$0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   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 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. count-2-rev N/A

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

       6. distribute-lft-in N/A

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

       7. count-2-rev N/A

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

       8. distribute-lft-in N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. count-2-rev N/A

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

       6. distribute-lft-in N/A

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

       7. count-2-rev N/A

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

       8. distribute-lft-in N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. rgt-mult-inverse N/A

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

      7. unpow2 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 98.6

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

   8. Applied rewrites 98.6%

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

   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 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. count-2-rev N/A

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

       6. distribute-lft-in N/A

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

       7. count-2-rev N/A

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

       8. distribute-lft-in N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

Alternative 4: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. rgt-mult-inverse N/A

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

      7. unpow2 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if +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 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. count-2-rev N/A

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

       6. distribute-lft-in N/A

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

       7. count-2-rev N/A

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

       8. distribute-lft-in N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

Alternative 5: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. rgt-mult-inverse N/A

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

      7. unpow2 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 98.5

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

   7. Applied rewrites 98.5%

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

   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 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in b around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. count-2-rev N/A

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

       6. distribute-lft-in N/A

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

       7. count-2-rev N/A

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

       8. distribute-lft-in N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

Alternative 6: 98.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, a, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), \mathsf{fma}\left(b, b, 4\right)\right), b \cdot b, \left(a \cdot a\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-20)
   (fma (* a a) (fma a a (fma 4.0 a 4.0)) -1.0)
   (-
    (fma (fma a (fma 2.0 a -12.0) (fma b b 4.0)) (* b b) (* (* a a) 4.0))
    1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-20) {
		tmp = fma((a * a), fma(a, a, fma(4.0, a, 4.0)), -1.0);
	} else {
		tmp = fma(fma(a, fma(2.0, a, -12.0), fma(b, b, 4.0)), (b * b), ((a * a) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-20)
		tmp = fma(Float64(a * a), fma(a, a, fma(4.0, a, 4.0)), -1.0);
	else
		tmp = Float64(fma(fma(a, fma(2.0, a, -12.0), fma(b, b, 4.0)), Float64(b * b), Float64(Float64(a * a) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-20], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(4.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(a * N[(2.0 * a + -12.0), $MachinePrecision] + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.99999999999999989e-20

   1. Initial program 83.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft1-in N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 1.99999999999999989e-20 < (*.f64 b b)

   1. Initial program 72.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.3%

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

Alternative 7: 98.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, a, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, -12\right), a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, \left(a \cdot a\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-20)
   (fma (* a a) (fma a a (fma 4.0 a 4.0)) -1.0)
   (-
    (fma (* (fma (fma 2.0 a -12.0) a (fma b b 4.0)) b) b (* (* a a) 4.0))
    1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-20) {
		tmp = fma((a * a), fma(a, a, fma(4.0, a, 4.0)), -1.0);
	} else {
		tmp = fma((fma(fma(2.0, a, -12.0), a, fma(b, b, 4.0)) * b), b, ((a * a) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-20)
		tmp = fma(Float64(a * a), fma(a, a, fma(4.0, a, 4.0)), -1.0);
	else
		tmp = Float64(fma(Float64(fma(fma(2.0, a, -12.0), a, fma(b, b, 4.0)) * b), b, Float64(Float64(a * a) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-20], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(4.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * a + -12.0), $MachinePrecision] * a + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.99999999999999989e-20

   1. Initial program 83.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft1-in N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 1.99999999999999989e-20 < (*.f64 b b)

   1. Initial program 72.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 97.3%

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

Alternative 8: 98.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, a, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 2, b \cdot b\right) \cdot b, b, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-20)
   (fma (* a a) (fma a a (fma 4.0 a 4.0)) -1.0)
   (fma (* b b) 4.0 (fma (* (fma (* a a) 2.0 (* b b)) b) b -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-20) {
		tmp = fma((a * a), fma(a, a, fma(4.0, a, 4.0)), -1.0);
	} else {
		tmp = fma((b * b), 4.0, fma((fma((a * a), 2.0, (b * b)) * b), b, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-20)
		tmp = fma(Float64(a * a), fma(a, a, fma(4.0, a, 4.0)), -1.0);
	else
		tmp = fma(Float64(b * b), 4.0, fma(Float64(fma(Float64(a * a), 2.0, Float64(b * b)) * b), b, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-20], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(4.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0 + N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.99999999999999989e-20

   1. Initial program 83.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft1-in N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 1.99999999999999989e-20 < (*.f64 b b)

   1. Initial program 72.3%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.6

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

   8. Applied rewrites 66.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 57.1%

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

   11. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. pow-sqr N/A

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

       6. distribute-rgt-in N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. lower-fma.f64 N/A

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

   13. Applied rewrites 97.1%

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

Alternative 9: 94.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000115 \lor \neg \left(a \leq 12\right):\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, a, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.000115) (not (<= a 12.0)))
   (fma (* a a) (fma a a (fma 4.0 a 4.0)) -1.0)
   (fma (fma b b 4.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.000115) || !(a <= 12.0)) {
		tmp = fma((a * a), fma(a, a, fma(4.0, a, 4.0)), -1.0);
	} else {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -0.000115) || !(a <= 12.0))
		tmp = fma(Float64(a * a), fma(a, a, fma(4.0, a, 4.0)), -1.0);
	else
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -0.000115], N[Not[LessEqual[a, 12.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(4.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.15e-4 or 12 < a

   1. Initial program 52.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft1-in N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 91.9

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

   8. Applied rewrites 91.9%

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

   if -1.15e-4 < a < 12

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000115 \lor \neg \left(a \leq 12\right):\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, a, 4\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+153} \lor \neg \left(a \leq 1.85 \cdot 10^{+147}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.5e+153) (not (<= a 1.85e+147)))
   (- (* (* a a) 4.0) 1.0)
   (fma (fma b b 4.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.5e+153) || !(a <= 1.85e+147)) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.5e+153) || !(a <= 1.85e+147))
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.5e+153], N[Not[LessEqual[a, 1.85e+147]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4999999999999999e153 or 1.85e147 < a

   1. Initial program 37.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 95.5%

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

   if -3.4999999999999999e153 < a < 1.85e147

   1. Initial program 90.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.1

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

   8. Applied rewrites 82.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+153} \lor \neg \left(a \leq 1.85 \cdot 10^{+147}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+153} \lor \neg \left(a \leq 1.85 \cdot 10^{+147}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.5e+153) (not (<= a 1.85e+147)))
   (- (* (* a a) 4.0) 1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.5e+153) || !(a <= 1.85e+147)) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.5e+153) || !(a <= 1.85e+147))
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.5e+153], N[Not[LessEqual[a, 1.85e+147]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4999999999999999e153 or 1.85e147 < a

   1. Initial program 37.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 95.5%

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

   if -3.4999999999999999e153 < a < 1.85e147

   1. Initial program 90.2%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. rgt-mult-inverse N/A

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

      7. unpow2 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 94.5

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

       14. unpow2 N/A

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

       15. lower-fma.f64 82.1

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

   11. Applied rewrites 82.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+153} \lor \neg \left(a \leq 1.85 \cdot 10^{+147}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+306) (- (* (* a a) 4.0) 1.0) (fma 4.0 (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+306) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = fma(4.0, (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+306)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = fma(4.0, Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+306], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(4.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000007e306

   1. Initial program 82.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 60.8%

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

   if 4.00000000000000007e306 < (*.f64 b b)

   1. Initial program 60.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 100.0%

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

Alternative 13: 52.5% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(4, b \cdot b, -1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma 4.0 (* b b) -1.0))

C

double code(double a, double b) {
	return fma(4.0, (b * b), -1.0);
}

Julia

function code(a, b)
	return fma(4.0, Float64(b * b), -1.0)
end

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+r+ N/A

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

5. Applied rewrites 86.8%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 68.8

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

8. Applied rewrites 68.8%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 50.2%

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

Alternative 14: 25.4% accurate, 160.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+r+ N/A

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

5. Applied rewrites 86.8%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 68.8

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

8. Applied rewrites 68.8%

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

9. Taylor expanded in b around 0

   \[\leadsto -1 \]
10. Step-by-step derivation

11. Applied rewrites 30.8%

    \[\leadsto -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))