Bouland and Aaronson, Equation (24)

Percentage Accurate: 72.4% → 97.6%

Time: 9.0s

Alternatives: 15

Speedup: 5.3×

• 61.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(Float64(Float64(3.0 + a) * Float64(b * b)) + Float64(Float64(1.0 - a) * Float64(a * a))) * 4.0) + (Float64(Float64(b * b) + Float64(a * a)) ^ 2.0))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 - 1.0);
	else
		tmp = fma(fma(Float64(b * b), fma(2.0, a, 4.0), Float64(4.0 * a)), a, fma(Float64(fma(b, b, 12.0) * b), b, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(N[(3.0 + a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(2.0 * a + 4.0), $MachinePrecision] + N[(4.0 * a), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 94.8%

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

4. Final simplification 98.5%

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

Alternative 2: 50.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(3 + a\right) \cdot \left(b \cdot b\right) + \left(1 - a\right) \cdot \left(a \cdot a\right)\right) \cdot 4 + {\left(b \cdot b + a \cdot a\right)}^{2} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (* (+ (* (+ 3.0 a) (* b b)) (* (- 1.0 a) (* a a))) 4.0)
       (pow (+ (* b b) (* a a)) 2.0))
      2e-13)
   -1.0
   (* 4.0 (* a a))))

C

double code(double a, double b) {
	double tmp;
	if ((((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + pow(((b * b) + (a * a)), 2.0)) <= 2e-13) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (a * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((3.0d0 + a) * (b * b)) + ((1.0d0 - a) * (a * a))) * 4.0d0) + (((b * b) + (a * a)) ** 2.0d0)) <= 2d-13) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (a * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + Math.pow(((b * b) + (a * a)), 2.0)) <= 2e-13) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (a * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + math.pow(((b * b) + (a * a)), 2.0)) <= 2e-13:
		tmp = -1.0
	else:
		tmp = 4.0 * (a * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(3.0 + a) * Float64(b * b)) + Float64(Float64(1.0 - a) * Float64(a * a))) * 4.0) + (Float64(Float64(b * b) + Float64(a * a)) ^ 2.0)) <= 2e-13)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((((3.0 + a) * (b * b)) + ((1.0 - a) * (a * a))) * 4.0) + (((b * b) + (a * a)) ^ 2.0)) <= 2e-13)
		tmp = -1.0;
	else
		tmp = 4.0 * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[(N[(N[(3.0 + a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-13], -1.0, N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 2.0000000000000001e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 99.6%

       \[\leadsto -1 \]

   if 2.0000000000000001e-13 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 64.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 78.1%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 68.4%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 36.8%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + a\right) \cdot \left(b \cdot b\right) + \left(1 - a\right) \cdot \left(a \cdot a\right)\right) \cdot 4 + {\left(b \cdot b + a \cdot a\right)}^{2} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, \mathsf{fma}\left(b, b, 12\right)\right), b \cdot b, \mathsf{fma}\left(a \cdot a, 4, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-30)
   (- (* (fma (- a 4.0) a 4.0) (* a a)) 1.0)
   (fma
    (fma (fma 2.0 a 4.0) a (fma b b 12.0))
    (* b b)
    (fma (* a a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-30) {
		tmp = (fma((a - 4.0), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = fma(fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)), (b * b), fma((a * a), 4.0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-30)
		tmp = Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = fma(fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)), Float64(b * b), fma(Float64(a * a), 4.0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-30], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e-30

   1. Initial program 80.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.9%

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

   if 1e-30 < (*.f64 b b)

   1. Initial program 66.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 84.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 96.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, \mathsf{fma}\left(b, b, 12\right)\right), b \cdot b, \mathsf{fma}\left(a \cdot a, 4, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot a, a, \mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-30)
   (- (* (fma (- a 4.0) a 4.0) (* a a)) 1.0)
   (fma (* (* 2.0 (* b b)) a) a (fma (* (fma b b 12.0) b) b -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-30) {
		tmp = (fma((a - 4.0), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = fma(((2.0 * (b * b)) * a), a, fma((fma(b, b, 12.0) * b), b, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-30)
		tmp = Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = fma(Float64(Float64(2.0 * Float64(b * b)) * a), a, fma(Float64(fma(b, b, 12.0) * b), b, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-30], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e-30

   1. Initial program 80.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.9%

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

   if 1e-30 < (*.f64 b b)

   1. Initial program 66.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 84.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 96.7%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 96.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot a, a, \mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, 4, 12\right)\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.2)
   (- (* (fma (- a 4.0) a 4.0) (* a a)) 1.0)
   (fma (* (fma b b (fma a 4.0 12.0)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.2) {
		tmp = (fma((a - 4.0), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = fma((fma(b, b, fma(a, 4.0, 12.0)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.2)
		tmp = Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, fma(a, 4.0, 12.0)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.2], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + N[(a * 4.0 + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.20000000000000001

   1. Initial program 79.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.8%

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

   if 0.20000000000000001 < (*.f64 b b)

   1. Initial program 67.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

   8. Applied rewrites 91.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, 4, 12\right)\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.2)
   (- (* (fma (- a 4.0) a 4.0) (* a a)) 1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.2) {
		tmp = (fma((a - 4.0), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.2)
		tmp = Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.2], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.20000000000000001

   1. Initial program 79.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.8%

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

   if 0.20000000000000001 < (*.f64 b b)

   1. Initial program 67.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 89.0

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

   8. Applied rewrites 89.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.2:\\ \;\;\;\;\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.2)
   (- (* (* (fma (- a 4.0) a 4.0) a) a) 1.0)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.2) {
		tmp = ((fma((a - 4.0), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.2)
		tmp = Float64(Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * a) * a) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.2], N[(N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.20000000000000001

   1. Initial program 79.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.7%

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

   if 0.20000000000000001 < (*.f64 b b)

   1. Initial program 67.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 89.0

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

   8. Applied rewrites 89.0%

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

Alternative 8: 93.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* (* a a) (* a a)) 1.0)))
   (if (<= a -4.1e+70)
     t_0
     (if (<= a 2.35e+67) (fma (* (fma b b 12.0) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = ((a * a) * (a * a)) - 1.0;
	double tmp;
	if (a <= -4.1e+70) {
		tmp = t_0;
	} else if (a <= 2.35e+67) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0)
	tmp = 0.0
	if (a <= -4.1e+70)
		tmp = t_0;
	elseif (a <= 2.35e+67)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -4.1e+70], t$95$0, If[LessEqual[a, 2.35e+67], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1000000000000002e70 or 2.35000000000000009e67 < a

   1. Initial program 37.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   if -4.1000000000000002e70 < a < 2.35000000000000009e67

   1. Initial program 96.5%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 9: 84.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* a a))))
   (if (<= a -6.8e+153)
     t_0
     (if (<= a 2e+134) (fma (* (fma b b 12.0) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = 4.0 * (a * a);
	double tmp;
	if (a <= -6.8e+153) {
		tmp = t_0;
	} else if (a <= 2e+134) {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(4.0 * Float64(a * a))
	tmp = 0.0
	if (a <= -6.8e+153)
		tmp = t_0;
	elseif (a <= 2e+134)
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.8e+153], t$95$0, If[LessEqual[a, 2e+134], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.7999999999999995e153 or 1.99999999999999984e134 < a

   1. Initial program 25.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.8%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 96.2%

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

   if -6.7999999999999995e153 < a < 1.99999999999999984e134

   1. Initial program 90.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 83.4

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

   8. Applied rewrites 83.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+153}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* a a))))
   (if (<= a -6.8e+153)
     t_0
     (if (<= a 2e+134) (fma (* (* b b) b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = 4.0 * (a * a);
	double tmp;
	if (a <= -6.8e+153) {
		tmp = t_0;
	} else if (a <= 2e+134) {
		tmp = fma(((b * b) * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(4.0 * Float64(a * a))
	tmp = 0.0
	if (a <= -6.8e+153)
		tmp = t_0;
	elseif (a <= 2e+134)
		tmp = fma(Float64(Float64(b * b) * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.8e+153], t$95$0, If[LessEqual[a, 2e+134], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.7999999999999995e153 or 1.99999999999999984e134 < a

   1. Initial program 25.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.8%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 96.2%

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

   if -6.7999999999999995e153 < a < 1.99999999999999984e134

   1. Initial program 90.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

   8. Applied rewrites 83.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 82.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+153}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* a a))))
   (if (<= a -6.8e+153)
     t_0
     (if (<= a 2e+134) (fma (* b b) (* b b) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = 4.0 * (a * a);
	double tmp;
	if (a <= -6.8e+153) {
		tmp = t_0;
	} else if (a <= 2e+134) {
		tmp = fma((b * b), (b * b), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(4.0 * Float64(a * a))
	tmp = 0.0
	if (a <= -6.8e+153)
		tmp = t_0;
	elseif (a <= 2e+134)
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.8e+153], t$95$0, If[LessEqual[a, 2e+134], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.7999999999999995e153 or 1.99999999999999984e134 < a

   1. Initial program 25.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 98.8%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 96.2%

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

   if -6.7999999999999995e153 < a < 1.99999999999999984e134

   1. Initial program 90.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

   8. Applied rewrites 83.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 82.2%

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

   13. Applied rewrites 82.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+153}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e+172) (fma (* a a) 4.0 -1.0) (* (* (* b b) a) 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.1e+172) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = ((b * b) * a) * 4.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.1e+172)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * a) * 4.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.1e+172], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1000000000000001e172

   1. Initial program 74.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 83.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 53.1%

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

   if 2.1000000000000001e172 < b

   1. Initial program 60.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 53.3%

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

Alternative 13: 51.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.3 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot a\right) \cdot 4\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.3e+172) (fma (* a a) 4.0 -1.0) (* (* (* b a) 4.0) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.3e+172) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = ((b * a) * 4.0) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.3e+172)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = Float64(Float64(Float64(b * a) * 4.0) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.3e+172], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.3e172

   1. Initial program 74.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 83.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 53.1%

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

   if 5.3e172 < b

   1. Initial program 60.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 0.6%

       \[\leadsto -1 \]

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 37.9%

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

Alternative 14: 50.8% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. distribute-lft-in N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 72.2%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

   9. associate-+l+ N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

8. Applied rewrites 82.9%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 50.2%

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

Alternative 15: 24.6% accurate, 155.0× 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 72.6%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. distribute-lft-in N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 72.2%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. pow-sqr N/A

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

   8. distribute-lft-in N/A

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

   9. associate-+r+ N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(12 + \left(4 \cdot a + {b}^{2}\right)\right) \cdot {b}^{2}} + -1 \]

8. Applied rewrites 72.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, 4, 12\right)\right) \cdot b, b, -1\right)} \]

9. Taylor expanded in b around 0

   \[\leadsto -1 \]
10. Step-by-step derivation

11. Applied rewrites 22.4%

    \[\leadsto -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))