Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.7% → 99.9%

Time: 13.6s

Alternatives: 18

Speedup: 5.9×

• 60.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 10^{+287}:\\ \;\;\;\;-1 + \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), \mathsf{fma}\left(b, b, 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{a + b}, \frac{t\_0 \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)}{a - b}, \mathsf{fma}\left(4, b \cdot b, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))))
   (if (<=
        (+
         (pow (+ (* a a) (* b b)) 2.0)
         (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
        1e+287)
     (+
      -1.0
      (fma
       (* a a)
       (fma a (+ a 4.0) 4.0)
       (* (* b b) (fma a (fma 2.0 a -12.0) (fma b b 4.0)))))
     (fma
      (/ t_0 (+ a b))
      (/ (* t_0 (* (+ a b) (- a b))) (- a b))
      (fma 4.0 (* b b) -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 1e+287) {
		tmp = -1.0 + fma((a * a), fma(a, (a + 4.0), 4.0), ((b * b) * fma(a, fma(2.0, a, -12.0), fma(b, b, 4.0))));
	} else {
		tmp = fma((t_0 / (a + b)), ((t_0 * ((a + b) * (a - b))) / (a - b)), fma(4.0, (b * b), -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= 1e+287)
		tmp = Float64(-1.0 + fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), Float64(Float64(b * b) * fma(a, fma(2.0, a, -12.0), fma(b, b, 4.0)))));
	else
		tmp = fma(Float64(t_0 / Float64(a + b)), Float64(Float64(t_0 * Float64(Float64(a + b) * Float64(a - b))) / Float64(a - b)), fma(4.0, Float64(b * b), -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+287], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a * N[(2.0 * a + -12.0), $MachinePrecision] + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(N[(a + b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a - b), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < 1.0000000000000001e287

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 99.9%

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

   if 1.0000000000000001e287 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 62.3%

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

   4. Applied egg-rr 65.9%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 10^{+287}:\\ \;\;\;\;-1 + \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), \mathsf{fma}\left(b, b, 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(a, a, b \cdot b\right)}{a + b}, \frac{\mathsf{fma}\left(a, a, b \cdot b\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)}{a - b}, \mathsf{fma}\left(4, b \cdot b, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{+64}:\\ \;\;\;\;-1 + \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), \mathsf{fma}\left(b, b, 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))))
   (if (<=
        (+
         (pow (+ (* a a) (* b b)) 2.0)
         (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
        5e+64)
     (+
      -1.0
      (fma
       (* a a)
       (fma a (+ a 4.0) 4.0)
       (* (* b b) (fma a (fma 2.0 a -12.0) (fma b b 4.0)))))
     (fma t_0 t_0 -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 5e+64) {
		tmp = -1.0 + fma((a * a), fma(a, (a + 4.0), 4.0), ((b * b) * fma(a, fma(2.0, a, -12.0), fma(b, b, 4.0))));
	} else {
		tmp = fma(t_0, t_0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= 5e+64)
		tmp = Float64(-1.0 + fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), Float64(Float64(b * b) * fma(a, fma(2.0, a, -12.0), fma(b, b, 4.0)))));
	else
		tmp = fma(t_0, t_0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+64], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a * N[(2.0 * a + -12.0), $MachinePrecision] + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < 5e64

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 100.0%

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

   if 5e64 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 66.1%

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

   4. Applied egg-rr 69.2%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

      3. difference-of-squares N/A

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

      4. difference-of-squares N/A

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

      5. difference-of-squares N/A

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

      6. flip-+ N/A

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

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

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in b around 0

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

   12. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{+64}:\\ \;\;\;\;-1 + \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), \mathsf{fma}\left(b, b, 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
      5e-6)
   -1.0
   (* (* a a) 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 5e-6) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (a + 1.0d0)) + ((b * b) * (1.0d0 - (a * 3.0d0)))))) <= 5d-6) then
        tmp = -1.0d0
    else
        tmp = (a * a) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 5e-6) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 5e-6:
		tmp = -1.0
	else:
		tmp = (a * a) * 4.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= 5e-6)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * a) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 5e-6)
		tmp = -1.0;
	else
		tmp = (a * a) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], -1.0, N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < 5.00000000000000041e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   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 96.6

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

   5. Simplified 96.6%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 96.6%

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

   if 5.00000000000000041e-6 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 67.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

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

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

   5. Simplified 77.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{4 \cdot {a}^{2} - 1} \]
   7. 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 38.4

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

   8. Simplified 38.4%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{4 \cdot {a}^{2}} \]
   10. 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 38.9

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

   11. Simplified 38.9%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathbf{if}\;b \cdot b \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), 4\right), \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(b, b \cdot 4, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))))
   (if (<= (* b b) 1e-7)
     (fma
      b
      (* b (fma a (fma 2.0 a -12.0) 4.0))
      (fma (* a a) (fma a (+ a 4.0) 4.0) -1.0))
     (fma t_0 t_0 (fma b (* b 4.0) -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double tmp;
	if ((b * b) <= 1e-7) {
		tmp = fma(b, (b * fma(a, fma(2.0, a, -12.0), 4.0)), fma((a * a), fma(a, (a + 4.0), 4.0), -1.0));
	} else {
		tmp = fma(t_0, t_0, fma(b, (b * 4.0), -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	tmp = 0.0
	if (Float64(b * b) <= 1e-7)
		tmp = fma(b, Float64(b * fma(a, fma(2.0, a, -12.0), 4.0)), fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), -1.0));
	else
		tmp = fma(t_0, t_0, fma(b, Float64(b * 4.0), -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 1e-7], N[(b * N[(b * N[(a * N[(2.0 * a + -12.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(b * N[(b * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.9999999999999995e-8

   1. Initial program 80.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

   5. Simplified 86.9%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

   8. Simplified 100.0%

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

   if 9.9999999999999995e-8 < (*.f64 b b)

   1. Initial program 70.1%

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

      3. difference-of-squares N/A

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

      4. difference-of-squares N/A

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

      5. difference-of-squares N/A

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

      6. flip-+ N/A

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

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

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), 4\right), \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(b, b \cdot 4, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathbf{if}\;b \cdot b \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(b, b \cdot 4, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))))
   (if (<= (* b b) 1e-7)
     (fma (* b b) (fma a -12.0 4.0) (fma (* a a) (fma a (+ a 4.0) 4.0) -1.0))
     (fma t_0 t_0 (fma b (* b 4.0) -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double tmp;
	if ((b * b) <= 1e-7) {
		tmp = fma((b * b), fma(a, -12.0, 4.0), fma((a * a), fma(a, (a + 4.0), 4.0), -1.0));
	} else {
		tmp = fma(t_0, t_0, fma(b, (b * 4.0), -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	tmp = 0.0
	if (Float64(b * b) <= 1e-7)
		tmp = fma(Float64(b * b), fma(a, -12.0, 4.0), fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), -1.0));
	else
		tmp = fma(t_0, t_0, fma(b, Float64(b * 4.0), -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 1e-7], N[(N[(b * b), $MachinePrecision] * N[(a * -12.0 + 4.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(b * N[(b * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.9999999999999995e-8

   1. Initial program 80.8%

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

   4. Applied egg-rr 80.8%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 86.9%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. +-commutative N/A

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

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

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

      18. +-lowering-+.f64 86.9

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

   10. Simplified 86.9%

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

   11. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 99.8

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

   13. Simplified 99.8%

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

   if 9.9999999999999995e-8 < (*.f64 b b)

   1. Initial program 70.1%

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

      3. difference-of-squares N/A

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

      4. difference-of-squares N/A

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

      5. difference-of-squares N/A

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

      6. flip-+ N/A

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

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

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(b, b \cdot 4, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))))
   (if (<= (* b b) 5e-7)
     (fma (* b b) (fma a -12.0 4.0) (fma (* a a) (fma a (+ a 4.0) 4.0) -1.0))
     (fma t_0 t_0 -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double tmp;
	if ((b * b) <= 5e-7) {
		tmp = fma((b * b), fma(a, -12.0, 4.0), fma((a * a), fma(a, (a + 4.0), 4.0), -1.0));
	} else {
		tmp = fma(t_0, t_0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	tmp = 0.0
	if (Float64(b * b) <= 5e-7)
		tmp = fma(Float64(b * b), fma(a, -12.0, 4.0), fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), -1.0));
	else
		tmp = fma(t_0, t_0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 5e-7], N[(N[(b * b), $MachinePrecision] * N[(a * -12.0 + 4.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999977e-7

   1. Initial program 81.0%

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

   4. Applied egg-rr 81.0%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 87.0%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. +-commutative N/A

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

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

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

      18. +-lowering-+.f64 86.9

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

   10. Simplified 86.9%

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

   11. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 99.7

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

   13. Simplified 99.7%

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

   if 4.99999999999999977e-7 < (*.f64 b b)

   1. Initial program 69.9%

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

   4. Applied egg-rr 74.2%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

      3. difference-of-squares N/A

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

      4. difference-of-squares N/A

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

      5. difference-of-squares N/A

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

      6. flip-+ N/A

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

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

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in b around 0

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

   12. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathbf{if}\;b \cdot b \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))))
   (if (<= (* b b) 1e-7)
     (fma (* a a) (fma a (+ a 4.0) 4.0) -1.0)
     (fma t_0 t_0 -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double tmp;
	if ((b * b) <= 1e-7) {
		tmp = fma((a * a), fma(a, (a + 4.0), 4.0), -1.0);
	} else {
		tmp = fma(t_0, t_0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	tmp = 0.0
	if (Float64(b * b) <= 1e-7)
		tmp = fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), -1.0);
	else
		tmp = fma(t_0, t_0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 1e-7], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.9999999999999995e-8

   1. Initial program 80.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

   5. Simplified 86.9%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-in N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   8. Simplified 99.8%

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

   if 9.9999999999999995e-8 < (*.f64 b b)

   1. Initial program 70.1%

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

      3. difference-of-squares N/A

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

      4. difference-of-squares N/A

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

      5. difference-of-squares N/A

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

      6. flip-+ N/A

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

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

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in b around 0

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

   12. Simplified 99.5%

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

4. Final simplification 99.6%

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

Alternative 8: 97.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(2, a \cdot a, 4\right)\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+16)
   (fma (* a a) (fma a (+ a 4.0) 4.0) -1.0)
   (fma (* b b) (fma b b (fma 2.0 (* a a) 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+16) {
		tmp = fma((a * a), fma(a, (a + 4.0), 4.0), -1.0);
	} else {
		tmp = fma((b * b), fma(b, b, fma(2.0, (a * a), 4.0)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+16)
		tmp = fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), -1.0);
	else
		tmp = fma(Float64(b * b), fma(b, b, fma(2.0, Float64(a * a), 4.0)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+16], 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 + N[(2.0 * N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e16

   1. Initial program 80.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

   5. Simplified 87.1%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-in N/A

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

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

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

      10. +-lowering-+.f64 99.4

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

   8. Simplified 99.4%

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

   if 1e16 < (*.f64 b b)

   1. Initial program 70.2%

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

   4. Applied egg-rr 74.5%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

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

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

   10. Simplified 97.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(2, a \cdot a, 4\right)\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+21)
   (fma (* a a) (fma a (+ a 4.0) 4.0) -1.0)
   (fma (* b b) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+21) {
		tmp = fma((a * a), fma(a, (a + 4.0), 4.0), -1.0);
	} else {
		tmp = fma((b * b), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+21)
		tmp = fma(Float64(a * a), fma(a, Float64(a + 4.0), 4.0), -1.0);
	else
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+21], 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), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e21

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

   5. Simplified 86.7%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-in N/A

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

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

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

      10. +-lowering-+.f64 98.5

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

   8. Simplified 98.5%

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

   if 4e21 < (*.f64 b b)

   1. Initial program 69.5%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 92.5

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

   5. Simplified 92.5%

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

   6. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 92.5

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

   8. Simplified 92.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a + 4, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -59:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(a, a + 4, 4\right)\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -59.0)
   (* a (* a (fma a (+ a 4.0) 4.0)))
   (if (<= a 1.4e+50) (fma (* b b) (fma b b 4.0) -1.0) (* a (* a (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -59.0) {
		tmp = a * (a * fma(a, (a + 4.0), 4.0));
	} else if (a <= 1.4e+50) {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	} else {
		tmp = a * (a * (a * a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -59.0)
		tmp = Float64(a * Float64(a * fma(a, Float64(a + 4.0), 4.0)));
	elseif (a <= 1.4e+50)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = Float64(a * Float64(a * Float64(a * a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -59.0], N[(a * N[(a * N[(a * N[(a + 4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.4e+50], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -59

   1. Initial program 31.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

   5. Simplified 83.8%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

      8. unpow2 N/A

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

      9. *-lft-identity N/A

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

      10. unpow2 N/A

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

      11. distribute-lft-out N/A

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

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

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

      13. +-lowering-+.f64 82.5

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

   8. Simplified 82.5%

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

   9. Taylor expanded in a around 0

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

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

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

       2. unpow2 N/A

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

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

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

       4. associate-+r+ N/A

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

       5. +-commutative N/A

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

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

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

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

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

       8. +-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 94.2

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

   11. Simplified 94.2%

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

   12. Taylor expanded in b around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

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

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

       7. +-commutative N/A

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

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

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

       9. +-lowering-+.f64 89.2

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

   14. Simplified 89.2%

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

   if -59 < a < 1.3999999999999999e50

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 96.8

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

   5. Simplified 96.8%

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

   if 1.3999999999999999e50 < a

   1. Initial program 57.7%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -59:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(a, a + 4, 4\right)\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -8.5 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -8.5e+69)
     t_0
     (if (<= a 1.55e+47) (fma (* b b) (fma b b 4.0) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -8.5e+69) {
		tmp = t_0;
	} else if (a <= 1.55e+47) {
		tmp = fma((b * b), fma(b, b, 4.0), -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 <= -8.5e+69)
		tmp = t_0;
	elseif (a <= 1.55e+47)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -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, -8.5e+69], t$95$0, If[LessEqual[a, 1.55e+47], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.5000000000000002e69 or 1.55e47 < a

   1. Initial program 37.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 99.1

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

   5. Simplified 99.1%

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

   if -8.5000000000000002e69 < a < 1.55e47

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 92.9

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

   5. Simplified 92.9%

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

Alternative 12: 82.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -40:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 380000:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -40.0) t_0 (if (<= a 380000.0) (fma b (* b 4.0) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -40.0) {
		tmp = t_0;
	} else if (a <= 380000.0) {
		tmp = fma(b, (b * 4.0), -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 <= -40.0)
		tmp = t_0;
	elseif (a <= 380000.0)
		tmp = fma(b, Float64(b * 4.0), -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, -40.0], t$95$0, If[LessEqual[a, 380000.0], N[(b * N[(b * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -40 or 3.8e5 < a

   1. Initial program 46.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 88.0

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

   5. Simplified 88.0%

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

   if -40 < a < 3.8e5

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

   5. Simplified 73.1%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-lowering-*.f64 72.0

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

   8. Simplified 72.0%

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

Alternative 13: 93.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot a\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+21) (fma (* a (* a a)) a -1.0) (fma (* b b) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+21) {
		tmp = fma((a * (a * a)), a, -1.0);
	} else {
		tmp = fma((b * b), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+21)
		tmp = fma(Float64(a * Float64(a * a)), a, -1.0);
	else
		tmp = fma(Float64(b * b), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+21], N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e21

   1. Initial program 80.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   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 94.9

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

   5. Simplified 94.9%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)} - 1 \]
   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 94.9

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

   7. Applied egg-rr 94.9%

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

   if 4e21 < (*.f64 b b)

   1. Initial program 69.5%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 92.5

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

   5. Simplified 92.5%

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

   6. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 92.5

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

   8. Simplified 92.5%

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

Alternative 14: 93.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot a\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+21) (fma (* a (* a a)) a -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+21) {
		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) <= 4e+21)
		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], 4e+21], 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) < 4e21

   1. Initial program 80.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   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 94.9

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

   5. Simplified 94.9%

      \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)} - 1 \]
   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 94.9

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

   7. Applied egg-rr 94.9%

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

   if 4e21 < (*.f64 b b)

   1. Initial program 69.5%

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

   3. Taylor expanded in 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 92.5

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

   5. Simplified 92.5%

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

Alternative 15: 82.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+21) (fma 4.0 (* a a) -1.0) (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+21) {
		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) <= 4e+21)
		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], 4e+21], 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) < 4e21

   1. Initial program 80.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

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

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

   5. Simplified 82.7%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{4 \cdot {a}^{2} - 1} \]
   7. 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 81.4

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

   8. Simplified 81.4%

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

   if 4e21 < (*.f64 b b)

   1. Initial program 69.5%

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

   3. Taylor expanded in 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 92.5

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

   5. Simplified 92.5%

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

Alternative 16: 70.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+303) (fma 4.0 (* a a) -1.0) (fma b (* b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+303) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = fma(b, (b * 4.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+303)
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = fma(b, Float64(b * 4.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+303], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(b * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e303

   1. Initial program 77.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

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

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

   5. Simplified 87.7%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{4 \cdot {a}^{2} - 1} \]
   7. 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 60.6

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

   8. Simplified 60.6%

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

   if 4e303 < (*.f64 b b)

   1. Initial program 66.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

Alternative 17: 51.8% accurate, 13.3× 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 74.9%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

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

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

5. Simplified 82.6%

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

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{4 \cdot {a}^{2} - 1} \]
7. 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 52.1

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

8. Simplified 52.1%

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

Alternative 18: 26.0% accurate, 160.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.9%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{{a}^{4}} - 1 \]
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 63.8

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

5. Simplified 63.8%

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

6. Taylor expanded in a around 0

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

8. Simplified 22.8%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))