Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.7% → 99.9%

Time: 9.4s

Alternatives: 15

Speedup: 5.5×

• 60.4% 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: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

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 (* (* (* a a) 2.0) b) b (fma (fma (+ 4.0 a) a 4.0) (* 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((((a * a) * 2.0) * b), b, fma(fma((4.0 + a), a, 4.0), (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(Float64(Float64(a * a) * 2.0) * b), b, fma(fma(Float64(4.0 + a), a, 4.0), 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[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $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 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 38.6

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

   5. Applied rewrites 38.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

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

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 (* (* (* a a) 2.0) b) b (fma (fma (+ 4.0 a) a 4.0) (* 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((((a * a) * 2.0) * b), b, fma(fma((4.0 + a), a, 4.0), (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(Float64(Float64(a * a) * 2.0) * b), b, fma(fma(Float64(4.0 + a), a, 4.0), 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[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $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 99.5

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

   5. Applied rewrites 99.5%

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

   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 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 38.6

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

   5. Applied rewrites 38.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

Alternative 3: 51.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

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))))))
      5e-8)
   -1.0
   (* (* a a) 4.0)))

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)))))) <= 5e-8) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) <= 5d-8) then
        tmp = -1.0d0
    else
        tmp = (a * a) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) <= 5e-8) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) <= 5e-8:
		tmp = -1.0
	else:
		tmp = (a * a) * 4.0
	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)))))) <= 5e-8)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * a) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) <= 5e-8)
		tmp = -1.0;
	else
		tmp = (a * a) * 4.0;
	end
	tmp_2 = 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], 5e-8], -1.0, N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < 4.9999999999999998e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in 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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in a around 0

      \[\leadsto -1 \]
   10. Step-by-step derivation

   11. Applied rewrites 98.6%

       \[\leadsto -1 \]

   if 4.9999999999999998e-8 < (+.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 68.0%

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

   3. Taylor expanded in a around 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 80.0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in b around 0

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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 61.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 33.2%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 33.7%

       \[\leadsto \left(a \cdot a\right) \cdot 4 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.1 \cdot 10^{-7} \lor \neg \left(a \leq 2.45 \cdot 10^{-26}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right), a \cdot a, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, {b}^{4}\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -8.1e-7) (not (<= a 2.45e-26)))
   (fma (* (* (* a a) 2.0) b) b (fma (fma (+ 4.0 a) a 4.0) (* a a) -1.0))
   (- (fma (* b b) 4.0 (pow b 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -8.1e-7) || !(a <= 2.45e-26)) {
		tmp = fma((((a * a) * 2.0) * b), b, fma(fma((4.0 + a), a, 4.0), (a * a), -1.0));
	} else {
		tmp = fma((b * b), 4.0, pow(b, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -8.1e-7) || !(a <= 2.45e-26))
		tmp = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, fma(fma(Float64(4.0 + a), a, 4.0), Float64(a * a), -1.0));
	else
		tmp = Float64(fma(Float64(b * b), 4.0, (b ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -8.1e-7], N[Not[LessEqual[a, 2.45e-26]], $MachinePrecision]], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.09999999999999974e-7 or 2.45e-26 < a

   1. Initial program 55.3%

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

   3. Taylor expanded in a around 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 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.4%

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

   if -8.09999999999999974e-7 < a < 2.45e-26

   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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.1 \cdot 10^{-7} \lor \neg \left(a \leq 2.45 \cdot 10^{-26}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right), a \cdot a, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, {b}^{4}\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.1 \cdot 10^{-7} \lor \neg \left(a \leq 2.45 \cdot 10^{-26}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right), a \cdot a, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -8.1e-7) (not (<= a 2.45e-26)))
   (fma (* (* (* a a) 2.0) b) b (fma (fma (+ 4.0 a) a 4.0) (* a a) -1.0))
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -8.1e-7) || !(a <= 2.45e-26)) {
		tmp = fma((((a * a) * 2.0) * b), b, fma(fma((4.0 + a), a, 4.0), (a * a), -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 <= -8.1e-7) || !(a <= 2.45e-26))
		tmp = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, fma(fma(Float64(4.0 + a), a, 4.0), Float64(a * a), -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, -8.1e-7], N[Not[LessEqual[a, 2.45e-26]], $MachinePrecision]], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $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 < -8.09999999999999974e-7 or 2.45e-26 < a

   1. Initial program 55.3%

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

   3. Taylor expanded in a around 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 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.4%

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

   if -8.09999999999999974e-7 < a < 2.45e-26

   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 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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 75.9%

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

   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. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 99.9

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.1 \cdot 10^{-7} \lor \neg \left(a \leq 2.45 \cdot 10^{-26}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, \mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right), a \cdot a, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 + a\right) \cdot a\\ \mathbf{if}\;a \leq -4500000000000:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 700000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (+ 4.0 a) a)))
   (if (<= a -4500000000000.0)
     (fma (* t_0 a) a -1.0)
     (if (<= a 700000.0)
       (fma (* (fma b b 4.0) b) b -1.0)
       (fma t_0 (* a a) -1.0)))))

C

double code(double a, double b) {
	double t_0 = (4.0 + a) * a;
	double tmp;
	if (a <= -4500000000000.0) {
		tmp = fma((t_0 * a), a, -1.0);
	} else if (a <= 700000.0) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma(t_0, (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(4.0 + a) * a)
	tmp = 0.0
	if (a <= -4500000000000.0)
		tmp = fma(Float64(t_0 * a), a, -1.0);
	elseif (a <= 700000.0)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(t_0, Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -4500000000000.0], N[(N[(t$95$0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 700000.0], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(t$95$0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.5e12

   1. Initial program 37.3%

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

   3. Taylor expanded in a around 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 37.3

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in b around 0

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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 89.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 89.9%

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

   13. Applied rewrites 90.0%

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

   if -4.5e12 < a < 7e5

   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 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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 77.7%

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

   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. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 99.2

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

   11. Applied rewrites 99.2%

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

   if 7e5 < a

   1. Initial program 64.4%

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

   3. Taylor expanded in a around 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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 96.1%

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

Alternative 7: 94.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4500000000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{elif}\;a \leq 700000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(4 + a\right) \cdot a, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4500000000000.0)
   (- (* (* a a) (* a a)) 1.0)
   (if (<= a 700000.0)
     (fma (* (fma b b 4.0) b) b -1.0)
     (fma (* (+ 4.0 a) a) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4500000000000.0) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else if (a <= 700000.0) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma(((4.0 + a) * a), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4500000000000.0)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	elseif (a <= 700000.0)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(Float64(Float64(4.0 + a) * a), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4500000000000.0], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 700000.0], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.5e12

   1. Initial program 37.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 90.1

         \[\leadsto \color{blue}{{a}^{4}} - 1 \]

   5. Applied rewrites 90.1%

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

   7. Applied rewrites 89.9%

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

   if -4.5e12 < a < 7e5

   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 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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 77.7%

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

   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. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 99.2

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

   11. Applied rewrites 99.2%

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

   if 7e5 < a

   1. Initial program 64.4%

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

   3. Taylor expanded in a around 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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 96.1%

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

Alternative 8: 93.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+118)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (fma (fma b b 4.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+118) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -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 (Float64(b * b) <= 2e+118)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+118], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -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 (*.f64 b b) < 1.99999999999999993e118

   1. Initial program 83.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 85.1

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

   5. Applied rewrites 85.1%

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

   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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 93.1%

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

   10. Applied rewrites 93.2%

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

   if 1.99999999999999993e118 < (*.f64 b b)

   1. Initial program 66.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   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. metadata-eval N/A

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

      6. *-commutative N/A

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

      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 98.9

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

   8. Applied rewrites 98.9%

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

Alternative 9: 94.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4500000000000 \lor \neg \left(a \leq 4100000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4500000000000.0) (not (<= a 4100000.0)))
   (- (* (* a a) (* a a)) 1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -4500000000000.0) || !(a <= 4100000.0)) {
		tmp = ((a * a) * (a * a)) - 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 <= -4500000000000.0) || !(a <= 4100000.0))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 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, -4500000000000.0], N[Not[LessEqual[a, 4100000.0]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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 < -4.5e12 or 4.1e6 < a

   1. Initial program 51.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 92.4

         \[\leadsto \color{blue}{{a}^{4}} - 1 \]

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.3%

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

   if -4.5e12 < a < 4.1e6

   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 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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 77.7%

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

   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. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 99.2

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

   11. Applied rewrites 99.2%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4500000000000 \lor \neg \left(a \leq 4100000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+138}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.5e+138)
   (* (* a a) 4.0)
   (if (<= a 3.6e+89)
     (fma (* (fma b b 4.0) b) b -1.0)
     (fma (* 4.0 a) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+138) {
		tmp = (a * a) * 4.0;
	} else if (a <= 3.6e+89) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma((4.0 * a), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.5e+138)
		tmp = Float64(Float64(a * a) * 4.0);
	elseif (a <= 3.6e+89)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(Float64(4.0 * a), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.5e+138], N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[a, 3.6e+89], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(4.0 * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.50000000000000054e138

   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 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 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in b around 0

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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 93.1%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 93.1%

       \[\leadsto \left(a \cdot a\right) \cdot 4 \]

   if -6.50000000000000054e138 < a < 3.6e89

   1. Initial program 92.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 82.3%

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

   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. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 82.9

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

   11. Applied rewrites 82.9%

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

   if 3.6e89 < a

   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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 95.9%

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

Alternative 11: 86.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+138}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.5e+138)
   (* (* a a) 4.0)
   (if (<= a 3.6e+89)
     (fma (* (* b b) b) b -1.0)
     (fma (* 4.0 a) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+138) {
		tmp = (a * a) * 4.0;
	} else if (a <= 3.6e+89) {
		tmp = fma(((b * b) * b), b, -1.0);
	} else {
		tmp = fma((4.0 * a), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.5e+138)
		tmp = Float64(Float64(a * a) * 4.0);
	elseif (a <= 3.6e+89)
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	else
		tmp = fma(Float64(4.0 * a), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.5e+138], N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[a, 3.6e+89], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(4.0 * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.50000000000000054e138

   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 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 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in b around 0

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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 93.1%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 93.1%

       \[\leadsto \left(a \cdot a\right) \cdot 4 \]

   if -6.50000000000000054e138 < a < 3.6e89

   1. Initial program 92.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 82.3%

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

   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. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 82.9

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

   11. Applied rewrites 82.9%

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

   12. Taylor expanded in b around inf

       \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 82.5%

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

   if 3.6e89 < a

   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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 95.9%

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

Alternative 12: 72.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+138}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.5e+138)
   (* (* a a) 4.0)
   (if (<= a 1.2e+89) (fma (* b b) 4.0 -1.0) (fma (* 4.0 a) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+138) {
		tmp = (a * a) * 4.0;
	} else if (a <= 1.2e+89) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = fma((4.0 * a), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.5e+138)
		tmp = Float64(Float64(a * a) * 4.0);
	elseif (a <= 1.2e+89)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = fma(Float64(4.0 * a), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.5e+138], N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[a, 1.2e+89], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(4.0 * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.50000000000000054e138

   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 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 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in b around 0

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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 93.1%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 93.1%

       \[\leadsto \left(a \cdot a\right) \cdot 4 \]

   if -6.50000000000000054e138 < a < 1.20000000000000002e89

   1. Initial program 93.3%

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

   3. Taylor expanded in a around 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 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 82.2%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 63.6%

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

   if 1.20000000000000002e89 < a

   1. Initial program 58.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 94.0%

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

Alternative 13: 69.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+138} \lor \neg \left(a \leq 3.2 \cdot 10^{+153}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -6.5e+138) (not (<= a 3.2e+153)))
   (* (* a a) 4.0)
   (fma (* b b) 4.0 -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -6.5e+138) || !(a <= 3.2e+153)) {
		tmp = (a * a) * 4.0;
	} else {
		tmp = fma((b * b), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -6.5e+138) || !(a <= 3.2e+153))
		tmp = Float64(Float64(a * a) * 4.0);
	else
		tmp = fma(Float64(b * b), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -6.5e+138], N[Not[LessEqual[a, 3.2e+153]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.50000000000000054e138 or 3.2000000000000001e153 < a

   1. Initial program 31.6%

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

   3. Taylor expanded in a around 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 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in 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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 96.8%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 96.8%

       \[\leadsto \left(a \cdot a\right) \cdot 4 \]

   if -6.50000000000000054e138 < a < 3.2000000000000001e153

   1. Initial program 90.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 61.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+138} \lor \neg \left(a \leq 3.2 \cdot 10^{+153}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.9 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 3.9e+307) (fma (* a a) 4.0 -1.0) (fma (* b b) 4.0 -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3.9e+307) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = fma((b * b), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 3.9e+307)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = fma(Float64(b * b), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 3.9e+307], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3.9000000000000001e307

   1. Initial program 82.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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 86.1

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

   5. Applied rewrites 86.1%

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

   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

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 60.9%

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

   if 3.9000000000000001e307 < (*.f64 b b)

   1. Initial program 59.3%

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

   3. Taylor expanded in a around 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 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

Alternative 15: 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 77.6%

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

3. Taylor expanded in a around 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 85.9

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

5. Applied rewrites 85.9%

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

6. Taylor expanded in b around 0

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

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

   3. associate-*l* N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt1-in N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. distribute-lft-in N/A

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

   10. associate-+r+ N/A

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

   11. unpow2 N/A

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

   12. distribute-rgt-in N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

8. Applied rewrites 73.2%

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

9. Taylor expanded in a around 0

   \[\leadsto -1 \]
10. Step-by-step derivation

11. Applied rewrites 30.1%

    \[\leadsto -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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))