Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.8% → 98.2%

Time: 12.4s

Alternatives: 13

Speedup: 5.7×

• 60.8% 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: 73.8% 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: 98.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\mathsf{fma}\left(a, a, b \cdot b\right)}{\frac{1}{b \cdot b}} + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-15)
   (fma (* a a) (fma a (+ a -4.0) 4.0) -1.0)
   (+ -1.0 (+ (/ (fma a a (* b b)) (/ 1.0 (* b b))) (* 4.0 (* (* b b) 3.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-15) {
		tmp = fma((a * a), fma(a, (a + -4.0), 4.0), -1.0);
	} else {
		tmp = -1.0 + ((fma(a, a, (b * b)) / (1.0 / (b * b))) + (4.0 * ((b * b) * 3.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-15)
		tmp = fma(Float64(a * a), fma(a, Float64(a + -4.0), 4.0), -1.0);
	else
		tmp = Float64(-1.0 + Float64(Float64(fma(a, a, Float64(b * b)) / Float64(1.0 / Float64(b * b))) + Float64(4.0 * Float64(Float64(b * b) * 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-15], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(N[(N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(b * b), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999999e-15

   1. Initial program 82.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 82.8

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

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-neg-out N/A

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

      10. unpow2 N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 82.8

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

   8. Simplified 82.8%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. +-lowering-+.f64 99.9

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

   11. Simplified 99.9%

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

   if 4.99999999999999999e-15 < (*.f64 b b)

   1. Initial program 73.4%

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

      1. unpow2 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-+ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 73.4

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

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 97.6

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

   10. Simplified 97.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\mathsf{fma}\left(a, a, b \cdot b\right)}{\frac{1}{b \cdot b}} + 4 \cdot \left(\left(b \cdot b\right) \cdot 3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 51.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + {\left(a \cdot a + b \cdot b\right)}^{2} \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0))))
       (pow (+ (* a a) (* b b)) 2.0))
      0.5)
   -1.0
   (* (* a a) 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (((4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))) + pow(((a * a) + (b * b)), 2.0)) <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (a + 3.0d0)))) + (((a * a) + (b * b)) ** 2.0d0)) <= 0.5d0) then
        tmp = -1.0d0
    else
        tmp = (a * a) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (((4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))) + Math.pow(((a * a) + (b * b)), 2.0)) <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))) + math.pow(((a * a) + (b * b)), 2.0)) <= 0.5:
		tmp = -1.0
	else:
		tmp = (a * a) * 4.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))) + (Float64(Float64(a * a) + Float64(b * b)) ^ 2.0)) <= 0.5)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * a) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))) + (((a * a) + (b * b)) ^ 2.0)) <= 0.5)
		tmp = -1.0;
	else
		tmp = (a * a) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 0.5], -1.0, N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < 0.5

   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 inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 97.2

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

   5. Simplified 97.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 97.2%

      \[\leadsto \color{blue}{-1} \]

   if 0.5 < (+.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 71.2%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 49.4

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

   5. Simplified 49.4%

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

   6. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-neg-out N/A

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

      10. unpow2 N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 37.3

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

   8. Simplified 37.3%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 27.0

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

   11. Simplified 27.0%

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

   12. Taylor expanded in a around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{4 \cdot {a}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 27.4

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

   14. Simplified 27.4%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + {\left(a \cdot a + b \cdot b\right)}^{2} \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+115}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-53)
   (fma (* a a) (fma a (+ a -4.0) 4.0) -1.0)
   (if (<= (* b b) 2e+115)
     (+
      -1.0
      (+
       (* b (* b (* b b)))
       (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0))))))
     (fma (* b b) (fma b b 12.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-53) {
		tmp = fma((a * a), fma(a, (a + -4.0), 4.0), -1.0);
	} else if ((b * b) <= 2e+115) {
		tmp = -1.0 + ((b * (b * (b * b))) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))));
	} else {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-53)
		tmp = fma(Float64(a * a), fma(a, Float64(a + -4.0), 4.0), -1.0);
	elseif (Float64(b * b) <= 2e+115)
		tmp = Float64(-1.0 + Float64(Float64(b * Float64(b * Float64(b * b))) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0))))));
	else
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-53], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 2e+115], N[(-1.0 + N[(N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 1.00000000000000003e-53

   1. Initial program 81.9%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 81.9

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

   5. Simplified 81.9%

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

   6. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-neg-out N/A

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

      10. unpow2 N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 81.9

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

   8. Simplified 81.9%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. +-lowering-+.f64 99.9

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

   11. Simplified 99.9%

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

   if 1.00000000000000003e-53 < (*.f64 b b) < 2e115

   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

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 85.0

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

   5. Simplified 85.0%

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

   if 2e115 < (*.f64 b b)

   1. Initial program 69.1%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+115}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathsf{fma}\left(\frac{-1}{\frac{-1}{t\_0}}, t\_0, \mathsf{fma}\left(b \cdot b, 12, -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (fma (/ -1.0 (/ -1.0 t_0)) t_0 (fma (* b b) 12.0 -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	return fma((-1.0 / (-1.0 / t_0)), t_0, fma((b * b), 12.0, -1.0));
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	return fma(Float64(-1.0 / Float64(-1.0 / t_0)), t_0, fma(Float64(b * b), 12.0, -1.0))
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, N[(N[(-1.0 / N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.4%

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

   1. unpow2 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-+ N/A

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

   10. /-lowering-/.f64 N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-lowering-*.f64 78.4

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

4. Applied egg-rr 78.4%

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

   1. remove-double-div N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. associate-/l/ N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. +-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 N/A

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

   14. +-commutative N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

   16. *-lowering-*.f64 78.3

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

6. Applied egg-rr 78.3%

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

7. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 98.5

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

9. Simplified 98.5%

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

    1. associate--l+ N/A

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

    2. *-commutative N/A

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

    3. div-inv N/A

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

    4. associate-/r/ N/A

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

    5. accelerator-lowering-fma.f64 N/A

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

    6. /-lowering-/.f64 N/A

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

    7. /-lowering-/.f64 N/A

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

    8. accelerator-lowering-fma.f64 N/A

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

    9. *-lowering-*.f64 N/A

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

    10. accelerator-lowering-fma.f64 N/A

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

    11. *-lowering-*.f64 N/A

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

    12. sub-neg N/A

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

11. Applied egg-rr 98.5%

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

Alternative 5: 94.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 12, \mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-15)
   (fma (* a a) (fma a (+ a -4.0) 4.0) -1.0)
   (fma b (* b 12.0) (fma b (* b (* b b)) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-15) {
		tmp = fma((a * a), fma(a, (a + -4.0), 4.0), -1.0);
	} else {
		tmp = fma(b, (b * 12.0), fma(b, (b * (b * b)), -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-15)
		tmp = fma(Float64(a * a), fma(a, Float64(a + -4.0), 4.0), -1.0);
	else
		tmp = fma(b, Float64(b * 12.0), fma(b, Float64(b * Float64(b * b)), -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-15], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * 12.0), $MachinePrecision] + N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999999e-15

   1. Initial program 82.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 82.8

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

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-neg-out N/A

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

      10. unpow2 N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 82.8

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

   8. Simplified 82.8%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. +-lowering-+.f64 99.9

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

   11. Simplified 99.9%

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

   if 4.99999999999999999e-15 < (*.f64 b b)

   1. Initial program 73.4%

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

      1. unpow2 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-+ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 73.4

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

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 91.2

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

   10. Simplified 91.2%

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

Alternative 6: 94.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + -4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-15)
   (fma (* a a) (fma a (+ a -4.0) 4.0) -1.0)
   (fma (* b b) (fma b b 12.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-15) {
		tmp = fma((a * a), fma(a, (a + -4.0), 4.0), -1.0);
	} else {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-15)
		tmp = fma(Float64(a * a), fma(a, Float64(a + -4.0), 4.0), -1.0);
	else
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-15], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999999e-15

   1. Initial program 82.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 82.8

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

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-neg-out N/A

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

      10. unpow2 N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 82.8

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

   8. Simplified 82.8%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. +-lowering-+.f64 99.9

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

   11. Simplified 99.9%

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

   if 4.99999999999999999e-15 < (*.f64 b b)

   1. Initial program 73.4%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 91.2

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

   5. Simplified 91.2%

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

Alternative 7: 93.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot a\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-15)
   (fma (* a (* a a)) a -1.0)
   (fma (* b b) (fma b b 12.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-15) {
		tmp = fma((a * (a * a)), a, -1.0);
	} else {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-15)
		tmp = fma(Float64(a * Float64(a * a)), a, -1.0);
	else
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-15], N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999999e-15

   1. Initial program 82.8%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 97.2

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

   5. Simplified 97.2%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 97.2

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

   7. Applied egg-rr 97.2%

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

   if 4.99999999999999999e-15 < (*.f64 b b)

   1. Initial program 73.4%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 91.2

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

   5. Simplified 91.2%

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

Alternative 8: 82.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -50000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.4:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 12, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -50000000000.0) t_0 (if (<= a 2.4) (fma b (* b 12.0) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -50000000000.0) {
		tmp = t_0;
	} else if (a <= 2.4) {
		tmp = fma(b, (b * 12.0), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * a)))
	tmp = 0.0
	if (a <= -50000000000.0)
		tmp = t_0;
	elseif (a <= 2.4)
		tmp = fma(b, Float64(b * 12.0), -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -50000000000.0], t$95$0, If[LessEqual[a, 2.4], N[(b * N[(b * 12.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -5e10 or 2.39999999999999991 < a

   1. Initial program 57.4%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

         \[\leadsto {a}^{\color{blue}{\left(3 + 1\right)}} \]

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 85.0

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

   5. Simplified 85.0%

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

   if -5e10 < a < 2.39999999999999991

   1. Initial program 98.4%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 72.3

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

   5. Simplified 72.3%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 72.1

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

   8. Simplified 72.1%

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

Alternative 9: 93.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot a\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-9) (fma (* a (* a a)) a -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-9) {
		tmp = fma((a * (a * a)), a, -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-9)
		tmp = fma(Float64(a * Float64(a * a)), a, -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-9], N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000012e-9

   1. Initial program 82.9%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 96.9

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

   5. Simplified 96.9%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 96.9

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

   7. Applied egg-rr 96.9%

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

   if 2.00000000000000012e-9 < (*.f64 b b)

   1. Initial program 73.2%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

         \[\leadsto b \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

      8. *-lowering-*.f64 89.6

         \[\leadsto b \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

   5. Simplified 89.6%

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

Alternative 10: 81.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-9) (fma 4.0 (* a a) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-9) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-9)
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = Float64(b * Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-9], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000012e-9

   1. Initial program 82.9%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 82.9

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

   5. Simplified 82.9%

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

   6. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-neg-out N/A

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

      10. unpow2 N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 82.6

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

   8. Simplified 82.6%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 71.8

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

   11. Simplified 71.8%

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

   if 2.00000000000000012e-9 < (*.f64 b b)

   1. Initial program 73.2%

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

   3. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

         \[\leadsto b \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

      8. *-lowering-*.f64 89.6

         \[\leadsto b \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

   5. Simplified 89.6%

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

Alternative 11: 69.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 12, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+276) (fma 4.0 (* a a) -1.0) (fma b (* b 12.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+276) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = fma(b, (b * 12.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+276)
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = fma(b, Float64(b * 12.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+276], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.0000000000000001e276

   1. Initial program 82.0%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 63.6

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

   5. Simplified 63.6%

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

   6. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-neg-out N/A

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

      10. unpow2 N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 64.5

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

   8. Simplified 64.5%

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

   9. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 55.4

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

   11. Simplified 55.4%

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

   if 2.0000000000000001e276 < (*.f64 b b)

   1. Initial program 66.1%

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

   3. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 56.9

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

   5. Simplified 56.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 89.4

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

   8. Simplified 89.4%

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

Alternative 12: 51.5% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in a around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 62.1

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

5. Simplified 62.1%

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

6. Taylor expanded in b around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. *-lowering-*.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. mul-1-neg N/A

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

   9. distribute-lft-neg-out N/A

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

   10. unpow2 N/A

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

   11. unsub-neg N/A

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

   12. --lowering--.f64 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 52.8

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

8. Simplified 52.8%

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

9. Taylor expanded in a around 0

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

    3. accelerator-lowering-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. *-lowering-*.f64 44.7

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

11. Simplified 44.7%

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

Alternative 13: 25.0% 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 78.4%

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

3. Taylor expanded in a around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

      \[\leadsto \color{blue}{{a}^{3} \cdot a} - 1 \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot {a}^{3}} - 1 \]

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 66.2

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

5. Simplified 66.2%

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

6. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 24.9%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))