Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.5% → 98.9%

Time: 12.7s

Alternatives: 14

Speedup: 5.7×

• 60.7% 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: 73.5% 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: 98.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathsf{fma}\left(b \cdot 4, b, -1 + t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b)))) (fma (* b 4.0) b (+ -1.0 (* t_0 t_0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	return fma((b * 4.0), b, (-1.0 + (t_0 * t_0)));
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	return fma(Float64(b * 4.0), b, Float64(-1.0 + Float64(t_0 * t_0)))
end

Wolfram

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

Derivation

1. Initial program 71.4%

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

4. Applied egg-rr 73.7%

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

5. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 99.4

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

7. Simplified 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. remove-double-div 99.4

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

   9. lift-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f64 N/A

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

   12. associate-+l+ N/A

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

9. Applied egg-rr 99.4%

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

Alternative 2: 51.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* b b) (* a a)) 2.0)
       (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
      2e-14)
   -1.0
   (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 2e-14) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((b * b) + (a * a)) ** 2.0d0) + (4.0d0 * (((a * a) * (a + 1.0d0)) + ((b * b) * (1.0d0 - (a * 3.0d0)))))) <= 2d-14) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 2e-14) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 2e-14:
		tmp = -1.0
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 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)))))) <= 2e-14)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((((b * b) + (a * a)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 2e-14)
		tmp = -1.0;
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $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], 2e-14], -1.0, N[(4.0 * N[(b * b), $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)))))) < 2e-14

   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 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. lower-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. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around 0

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

   11. Simplified 99.7%

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

   if 2e-14 < (+.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 61.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. lower-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 75.7%

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

   6. Taylor expanded in 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. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 35.2

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

   8. Simplified 35.2%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 35.7

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

   11. Simplified 35.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b \cdot 2, a \cdot a\right)\\ \mathbf{if}\;a \leq -820:\\ \;\;\;\;a \cdot \left(a \cdot t\_0\right)\\ \mathbf{elif}\;a \leq 3000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, b, 4\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, \mathsf{fma}\left(a \cdot a, t\_0, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b (* b 2.0) (* a a))))
   (if (<= a -820.0)
     (* a (* a t_0))
     (if (<= a 3000000000000.0)
       (fma (* b (fma b b 4.0)) b -1.0)
       (fma 4.0 (* b b) (fma (* a a) t_0 -1.0))))))

C

double code(double a, double b) {
	double t_0 = fma(b, (b * 2.0), (a * a));
	double tmp;
	if (a <= -820.0) {
		tmp = a * (a * t_0);
	} else if (a <= 3000000000000.0) {
		tmp = fma((b * fma(b, b, 4.0)), b, -1.0);
	} else {
		tmp = fma(4.0, (b * b), fma((a * a), t_0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, Float64(b * 2.0), Float64(a * a))
	tmp = 0.0
	if (a <= -820.0)
		tmp = Float64(a * Float64(a * t_0));
	elseif (a <= 3000000000000.0)
		tmp = fma(Float64(b * fma(b, b, 4.0)), b, -1.0);
	else
		tmp = fma(4.0, Float64(b * b), fma(Float64(a * a), t_0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * 2.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -820.0], N[(a * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3000000000000.0], N[(N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(4.0 * N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -820

   1. Initial program 27.2%

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

   4. Applied egg-rr 36.3%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.0

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

   7. Simplified 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. remove-double-div 99.0

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

      9. lift-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-+l+ N/A

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

   9. Applied egg-rr 99.0%

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

   10. Taylor expanded in a around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

       7. associate-*r/ N/A

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

       8. associate-*l/ N/A

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

       9. associate-/l* N/A

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

       10. *-rgt-identity N/A

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

       11. associate-*r/ N/A

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

       12. rgt-mult-inverse N/A

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

       13. *-rgt-identity N/A

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

       14. +-commutative N/A

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

       15. associate-*l* N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

   12. Simplified 97.7%

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

   if -820 < a < 3e12

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   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. lower-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. lower-*.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. lower-fma.f64 99.1

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

   5. Simplified 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.2

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

   7. Applied egg-rr 99.2%

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

   if 3e12 < a

   1. Initial program 60.9%

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

   4. Applied egg-rr 60.8%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.7

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

   7. Simplified 99.7%

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

   8. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-+r- N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

   10. Simplified 99.3%

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

Alternative 4: 98.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(b, b \cdot 4, -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b)))) (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));
	return fma(t_0, t_0, fma(b, (b * 4.0), -1.0));
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	return fma(t_0, t_0, fma(b, Float64(b * 4.0), -1.0))
end

Wolfram

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

Derivation

1. Initial program 71.4%

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

4. Applied egg-rr 73.7%

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

5. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 99.4

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

7. Simplified 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. remove-double-div 99.4

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

   9. lift-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.4

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

9. Applied egg-rr 99.4%

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

Alternative 5: 97.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \mathsf{fma}\left(b, b \cdot 2, a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -820:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, b, 4\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (fma b (* b 2.0) (* a a))))))
   (if (<= a -820.0)
     t_0
     (if (<= a 3000000000000.0) (fma (* b (fma b b 4.0)) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * fma(b, (b * 2.0), (a * a)));
	double tmp;
	if (a <= -820.0) {
		tmp = t_0;
	} else if (a <= 3000000000000.0) {
		tmp = fma((b * fma(b, b, 4.0)), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * fma(b, Float64(b * 2.0), Float64(a * a))))
	tmp = 0.0
	if (a <= -820.0)
		tmp = t_0;
	elseif (a <= 3000000000000.0)
		tmp = fma(Float64(b * fma(b, b, 4.0)), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(b * N[(b * 2.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -820.0], t$95$0, If[LessEqual[a, 3000000000000.0], N[(N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -820 or 3e12 < a

   1. Initial program 43.8%

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

   4. Applied egg-rr 48.4%

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

   5. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.4

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

   7. Simplified 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. remove-double-div 99.4

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

      9. lift-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-+l+ N/A

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in a around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

       7. associate-*r/ N/A

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

       8. associate-*l/ N/A

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

       9. associate-/l* N/A

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

       10. *-rgt-identity N/A

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

       11. associate-*r/ N/A

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

       12. rgt-mult-inverse N/A

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

       13. *-rgt-identity N/A

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

       14. +-commutative N/A

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

       15. associate-*l* N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

   12. Simplified 98.5%

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

   if -820 < a < 3e12

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   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. lower-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. lower-*.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. lower-fma.f64 99.1

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

   5. Simplified 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.2

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

   7. Applied egg-rr 99.2%

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

Alternative 6: 94.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(a, 4 + a, 4\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+67)
   (fma (* a (fma a (+ 4.0 a) 4.0)) a -1.0)
   (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+67) {
		tmp = fma((a * fma(a, (4.0 + a), 4.0)), a, -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+67)
		tmp = fma(Float64(a * fma(a, Float64(4.0 + a), 4.0)), 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], 1e+67], N[(N[(a * N[(a * N[(4.0 + a), $MachinePrecision] + 4.0), $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) < 9.99999999999999983e66

   1. Initial program 80.7%

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

   3. Taylor expanded in 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. lower-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 86.5%

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

   7. Applied egg-rr 86.5%

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

   8. Taylor expanded in b around 0

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

   10. Simplified 93.6%

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

   if 9.99999999999999983e66 < (*.f64 b b)

   1. Initial program 58.0%

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

   3. Taylor expanded in b around 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. lower-*.f64 N/A

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

      6. lower-*.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. lower-*.f64 95.5

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

   5. Simplified 95.5%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(a, 4 + a, 4\right), a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, 4 + a, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+67)
   (fma (* a a) (fma a (+ 4.0 a) 4.0) -1.0)
   (* b (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+67) {
		tmp = fma((a * a), fma(a, (4.0 + a), 4.0), -1.0);
	} else {
		tmp = b * (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+67)
		tmp = fma(Float64(a * a), fma(a, Float64(4.0 + a), 4.0), -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], 1e+67], N[(N[(a * a), $MachinePrecision] * N[(a * N[(4.0 + a), $MachinePrecision] + 4.0), $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) < 9.99999999999999983e66

   1. Initial program 80.7%

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

   3. Taylor expanded in 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. lower-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 86.5%

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

   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. lower-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. lower-*.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. lower-fma.f64 N/A

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

      10. lower-+.f64 93.6

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

   8. Simplified 93.6%

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

   if 9.99999999999999983e66 < (*.f64 b b)

   1. Initial program 58.0%

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

   3. Taylor expanded in b around 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. lower-*.f64 N/A

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

      6. lower-*.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. lower-*.f64 95.5

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

   5. Simplified 95.5%

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

Alternative 8: 94.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot \left(4 + a\right)\right)\right)\\ \mathbf{if}\;a \leq -2300:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 46000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, b, 4\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a (+ 4.0 a))))))
   (if (<= a -2300.0)
     t_0
     (if (<= a 46000000000000.0) (fma (* b (fma b b 4.0)) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * (4.0 + a)));
	double tmp;
	if (a <= -2300.0) {
		tmp = t_0;
	} else if (a <= 46000000000000.0) {
		tmp = fma((b * fma(b, b, 4.0)), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * Float64(a * Float64(4.0 + a))))
	tmp = 0.0
	if (a <= -2300.0)
		tmp = t_0;
	elseif (a <= 46000000000000.0)
		tmp = fma(Float64(b * fma(b, b, 4.0)), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * N[(a * N[(4.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2300.0], t$95$0, If[LessEqual[a, 46000000000000.0], N[(N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2300 or 4.6e13 < a

   1. Initial program 43.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 91.3

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

   5. Simplified 91.3%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 91.3

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

   8. Simplified 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-*.f64 91.3

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 91.3

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

   10. Applied egg-rr 91.3%

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

   if -2300 < a < 4.6e13

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   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. lower-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. lower-*.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. lower-fma.f64 99.1

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

   5. Simplified 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.2

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

   7. Applied egg-rr 99.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2300:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(4 + a\right)\right)\right)\\ \mathbf{elif}\;a \leq 46000000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, b, 4\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot \left(4 + a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -430000:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, b, 4\right), b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -430000.0)
   (* a (* a (* a a)))
   (if (<= a 7.5e+18) (fma (* b (fma b b 4.0)) b -1.0) (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -430000.0) {
		tmp = a * (a * (a * a));
	} else if (a <= 7.5e+18) {
		tmp = fma((b * fma(b, b, 4.0)), b, -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -430000.0)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	elseif (a <= 7.5e+18)
		tmp = fma(Float64(b * fma(b, b, 4.0)), b, -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -430000.0], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7.5e+18], N[(N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.3e5

   1. Initial program 27.2%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   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. lower-*.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. lower-*.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. lower-*.f64 91.9

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

   5. Simplified 91.9%

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

   if -4.3e5 < a < 7.5e18

   1. Initial program 99.1%

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

   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. lower-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. lower-*.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. lower-fma.f64 98.3

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

   5. Simplified 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 98.4

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

   7. Applied egg-rr 98.4%

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

   if 7.5e18 < a

   1. Initial program 61.2%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   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. lower-*.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. lower-*.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. lower-*.f64 91.5

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

   5. Simplified 91.5%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 91.5

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

   7. Applied egg-rr 91.5%

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

Alternative 10: 94.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -430000:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -430000.0)
   (* a (* a (* a a)))
   (if (<= a 7.5e+18) (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 <= -430000.0) {
		tmp = a * (a * (a * a));
	} else if (a <= 7.5e+18) {
		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 <= -430000.0)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	elseif (a <= 7.5e+18)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -430000.0], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7.5e+18], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.3e5

   1. Initial program 27.2%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   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. lower-*.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. lower-*.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. lower-*.f64 91.9

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

   5. Simplified 91.9%

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

   if -4.3e5 < a < 7.5e18

   1. Initial program 99.1%

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

   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. lower-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. lower-*.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. lower-fma.f64 98.3

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

   5. Simplified 98.3%

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

   if 7.5e18 < a

   1. Initial program 61.2%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   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. lower-*.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. lower-*.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. lower-*.f64 91.5

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

   5. Simplified 91.5%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 91.5

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

   7. Applied egg-rr 91.5%

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

Alternative 11: 82.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2050:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 420:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2050.0)
   (* a (* a (* a a)))
   (if (<= a 420.0) (fma 4.0 (* b b) -1.0) (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2050.0) {
		tmp = a * (a * (a * a));
	} else if (a <= 420.0) {
		tmp = fma(4.0, (b * b), -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2050.0)
		tmp = Float64(a * Float64(a * Float64(a * a)));
	elseif (a <= 420.0)
		tmp = fma(4.0, Float64(b * b), -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2050.0], N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 420.0], N[(4.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2050

   1. Initial program 27.2%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   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. lower-*.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. lower-*.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. lower-*.f64 91.9

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

   5. Simplified 91.9%

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

   if -2050 < a < 420

   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. lower-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 74.4%

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

   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. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.4

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

   8. Simplified 74.4%

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

   if 420 < a

   1. Initial program 62.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. lower-*.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. lower-*.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. lower-*.f64 86.7

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

   5. Simplified 86.7%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 86.7

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

   7. Applied egg-rr 86.7%

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

Alternative 12: 82.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -2050:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 420:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a (* a a)))))
   (if (<= a -2050.0) t_0 (if (<= a 420.0) (fma 4.0 (* b b) -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (a * (a * a));
	double tmp;
	if (a <= -2050.0) {
		tmp = t_0;
	} else if (a <= 420.0) {
		tmp = fma(4.0, (b * b), -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 <= -2050.0)
		tmp = t_0;
	elseif (a <= 420.0)
		tmp = fma(4.0, Float64(b * b), -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, -2050.0], t$95$0, If[LessEqual[a, 420.0], N[(4.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2050 or 420 < a

   1. Initial program 45.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. lower-*.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. lower-*.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. lower-*.f64 89.3

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

   5. Simplified 89.3%

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

   if -2050 < a < 420

   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. lower-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 74.4%

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

   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. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.4

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

   8. Simplified 74.4%

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

Alternative 13: 52.0% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

3. Taylor expanded in 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. lower-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 81.8%

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

6. Taylor expanded in a around 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. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 51.4

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

8. Simplified 51.4%

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

Alternative 14: 25.1% 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 71.4%

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

3. Taylor expanded in 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. lower-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 81.8%

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

6. Taylor expanded in a around 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. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 51.4

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

8. Simplified 51.4%

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

9. Taylor expanded in b around 0

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

11. Simplified 25.5%

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

Reproduce

herbie shell --seed 2024208 
(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))