Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.6% → 99.9%

Time: 10.2s

Alternatives: 12

Speedup: 8.6×

• 61.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\\ \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right) \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \mathsf{fma}\left(a, -3, 1\right)\right)\right), t_0\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + e^{\mathsf{log1p}\left(t_0\right)}\right) + \left(-1 + \left(b \cdot b\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (pow (hypot a b) 4.0)))
   (if (<=
        (+
         (pow (+ (* a a) (* b b)) 2.0)
         (* 4.0 (+ (* (* b b) (- 1.0 (* a 3.0))) (* (* a a) (+ a 1.0)))))
        2e+293)
     (+ (fma 4.0 (fma a (fma a a a) (* b (* b (fma a -3.0 1.0)))) t_0) -1.0)
     (+ (+ -1.0 (exp (log1p t_0))) (+ -1.0 (* (* b b) 4.0))))))

C

double code(double a, double b) {
	double t_0 = pow(hypot(a, b), 4.0);
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))))) <= 2e+293) {
		tmp = fma(4.0, fma(a, fma(a, a, a), (b * (b * fma(a, -3.0, 1.0)))), t_0) + -1.0;
	} else {
		tmp = (-1.0 + exp(log1p(t_0))) + (-1.0 + ((b * b) * 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = hypot(a, b) ^ 4.0
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))) + Float64(Float64(a * a) * Float64(a + 1.0))))) <= 2e+293)
		tmp = Float64(fma(4.0, fma(a, fma(a, a, a), Float64(b * Float64(b * fma(a, -3.0, 1.0)))), t_0) + -1.0);
	else
		tmp = Float64(Float64(-1.0 + exp(log1p(t_0))) + Float64(-1.0 + Float64(Float64(b * b) * 4.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+293], N[(N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(-1.0 + N[Exp[N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))))) < 1.9999999999999998e293

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

   3. Simplified 100.0%

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

   if 1.9999999999999998e293 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a))))))

   1. Initial program 52.7%

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

      1. associate--l+ 52.7%

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

      2. fma-def 52.7%

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

   3. Simplified 58.2%

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

      1. fma-def 58.2%

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

      2. add-sqr-sqrt 58.2%

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

      3. hypot-udef 58.2%

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

      4. hypot-udef 58.2%

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

      5. pow-prod-down 58.2%

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

      6. pow-prod-up 58.2%

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

      7. metadata-eval 58.2%

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

      8. expm1-log1p-u 58.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

      9. expm1-udef 58.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 100.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{{b}^{2}} - 1\right) \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right) \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \mathsf{fma}\left(a, -3, 1\right)\right)\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)}\right) + \left(-1 + \left(b \cdot b\right) \cdot 4\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;t_0 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)}\right) + \left(-1 + \left(b \cdot b\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* b b) (- 1.0 (* a 3.0))) (* (* a a) (+ a 1.0)))))))
   (if (<= t_0 2e+293)
     (+ t_0 -1.0)
     (+
      (+ -1.0 (exp (log1p (pow (hypot a b) 4.0))))
      (+ -1.0 (* (* b b) 4.0))))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))));
	double tmp;
	if (t_0 <= 2e+293) {
		tmp = t_0 + -1.0;
	} else {
		tmp = (-1.0 + exp(log1p(pow(hypot(a, b), 4.0)))) + (-1.0 + ((b * b) * 4.0));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))));
	double tmp;
	if (t_0 <= 2e+293) {
		tmp = t_0 + -1.0;
	} else {
		tmp = (-1.0 + Math.exp(Math.log1p(Math.pow(Math.hypot(a, b), 4.0)))) + (-1.0 + ((b * b) * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))))
	tmp = 0
	if t_0 <= 2e+293:
		tmp = t_0 + -1.0
	else:
		tmp = (-1.0 + math.exp(math.log1p(math.pow(math.hypot(a, b), 4.0)))) + (-1.0 + ((b * b) * 4.0))
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))) + Float64(Float64(a * a) * Float64(a + 1.0)))))
	tmp = 0.0
	if (t_0 <= 2e+293)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(Float64(-1.0 + exp(log1p((hypot(a, b) ^ 4.0)))) + Float64(-1.0 + Float64(Float64(b * b) * 4.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+293], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(-1.0 + N[Exp[N[Log[1 + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))))) < 1.9999999999999998e293

   1. Initial program 99.7%

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

   if 1.9999999999999998e293 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a))))))

   1. Initial program 52.7%

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

      1. associate--l+ 52.7%

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

      2. fma-def 52.7%

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

   3. Simplified 58.2%

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

      1. fma-def 58.2%

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

      2. add-sqr-sqrt 58.2%

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

      3. hypot-udef 58.2%

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

      4. hypot-udef 58.2%

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

      5. pow-prod-down 58.2%

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

      6. pow-prod-up 58.2%

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

      7. metadata-eval 58.2%

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

      8. expm1-log1p-u 58.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

      9. expm1-udef 58.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 100.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{{b}^{2}} - 1\right) \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right) \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)}\right) + \left(-1 + \left(b \cdot b\right) \cdot 4\right)\\ \end{array} \]

Alternative 3: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 11.6%

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

   4. Taylor expanded in a around inf 93.2%

      \[\leadsto \color{blue}{{a}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

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

Alternative 4: 94.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1800000000 \lor \neg \left(a \leq 8.2 \cdot 10^{+21}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + \left(b \cdot b\right) \cdot 4\right) + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1800000000.0) (not (<= a 8.2e+21)))
   (pow a 4.0)
   (+ (+ -1.0 (* (* b b) 4.0)) (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1800000000.0) || !(a <= 8.2e+21)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (-1.0 + ((b * b) * 4.0)) + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1800000000.0d0)) .or. (.not. (a <= 8.2d+21))) then
        tmp = a ** 4.0d0
    else
        tmp = ((-1.0d0) + ((b * b) * 4.0d0)) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1800000000.0) || !(a <= 8.2e+21)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (-1.0 + ((b * b) * 4.0)) + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1800000000.0) or not (a <= 8.2e+21):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (-1.0 + ((b * b) * 4.0)) + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1800000000.0) || !(a <= 8.2e+21))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(-1.0 + Float64(Float64(b * b) * 4.0)) + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1800000000.0) || ~((a <= 8.2e+21)))
		tmp = a ^ 4.0;
	else
		tmp = (-1.0 + ((b * b) * 4.0)) + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1800000000.0], N[Not[LessEqual[a, 8.2e+21]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8e9 or 8.2e21 < a

   1. Initial program 47.6%

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

      1. associate--l+ 47.6%

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

      2. fma-def 47.6%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in a around inf 94.7%

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

   if -1.8e9 < a < 8.2e21

   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. Step-by-step derivation

      1. associate--l+ 99.1%

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

      2. fma-def 99.1%

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

   3. Simplified 99.1%

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

      1. fma-def 99.1%

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

      2. add-sqr-sqrt 99.1%

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

      3. hypot-udef 99.1%

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

      4. hypot-udef 99.1%

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

      5. pow-prod-down 99.1%

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

      6. pow-prod-up 99.2%

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

      7. metadata-eval 99.2%

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

      8. expm1-log1p-u 97.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

      9. expm1-udef 97.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 97.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{{b}^{2}} - 1\right) \]
   7. Step-by-step derivation

      1. unpow2 97.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 97.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   9. Taylor expanded in a around 0 99.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1800000000 \lor \neg \left(a \leq 8.2 \cdot 10^{+21}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + \left(b \cdot b\right) \cdot 4\right) + {b}^{4}\\ \end{array} \]

Alternative 5: 94.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1550000000 \lor \neg \left(a \leq 9.2 \cdot 10^{+26}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1550000000.0) (not (<= a 9.2e+26)))
   (pow a 4.0)
   (+ -1.0 (* (* b b) (+ (* b b) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1550000000.0) || !(a <= 9.2e+26)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1550000000.0d0)) .or. (.not. (a <= 9.2d+26))) then
        tmp = a ** 4.0d0
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1550000000.0) || !(a <= 9.2e+26)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1550000000.0) or not (a <= 9.2e+26):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1550000000.0) || !(a <= 9.2e+26))
		tmp = a ^ 4.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1550000000.0) || ~((a <= 9.2e+26)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1550000000.0], N[Not[LessEqual[a, 9.2e+26]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.55e9 or 9.2000000000000002e26 < a

   1. Initial program 47.6%

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

      1. associate--l+ 47.6%

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

      2. fma-def 47.6%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in a around inf 94.7%

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

   if -1.55e9 < a < 9.2000000000000002e26

   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. Step-by-step derivation

      1. associate--l+ 99.1%

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

      2. fma-def 99.1%

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

   3. Simplified 99.1%

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

      1. fma-def 99.1%

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

      2. add-sqr-sqrt 99.1%

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

      3. hypot-udef 99.1%

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

      4. hypot-udef 99.1%

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

      5. pow-prod-down 99.1%

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

      6. pow-prod-up 99.2%

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

      7. metadata-eval 99.2%

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

      8. expm1-log1p-u 97.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

      9. expm1-udef 97.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 97.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{{b}^{2}} - 1\right) \]
   7. Step-by-step derivation

      1. unpow2 97.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 97.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   9. Taylor expanded in a around 0 99.2%

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

       1. associate-+r- 99.2%

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

       2. sqr-pow 99.1%

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

       3. metadata-eval 99.1%

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

       4. pow2 99.1%

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

       5. metadata-eval 99.1%

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

       6. pow2 99.1%

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

       7. distribute-rgt-out 99.1%

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

   11. Applied egg-rr 99.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1550000000 \lor \neg \left(a \leq 9.2 \cdot 10^{+26}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 6: 58.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot 4\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -5.1 \cdot 10^{-20}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot -12\right)\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{-214}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) 4.0)))
   (if (<= a -2.5e+153)
     (* a (* a 4.0))
     (if (<= a -5.1e-20)
       (* (* b b) (* a -12.0))
       (if (<= a 9.2e-214)
         -1.0
         (if (<= a 1.05e-165)
           t_0
           (if (<= a 3e-9)
             -1.0
             (if (<= a 3.5e+102) t_0 (* 4.0 (* (* a a) (+ a 1.0)))))))))))

C

double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if (a <= -2.5e+153) {
		tmp = a * (a * 4.0);
	} else if (a <= -5.1e-20) {
		tmp = (b * b) * (a * -12.0);
	} else if (a <= 9.2e-214) {
		tmp = -1.0;
	} else if (a <= 1.05e-165) {
		tmp = t_0;
	} else if (a <= 3e-9) {
		tmp = -1.0;
	} else if (a <= 3.5e+102) {
		tmp = t_0;
	} else {
		tmp = 4.0 * ((a * a) * (a + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) * 4.0d0
    if (a <= (-2.5d+153)) then
        tmp = a * (a * 4.0d0)
    else if (a <= (-5.1d-20)) then
        tmp = (b * b) * (a * (-12.0d0))
    else if (a <= 9.2d-214) then
        tmp = -1.0d0
    else if (a <= 1.05d-165) then
        tmp = t_0
    else if (a <= 3d-9) then
        tmp = -1.0d0
    else if (a <= 3.5d+102) then
        tmp = t_0
    else
        tmp = 4.0d0 * ((a * a) * (a + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if (a <= -2.5e+153) {
		tmp = a * (a * 4.0);
	} else if (a <= -5.1e-20) {
		tmp = (b * b) * (a * -12.0);
	} else if (a <= 9.2e-214) {
		tmp = -1.0;
	} else if (a <= 1.05e-165) {
		tmp = t_0;
	} else if (a <= 3e-9) {
		tmp = -1.0;
	} else if (a <= 3.5e+102) {
		tmp = t_0;
	} else {
		tmp = 4.0 * ((a * a) * (a + 1.0));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b * b) * 4.0
	tmp = 0
	if a <= -2.5e+153:
		tmp = a * (a * 4.0)
	elif a <= -5.1e-20:
		tmp = (b * b) * (a * -12.0)
	elif a <= 9.2e-214:
		tmp = -1.0
	elif a <= 1.05e-165:
		tmp = t_0
	elif a <= 3e-9:
		tmp = -1.0
	elif a <= 3.5e+102:
		tmp = t_0
	else:
		tmp = 4.0 * ((a * a) * (a + 1.0))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * 4.0)
	tmp = 0.0
	if (a <= -2.5e+153)
		tmp = Float64(a * Float64(a * 4.0));
	elseif (a <= -5.1e-20)
		tmp = Float64(Float64(b * b) * Float64(a * -12.0));
	elseif (a <= 9.2e-214)
		tmp = -1.0;
	elseif (a <= 1.05e-165)
		tmp = t_0;
	elseif (a <= 3e-9)
		tmp = -1.0;
	elseif (a <= 3.5e+102)
		tmp = t_0;
	else
		tmp = Float64(4.0 * Float64(Float64(a * a) * Float64(a + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b * b) * 4.0;
	tmp = 0.0;
	if (a <= -2.5e+153)
		tmp = a * (a * 4.0);
	elseif (a <= -5.1e-20)
		tmp = (b * b) * (a * -12.0);
	elseif (a <= 9.2e-214)
		tmp = -1.0;
	elseif (a <= 1.05e-165)
		tmp = t_0;
	elseif (a <= 3e-9)
		tmp = -1.0;
	elseif (a <= 3.5e+102)
		tmp = t_0;
	else
		tmp = 4.0 * ((a * a) * (a + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[a, -2.5e+153], N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -5.1e-20], N[(N[(b * b), $MachinePrecision] * N[(a * -12.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.2e-214], -1.0, If[LessEqual[a, 1.05e-165], t$95$0, If[LessEqual[a, 3e-9], -1.0, If[LessEqual[a, 3.5e+102], t$95$0, N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if a < -2.50000000000000009e153

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 0.0%

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

      1. fma-def 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. hypot-udef 0.0%

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

      4. hypot-udef 0.0%

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

      5. pow-prod-down 0.0%

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

      6. pow-prod-up 0.0%

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

      7. metadata-eval 0.0%

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

      8. add-cube-cbrt 0.0%

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

      9. unpow-prod-down 0.0%

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

      10. pow2 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. *-commutative 0.0%

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

      2. metadata-eval 0.0%

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

      3. pow-sqr 0.0%

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

      4. sqr-pow 0.0%

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

      5. unpow2 0.0%

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

      6. metadata-eval 0.0%

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

      7. cube-unmult 0.0%

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

      8. metadata-eval 0.0%

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

      9. unpow2 0.0%

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

      10. pow-sqr 0.0%

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

      11. metadata-eval 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in a around inf 0.0%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} + 4 \cdot {a}^{3}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 0.0%

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

      2. *-lft-identity 0.0%

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

      3. cube-mult 0.0%

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

      4. unpow2 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. distribute-rgt-in 0.0%

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

      7. *-lft-identity 0.0%

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

      8. unpow2 0.0%

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

      9. unpow2 0.0%

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

      10. cube-mult 0.0%

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

   10. Simplified 0.0%

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

   11. Taylor expanded in a around 0 100.0%

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

       1. unpow2 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*r* 100.0%

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

       4. *-commutative 100.0%

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

   13. Simplified 100.0%

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

   if -2.50000000000000009e153 < a < -5.10000000000000019e-20

   1. Initial program 62.5%

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

      1. associate--l+ 62.5%

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

      2. fma-def 62.5%

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

   3. Simplified 81.1%

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

      1. fma-def 81.1%

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

      2. add-sqr-sqrt 81.1%

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

      3. hypot-udef 81.1%

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

      4. hypot-udef 81.1%

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

      5. pow-prod-down 81.1%

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

      6. pow-prod-up 81.4%

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

      7. metadata-eval 81.4%

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

      8. add-cube-cbrt 79.7%

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

      9. unpow-prod-down 79.6%

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

      10. pow2 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. metadata-eval 79.6%

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

      3. pow-sqr 79.6%

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

      4. sqr-pow 79.7%

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

      5. unpow2 79.7%

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

      6. metadata-eval 79.7%

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

      7. cube-unmult 79.7%

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

      8. metadata-eval 79.7%

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

      9. unpow2 79.7%

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

      10. pow-sqr 79.6%

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

      11. metadata-eval 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in b around inf 32.7%

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

      1. *-commutative 32.7%

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

      2. *-commutative 32.7%

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

      3. associate-*r* 32.7%

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

      4. unpow2 32.7%

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

      5. *-commutative 32.7%

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

      6. +-commutative 32.7%

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

      7. distribute-lft-in 32.7%

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

      8. metadata-eval 32.7%

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

      9. associate-*r* 32.7%

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

      10. metadata-eval 32.7%

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

   10. Simplified 32.7%

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

   11. Taylor expanded in a around inf 32.6%

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

       1. *-commutative 32.6%

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

   13. Simplified 32.6%

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

   if -5.10000000000000019e-20 < a < 9.20000000000000044e-214 or 1.04999999999999997e-165 < a < 2.99999999999999998e-9

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 60.0%

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

      1. associate--l+ 60.0%

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

      2. associate-*r* 60.0%

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

      3. unpow2 60.0%

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

   6. Simplified 60.0%

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

   7. Taylor expanded in a around 0 60.0%

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

   if 9.20000000000000044e-214 < a < 1.04999999999999997e-165 or 2.99999999999999998e-9 < a < 3.50000000000000011e102

   1. Initial program 84.8%

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

      1. associate--l+ 84.8%

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

      2. fma-def 84.8%

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

   3. Simplified 84.8%

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

      1. fma-def 84.8%

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

      2. add-sqr-sqrt 84.8%

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

      3. hypot-udef 84.8%

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

      4. hypot-udef 84.8%

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

      5. pow-prod-down 84.8%

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

      6. pow-prod-up 85.0%

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

      7. metadata-eval 85.0%

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

      8. add-cube-cbrt 83.9%

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

      9. unpow-prod-down 83.8%

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

      10. pow2 83.8%

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

   5. Applied egg-rr 83.8%

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

      1. *-commutative 83.8%

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

      2. metadata-eval 83.8%

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

      3. pow-sqr 83.7%

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

      4. sqr-pow 83.6%

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

      5. unpow2 83.6%

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

      6. metadata-eval 83.6%

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

      7. cube-unmult 83.6%

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

      8. metadata-eval 83.6%

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

      9. unpow2 83.6%

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

      10. pow-sqr 83.8%

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

      11. metadata-eval 83.8%

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

   7. Simplified 83.8%

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

   8. Taylor expanded in b around inf 21.9%

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

      1. *-commutative 21.9%

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

      2. *-commutative 21.9%

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

      3. associate-*r* 21.9%

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

      4. unpow2 21.9%

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

      5. *-commutative 21.9%

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

      6. +-commutative 21.9%

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

      7. distribute-lft-in 21.9%

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

      8. metadata-eval 21.9%

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

      9. associate-*r* 21.9%

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

      10. metadata-eval 21.9%

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

   10. Simplified 21.9%

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

   11. Taylor expanded in a around 0 38.0%

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

   if 3.50000000000000011e102 < a

   1. Initial program 70.3%

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

      1. associate--l+ 70.3%

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

      2. fma-def 70.3%

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

   3. Simplified 70.3%

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

      1. fma-def 70.3%

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

      2. add-sqr-sqrt 70.3%

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

      3. hypot-udef 70.3%

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

      4. hypot-udef 70.3%

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

      5. pow-prod-down 70.3%

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

      6. pow-prod-up 70.3%

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

      7. metadata-eval 70.3%

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

      8. add-cube-cbrt 70.3%

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

      9. unpow-prod-down 70.3%

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

      10. pow2 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. *-commutative 70.3%

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

      2. metadata-eval 70.3%

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

      3. pow-sqr 70.3%

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

      4. sqr-pow 70.3%

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

      5. unpow2 70.3%

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

      6. metadata-eval 70.3%

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

      7. cube-unmult 70.3%

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

      8. metadata-eval 70.3%

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

      9. unpow2 70.3%

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

      10. pow-sqr 70.3%

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

      11. metadata-eval 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in a around inf 100.0%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} + 4 \cdot {a}^{3}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

      2. *-lft-identity 100.0%

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

      3. cube-mult 100.0%

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

      4. unpow2 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-lft-identity 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. cube-mult 100.0%

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

   10. Simplified 100.0%

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

       1. cube-mult 100.0%

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

       2. distribute-rgt1-in 100.0%

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

   12. Applied egg-rr 100.0%

       \[\leadsto 4 \cdot \color{blue}{\left(\left(a + 1\right) \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -5.1 \cdot 10^{-20}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot -12\right)\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{-214}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-165}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\\ \end{array} \]

Alternative 7: 58.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot 4\\ \mathbf{if}\;a \leq -5 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -2.25 \cdot 10^{-20}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) 4.0)))
   (if (<= a -5e+153)
     (* a (* a 4.0))
     (if (<= a -2.25e-20)
       (* (* b b) (+ 4.0 (* a -12.0)))
       (if (<= a 1.25e-213)
         -1.0
         (if (<= a 1.05e-165)
           t_0
           (if (<= a 3e-9)
             -1.0
             (if (<= a 1.15e+101) t_0 (* 4.0 (* (* a a) (+ a 1.0)))))))))))

C

double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if (a <= -5e+153) {
		tmp = a * (a * 4.0);
	} else if (a <= -2.25e-20) {
		tmp = (b * b) * (4.0 + (a * -12.0));
	} else if (a <= 1.25e-213) {
		tmp = -1.0;
	} else if (a <= 1.05e-165) {
		tmp = t_0;
	} else if (a <= 3e-9) {
		tmp = -1.0;
	} else if (a <= 1.15e+101) {
		tmp = t_0;
	} else {
		tmp = 4.0 * ((a * a) * (a + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) * 4.0d0
    if (a <= (-5d+153)) then
        tmp = a * (a * 4.0d0)
    else if (a <= (-2.25d-20)) then
        tmp = (b * b) * (4.0d0 + (a * (-12.0d0)))
    else if (a <= 1.25d-213) then
        tmp = -1.0d0
    else if (a <= 1.05d-165) then
        tmp = t_0
    else if (a <= 3d-9) then
        tmp = -1.0d0
    else if (a <= 1.15d+101) then
        tmp = t_0
    else
        tmp = 4.0d0 * ((a * a) * (a + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if (a <= -5e+153) {
		tmp = a * (a * 4.0);
	} else if (a <= -2.25e-20) {
		tmp = (b * b) * (4.0 + (a * -12.0));
	} else if (a <= 1.25e-213) {
		tmp = -1.0;
	} else if (a <= 1.05e-165) {
		tmp = t_0;
	} else if (a <= 3e-9) {
		tmp = -1.0;
	} else if (a <= 1.15e+101) {
		tmp = t_0;
	} else {
		tmp = 4.0 * ((a * a) * (a + 1.0));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b * b) * 4.0
	tmp = 0
	if a <= -5e+153:
		tmp = a * (a * 4.0)
	elif a <= -2.25e-20:
		tmp = (b * b) * (4.0 + (a * -12.0))
	elif a <= 1.25e-213:
		tmp = -1.0
	elif a <= 1.05e-165:
		tmp = t_0
	elif a <= 3e-9:
		tmp = -1.0
	elif a <= 1.15e+101:
		tmp = t_0
	else:
		tmp = 4.0 * ((a * a) * (a + 1.0))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * 4.0)
	tmp = 0.0
	if (a <= -5e+153)
		tmp = Float64(a * Float64(a * 4.0));
	elseif (a <= -2.25e-20)
		tmp = Float64(Float64(b * b) * Float64(4.0 + Float64(a * -12.0)));
	elseif (a <= 1.25e-213)
		tmp = -1.0;
	elseif (a <= 1.05e-165)
		tmp = t_0;
	elseif (a <= 3e-9)
		tmp = -1.0;
	elseif (a <= 1.15e+101)
		tmp = t_0;
	else
		tmp = Float64(4.0 * Float64(Float64(a * a) * Float64(a + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b * b) * 4.0;
	tmp = 0.0;
	if (a <= -5e+153)
		tmp = a * (a * 4.0);
	elseif (a <= -2.25e-20)
		tmp = (b * b) * (4.0 + (a * -12.0));
	elseif (a <= 1.25e-213)
		tmp = -1.0;
	elseif (a <= 1.05e-165)
		tmp = t_0;
	elseif (a <= 3e-9)
		tmp = -1.0;
	elseif (a <= 1.15e+101)
		tmp = t_0;
	else
		tmp = 4.0 * ((a * a) * (a + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[a, -5e+153], N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -2.25e-20], N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(a * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.25e-213], -1.0, If[LessEqual[a, 1.05e-165], t$95$0, If[LessEqual[a, 3e-9], -1.0, If[LessEqual[a, 1.15e+101], t$95$0, N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if a < -5.00000000000000018e153

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 0.0%

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

      1. fma-def 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. hypot-udef 0.0%

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

      4. hypot-udef 0.0%

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

      5. pow-prod-down 0.0%

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

      6. pow-prod-up 0.0%

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

      7. metadata-eval 0.0%

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

      8. add-cube-cbrt 0.0%

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

      9. unpow-prod-down 0.0%

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

      10. pow2 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. *-commutative 0.0%

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

      2. metadata-eval 0.0%

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

      3. pow-sqr 0.0%

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

      4. sqr-pow 0.0%

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

      5. unpow2 0.0%

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

      6. metadata-eval 0.0%

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

      7. cube-unmult 0.0%

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

      8. metadata-eval 0.0%

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

      9. unpow2 0.0%

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

      10. pow-sqr 0.0%

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

      11. metadata-eval 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in a around inf 0.0%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} + 4 \cdot {a}^{3}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 0.0%

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

      2. *-lft-identity 0.0%

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

      3. cube-mult 0.0%

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

      4. unpow2 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. distribute-rgt-in 0.0%

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

      7. *-lft-identity 0.0%

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

      8. unpow2 0.0%

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

      9. unpow2 0.0%

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

      10. cube-mult 0.0%

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

   10. Simplified 0.0%

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

   11. Taylor expanded in a around 0 100.0%

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

       1. unpow2 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*r* 100.0%

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

       4. *-commutative 100.0%

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

   13. Simplified 100.0%

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

   if -5.00000000000000018e153 < a < -2.2500000000000001e-20

   1. Initial program 62.5%

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

      1. associate--l+ 62.5%

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

      2. fma-def 62.5%

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

   3. Simplified 81.1%

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

      1. fma-def 81.1%

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

      2. add-sqr-sqrt 81.1%

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

      3. hypot-udef 81.1%

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

      4. hypot-udef 81.1%

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

      5. pow-prod-down 81.1%

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

      6. pow-prod-up 81.4%

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

      7. metadata-eval 81.4%

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

      8. add-cube-cbrt 79.7%

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

      9. unpow-prod-down 79.6%

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

      10. pow2 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. metadata-eval 79.6%

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

      3. pow-sqr 79.6%

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

      4. sqr-pow 79.7%

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

      5. unpow2 79.7%

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

      6. metadata-eval 79.7%

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

      7. cube-unmult 79.7%

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

      8. metadata-eval 79.7%

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

      9. unpow2 79.7%

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

      10. pow-sqr 79.6%

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

      11. metadata-eval 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in b around inf 32.7%

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

      1. *-commutative 32.7%

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

      2. *-commutative 32.7%

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

      3. associate-*r* 32.7%

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

      4. unpow2 32.7%

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

      5. *-commutative 32.7%

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

      6. +-commutative 32.7%

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

      7. distribute-lft-in 32.7%

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

      8. metadata-eval 32.7%

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

      9. associate-*r* 32.7%

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

      10. metadata-eval 32.7%

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

   10. Simplified 32.7%

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

   if -2.2500000000000001e-20 < a < 1.24999999999999994e-213 or 1.04999999999999997e-165 < a < 2.99999999999999998e-9

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 60.0%

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

      1. associate--l+ 60.0%

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

      2. associate-*r* 60.0%

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

      3. unpow2 60.0%

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

   6. Simplified 60.0%

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

   7. Taylor expanded in a around 0 60.0%

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

   if 1.24999999999999994e-213 < a < 1.04999999999999997e-165 or 2.99999999999999998e-9 < a < 1.1500000000000001e101

   1. Initial program 84.8%

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

      1. associate--l+ 84.8%

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

      2. fma-def 84.8%

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

   3. Simplified 84.8%

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

      1. fma-def 84.8%

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

      2. add-sqr-sqrt 84.8%

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

      3. hypot-udef 84.8%

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

      4. hypot-udef 84.8%

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

      5. pow-prod-down 84.8%

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

      6. pow-prod-up 85.0%

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

      7. metadata-eval 85.0%

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

      8. add-cube-cbrt 83.9%

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

      9. unpow-prod-down 83.8%

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

      10. pow2 83.8%

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

   5. Applied egg-rr 83.8%

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

      1. *-commutative 83.8%

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

      2. metadata-eval 83.8%

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

      3. pow-sqr 83.7%

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

      4. sqr-pow 83.6%

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

      5. unpow2 83.6%

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

      6. metadata-eval 83.6%

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

      7. cube-unmult 83.6%

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

      8. metadata-eval 83.6%

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

      9. unpow2 83.6%

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

      10. pow-sqr 83.8%

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

      11. metadata-eval 83.8%

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

   7. Simplified 83.8%

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

   8. Taylor expanded in b around inf 21.9%

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

      1. *-commutative 21.9%

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

      2. *-commutative 21.9%

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

      3. associate-*r* 21.9%

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

      4. unpow2 21.9%

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

      5. *-commutative 21.9%

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

      6. +-commutative 21.9%

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

      7. distribute-lft-in 21.9%

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

      8. metadata-eval 21.9%

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

      9. associate-*r* 21.9%

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

      10. metadata-eval 21.9%

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

   10. Simplified 21.9%

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

   11. Taylor expanded in a around 0 38.0%

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

   if 1.1500000000000001e101 < a

   1. Initial program 70.3%

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

      1. associate--l+ 70.3%

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

      2. fma-def 70.3%

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

   3. Simplified 70.3%

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

      1. fma-def 70.3%

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

      2. add-sqr-sqrt 70.3%

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

      3. hypot-udef 70.3%

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

      4. hypot-udef 70.3%

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

      5. pow-prod-down 70.3%

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

      6. pow-prod-up 70.3%

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

      7. metadata-eval 70.3%

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

      8. add-cube-cbrt 70.3%

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

      9. unpow-prod-down 70.3%

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

      10. pow2 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. *-commutative 70.3%

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

      2. metadata-eval 70.3%

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

      3. pow-sqr 70.3%

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

      4. sqr-pow 70.3%

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

      5. unpow2 70.3%

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

      6. metadata-eval 70.3%

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

      7. cube-unmult 70.3%

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

      8. metadata-eval 70.3%

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

      9. unpow2 70.3%

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

      10. pow-sqr 70.3%

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

      11. metadata-eval 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in a around inf 100.0%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} + 4 \cdot {a}^{3}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

      2. *-lft-identity 100.0%

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

      3. cube-mult 100.0%

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

      4. unpow2 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-lft-identity 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. cube-mult 100.0%

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

   10. Simplified 100.0%

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

       1. cube-mult 100.0%

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

       2. distribute-rgt1-in 100.0%

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

   12. Applied egg-rr 100.0%

       \[\leadsto 4 \cdot \color{blue}{\left(\left(a + 1\right) \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -2.25 \cdot 10^{-20}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-165}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+101}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\\ \end{array} \]

Alternative 8: 54.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot 4\right)\\ t_1 := \left(b \cdot b\right) \cdot 4\\ \mathbf{if}\;a \leq -1.65 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -5.1 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a 4.0))) (t_1 (* (* b b) 4.0)))
   (if (<= a -1.65e+153)
     t_0
     (if (<= a -5.1e-20)
       t_1
       (if (<= a 1.25e-213)
         -1.0
         (if (<= a 4.5e-165)
           t_1
           (if (<= a 2.9e-9) -1.0 (if (<= a 1.1e+123) t_1 t_0))))))))

C

double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double t_1 = (b * b) * 4.0;
	double tmp;
	if (a <= -1.65e+153) {
		tmp = t_0;
	} else if (a <= -5.1e-20) {
		tmp = t_1;
	} else if (a <= 1.25e-213) {
		tmp = -1.0;
	} else if (a <= 4.5e-165) {
		tmp = t_1;
	} else if (a <= 2.9e-9) {
		tmp = -1.0;
	} else if (a <= 1.1e+123) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (a * 4.0d0)
    t_1 = (b * b) * 4.0d0
    if (a <= (-1.65d+153)) then
        tmp = t_0
    else if (a <= (-5.1d-20)) then
        tmp = t_1
    else if (a <= 1.25d-213) then
        tmp = -1.0d0
    else if (a <= 4.5d-165) then
        tmp = t_1
    else if (a <= 2.9d-9) then
        tmp = -1.0d0
    else if (a <= 1.1d+123) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double t_1 = (b * b) * 4.0;
	double tmp;
	if (a <= -1.65e+153) {
		tmp = t_0;
	} else if (a <= -5.1e-20) {
		tmp = t_1;
	} else if (a <= 1.25e-213) {
		tmp = -1.0;
	} else if (a <= 4.5e-165) {
		tmp = t_1;
	} else if (a <= 2.9e-9) {
		tmp = -1.0;
	} else if (a <= 1.1e+123) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * 4.0)
	t_1 = (b * b) * 4.0
	tmp = 0
	if a <= -1.65e+153:
		tmp = t_0
	elif a <= -5.1e-20:
		tmp = t_1
	elif a <= 1.25e-213:
		tmp = -1.0
	elif a <= 4.5e-165:
		tmp = t_1
	elif a <= 2.9e-9:
		tmp = -1.0
	elif a <= 1.1e+123:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * 4.0))
	t_1 = Float64(Float64(b * b) * 4.0)
	tmp = 0.0
	if (a <= -1.65e+153)
		tmp = t_0;
	elseif (a <= -5.1e-20)
		tmp = t_1;
	elseif (a <= 1.25e-213)
		tmp = -1.0;
	elseif (a <= 4.5e-165)
		tmp = t_1;
	elseif (a <= 2.9e-9)
		tmp = -1.0;
	elseif (a <= 1.1e+123)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * 4.0);
	t_1 = (b * b) * 4.0;
	tmp = 0.0;
	if (a <= -1.65e+153)
		tmp = t_0;
	elseif (a <= -5.1e-20)
		tmp = t_1;
	elseif (a <= 1.25e-213)
		tmp = -1.0;
	elseif (a <= 4.5e-165)
		tmp = t_1;
	elseif (a <= 2.9e-9)
		tmp = -1.0;
	elseif (a <= 1.1e+123)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[a, -1.65e+153], t$95$0, If[LessEqual[a, -5.1e-20], t$95$1, If[LessEqual[a, 1.25e-213], -1.0, If[LessEqual[a, 4.5e-165], t$95$1, If[LessEqual[a, 2.9e-9], -1.0, If[LessEqual[a, 1.1e+123], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.64999999999999997e153 or 1.09999999999999996e123 < a

   1. Initial program 33.8%

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

      1. associate--l+ 33.8%

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

      2. fma-def 33.8%

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

   3. Simplified 33.8%

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

      1. fma-def 33.8%

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

      2. add-sqr-sqrt 33.8%

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

      3. hypot-udef 33.8%

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

      4. hypot-udef 33.8%

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

      5. pow-prod-down 33.8%

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

      6. pow-prod-up 33.8%

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

      7. metadata-eval 33.8%

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

      8. add-cube-cbrt 33.8%

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

      9. unpow-prod-down 33.8%

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

      10. pow2 33.8%

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

   5. Applied egg-rr 33.8%

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

      1. *-commutative 33.8%

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

      2. metadata-eval 33.8%

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

      3. pow-sqr 33.8%

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

      4. sqr-pow 33.8%

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

      5. unpow2 33.8%

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

      6. metadata-eval 33.8%

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

      7. cube-unmult 33.8%

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

      8. metadata-eval 33.8%

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

      9. unpow2 33.8%

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

      10. pow-sqr 33.8%

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

      11. metadata-eval 33.8%

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

   7. Simplified 33.8%

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

   8. Taylor expanded in a around inf 47.1%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} + 4 \cdot {a}^{3}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 47.1%

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

      2. *-lft-identity 47.1%

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

      3. cube-mult 47.1%

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

      4. unpow2 47.1%

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

      5. distribute-rgt-in 47.1%

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

      6. distribute-rgt-in 47.1%

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

      7. *-lft-identity 47.1%

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

      8. unpow2 47.1%

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

      9. unpow2 47.1%

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

      10. cube-mult 47.1%

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

   10. Simplified 47.1%

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

   11. Taylor expanded in a around 0 93.2%

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

       1. unpow2 93.2%

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

       2. *-commutative 93.2%

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

       3. associate-*r* 93.2%

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

       4. *-commutative 93.2%

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

   13. Simplified 93.2%

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

   if -1.64999999999999997e153 < a < -5.10000000000000019e-20 or 1.24999999999999994e-213 < a < 4.49999999999999992e-165 or 2.89999999999999991e-9 < a < 1.09999999999999996e123

   1. Initial program 72.5%

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

      1. associate--l+ 72.5%

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

      2. fma-def 72.5%

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

   3. Simplified 81.6%

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

      1. fma-def 81.6%

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

      2. add-sqr-sqrt 81.6%

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

      3. hypot-udef 81.6%

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

      4. hypot-udef 81.6%

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

      5. pow-prod-down 81.6%

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

      6. pow-prod-up 81.8%

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

      7. metadata-eval 81.8%

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

      8. add-cube-cbrt 80.5%

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

      9. unpow-prod-down 80.4%

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

      10. pow2 80.4%

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

   5. Applied egg-rr 80.4%

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

      1. *-commutative 80.4%

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

      2. metadata-eval 80.4%

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

      3. pow-sqr 80.4%

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

      4. sqr-pow 80.4%

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

      5. unpow2 80.4%

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

      6. metadata-eval 80.4%

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

      7. cube-unmult 80.4%

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

      8. metadata-eval 80.4%

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

      9. unpow2 80.4%

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

      10. pow-sqr 80.4%

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

      11. metadata-eval 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in b around inf 25.9%

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

      1. *-commutative 25.9%

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

      2. *-commutative 25.9%

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

      3. associate-*r* 25.9%

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

      4. unpow2 25.9%

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

      5. *-commutative 25.9%

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

      6. +-commutative 25.9%

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

      7. distribute-lft-in 25.9%

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

      8. metadata-eval 25.9%

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

      9. associate-*r* 25.9%

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

      10. metadata-eval 25.9%

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

   10. Simplified 25.9%

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

   11. Taylor expanded in a around 0 32.4%

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

   if -5.10000000000000019e-20 < a < 1.24999999999999994e-213 or 4.49999999999999992e-165 < a < 2.89999999999999991e-9

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 60.0%

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

      1. associate--l+ 60.0%

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

      2. associate-*r* 60.0%

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

      3. unpow2 60.0%

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

   6. Simplified 60.0%

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

   7. Taylor expanded in a around 0 60.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -5.1 \cdot 10^{-20}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-165}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+123}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 9: 55.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot 4\right)\\ t_1 := \left(b \cdot b\right) \cdot 4\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -5.1 \cdot 10^{-20}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot -12\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.82 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a 4.0))) (t_1 (* (* b b) 4.0)))
   (if (<= a -6.8e+153)
     t_0
     (if (<= a -5.1e-20)
       (* (* b b) (* a -12.0))
       (if (<= a 1.25e-213)
         -1.0
         (if (<= a 1.05e-165)
           t_1
           (if (<= a 1.82e-9) -1.0 (if (<= a 1.1e+123) t_1 t_0))))))))

C

double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double t_1 = (b * b) * 4.0;
	double tmp;
	if (a <= -6.8e+153) {
		tmp = t_0;
	} else if (a <= -5.1e-20) {
		tmp = (b * b) * (a * -12.0);
	} else if (a <= 1.25e-213) {
		tmp = -1.0;
	} else if (a <= 1.05e-165) {
		tmp = t_1;
	} else if (a <= 1.82e-9) {
		tmp = -1.0;
	} else if (a <= 1.1e+123) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (a * 4.0d0)
    t_1 = (b * b) * 4.0d0
    if (a <= (-6.8d+153)) then
        tmp = t_0
    else if (a <= (-5.1d-20)) then
        tmp = (b * b) * (a * (-12.0d0))
    else if (a <= 1.25d-213) then
        tmp = -1.0d0
    else if (a <= 1.05d-165) then
        tmp = t_1
    else if (a <= 1.82d-9) then
        tmp = -1.0d0
    else if (a <= 1.1d+123) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double t_1 = (b * b) * 4.0;
	double tmp;
	if (a <= -6.8e+153) {
		tmp = t_0;
	} else if (a <= -5.1e-20) {
		tmp = (b * b) * (a * -12.0);
	} else if (a <= 1.25e-213) {
		tmp = -1.0;
	} else if (a <= 1.05e-165) {
		tmp = t_1;
	} else if (a <= 1.82e-9) {
		tmp = -1.0;
	} else if (a <= 1.1e+123) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * 4.0)
	t_1 = (b * b) * 4.0
	tmp = 0
	if a <= -6.8e+153:
		tmp = t_0
	elif a <= -5.1e-20:
		tmp = (b * b) * (a * -12.0)
	elif a <= 1.25e-213:
		tmp = -1.0
	elif a <= 1.05e-165:
		tmp = t_1
	elif a <= 1.82e-9:
		tmp = -1.0
	elif a <= 1.1e+123:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * 4.0))
	t_1 = Float64(Float64(b * b) * 4.0)
	tmp = 0.0
	if (a <= -6.8e+153)
		tmp = t_0;
	elseif (a <= -5.1e-20)
		tmp = Float64(Float64(b * b) * Float64(a * -12.0));
	elseif (a <= 1.25e-213)
		tmp = -1.0;
	elseif (a <= 1.05e-165)
		tmp = t_1;
	elseif (a <= 1.82e-9)
		tmp = -1.0;
	elseif (a <= 1.1e+123)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * 4.0);
	t_1 = (b * b) * 4.0;
	tmp = 0.0;
	if (a <= -6.8e+153)
		tmp = t_0;
	elseif (a <= -5.1e-20)
		tmp = (b * b) * (a * -12.0);
	elseif (a <= 1.25e-213)
		tmp = -1.0;
	elseif (a <= 1.05e-165)
		tmp = t_1;
	elseif (a <= 1.82e-9)
		tmp = -1.0;
	elseif (a <= 1.1e+123)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[a, -6.8e+153], t$95$0, If[LessEqual[a, -5.1e-20], N[(N[(b * b), $MachinePrecision] * N[(a * -12.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.25e-213], -1.0, If[LessEqual[a, 1.05e-165], t$95$1, If[LessEqual[a, 1.82e-9], -1.0, If[LessEqual[a, 1.1e+123], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.7999999999999995e153 or 1.09999999999999996e123 < a

   1. Initial program 33.8%

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

      1. associate--l+ 33.8%

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

      2. fma-def 33.8%

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

   3. Simplified 33.8%

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

      1. fma-def 33.8%

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

      2. add-sqr-sqrt 33.8%

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

      3. hypot-udef 33.8%

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

      4. hypot-udef 33.8%

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

      5. pow-prod-down 33.8%

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

      6. pow-prod-up 33.8%

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

      7. metadata-eval 33.8%

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

      8. add-cube-cbrt 33.8%

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

      9. unpow-prod-down 33.8%

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

      10. pow2 33.8%

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

   5. Applied egg-rr 33.8%

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

      1. *-commutative 33.8%

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

      2. metadata-eval 33.8%

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

      3. pow-sqr 33.8%

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

      4. sqr-pow 33.8%

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

      5. unpow2 33.8%

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

      6. metadata-eval 33.8%

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

      7. cube-unmult 33.8%

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

      8. metadata-eval 33.8%

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

      9. unpow2 33.8%

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

      10. pow-sqr 33.8%

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

      11. metadata-eval 33.8%

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

   7. Simplified 33.8%

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

   8. Taylor expanded in a around inf 47.1%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} + 4 \cdot {a}^{3}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 47.1%

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

      2. *-lft-identity 47.1%

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

      3. cube-mult 47.1%

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

      4. unpow2 47.1%

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

      5. distribute-rgt-in 47.1%

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

      6. distribute-rgt-in 47.1%

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

      7. *-lft-identity 47.1%

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

      8. unpow2 47.1%

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

      9. unpow2 47.1%

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

      10. cube-mult 47.1%

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

   10. Simplified 47.1%

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

   11. Taylor expanded in a around 0 93.2%

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

       1. unpow2 93.2%

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

       2. *-commutative 93.2%

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

       3. associate-*r* 93.2%

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

       4. *-commutative 93.2%

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

   13. Simplified 93.2%

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

   if -6.7999999999999995e153 < a < -5.10000000000000019e-20

   1. Initial program 62.5%

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

      1. associate--l+ 62.5%

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

      2. fma-def 62.5%

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

   3. Simplified 81.1%

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

      1. fma-def 81.1%

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

      2. add-sqr-sqrt 81.1%

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

      3. hypot-udef 81.1%

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

      4. hypot-udef 81.1%

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

      5. pow-prod-down 81.1%

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

      6. pow-prod-up 81.4%

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

      7. metadata-eval 81.4%

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

      8. add-cube-cbrt 79.7%

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

      9. unpow-prod-down 79.6%

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

      10. pow2 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. metadata-eval 79.6%

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

      3. pow-sqr 79.6%

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

      4. sqr-pow 79.7%

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

      5. unpow2 79.7%

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

      6. metadata-eval 79.7%

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

      7. cube-unmult 79.7%

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

      8. metadata-eval 79.7%

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

      9. unpow2 79.7%

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

      10. pow-sqr 79.6%

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

      11. metadata-eval 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in b around inf 32.7%

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

      1. *-commutative 32.7%

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

      2. *-commutative 32.7%

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

      3. associate-*r* 32.7%

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

      4. unpow2 32.7%

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

      5. *-commutative 32.7%

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

      6. +-commutative 32.7%

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

      7. distribute-lft-in 32.7%

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

      8. metadata-eval 32.7%

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

      9. associate-*r* 32.7%

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

      10. metadata-eval 32.7%

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

   10. Simplified 32.7%

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

   11. Taylor expanded in a around inf 32.6%

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

       1. *-commutative 32.6%

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

   13. Simplified 32.6%

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

   if -5.10000000000000019e-20 < a < 1.24999999999999994e-213 or 1.04999999999999997e-165 < a < 1.8199999999999999e-9

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 60.0%

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

      1. associate--l+ 60.0%

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

      2. associate-*r* 60.0%

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

      3. unpow2 60.0%

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

   6. Simplified 60.0%

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

   7. Taylor expanded in a around 0 60.0%

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

   if 1.24999999999999994e-213 < a < 1.04999999999999997e-165 or 1.8199999999999999e-9 < a < 1.09999999999999996e123

   1. Initial program 82.0%

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

      1. associate--l+ 82.0%

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

      2. fma-def 82.0%

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

   3. Simplified 82.0%

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

      1. fma-def 82.0%

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

      2. add-sqr-sqrt 82.0%

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

      3. hypot-udef 82.0%

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

      4. hypot-udef 82.0%

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

      5. pow-prod-down 82.0%

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

      6. pow-prod-up 82.2%

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

      7. metadata-eval 82.2%

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

      8. add-cube-cbrt 81.2%

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

      9. unpow-prod-down 81.2%

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

      10. pow2 81.2%

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

   5. Applied egg-rr 81.2%

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

      1. *-commutative 81.2%

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

      2. metadata-eval 81.2%

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

      3. pow-sqr 81.1%

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

      4. sqr-pow 81.0%

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

      5. unpow2 81.0%

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

      6. metadata-eval 81.0%

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

      7. cube-unmult 81.0%

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

      8. metadata-eval 81.0%

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

      9. unpow2 81.0%

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

      10. pow-sqr 81.2%

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

      11. metadata-eval 81.2%

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

   7. Simplified 81.2%

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

   8. Taylor expanded in b around inf 19.5%

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

      1. *-commutative 19.5%

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

      2. *-commutative 19.5%

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

      3. associate-*r* 19.5%

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

      4. unpow2 19.5%

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

      5. *-commutative 19.5%

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

      6. +-commutative 19.5%

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

      7. distribute-lft-in 19.5%

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

      8. metadata-eval 19.5%

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

      9. associate-*r* 19.5%

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

      10. metadata-eval 19.5%

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

   10. Simplified 19.5%

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

   11. Taylor expanded in a around 0 36.4%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -5.1 \cdot 10^{-20}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot -12\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-213}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-165}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{elif}\;a \leq 1.82 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+123}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 10: 87.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+102}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.2e+153)
   (* a (* a 4.0))
   (if (<= a 3.4e+102)
     (+ -1.0 (* (* b b) (+ (* b b) 4.0)))
     (* 4.0 (* (* a a) (+ a 1.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.2e+153) {
		tmp = a * (a * 4.0);
	} else if (a <= 3.4e+102) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} else {
		tmp = 4.0 * ((a * a) * (a + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-6.2d+153)) then
        tmp = a * (a * 4.0d0)
    else if (a <= 3.4d+102) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 4.0d0))
    else
        tmp = 4.0d0 * ((a * a) * (a + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.2e+153) {
		tmp = a * (a * 4.0);
	} else if (a <= 3.4e+102) {
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	} else {
		tmp = 4.0 * ((a * a) * (a + 1.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.2e+153:
		tmp = a * (a * 4.0)
	elif a <= 3.4e+102:
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0))
	else:
		tmp = 4.0 * ((a * a) * (a + 1.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.2e+153)
		tmp = Float64(a * Float64(a * 4.0));
	elseif (a <= 3.4e+102)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0)));
	else
		tmp = Float64(4.0 * Float64(Float64(a * a) * Float64(a + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.2e+153)
		tmp = a * (a * 4.0);
	elseif (a <= 3.4e+102)
		tmp = -1.0 + ((b * b) * ((b * b) + 4.0));
	else
		tmp = 4.0 * ((a * a) * (a + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.2e+153], N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.4e+102], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.2e153

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 0.0%

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

      1. fma-def 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. hypot-udef 0.0%

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

      4. hypot-udef 0.0%

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

      5. pow-prod-down 0.0%

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

      6. pow-prod-up 0.0%

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

      7. metadata-eval 0.0%

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

      8. add-cube-cbrt 0.0%

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

      9. unpow-prod-down 0.0%

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

      10. pow2 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. *-commutative 0.0%

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

      2. metadata-eval 0.0%

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

      3. pow-sqr 0.0%

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

      4. sqr-pow 0.0%

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

      5. unpow2 0.0%

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

      6. metadata-eval 0.0%

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

      7. cube-unmult 0.0%

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

      8. metadata-eval 0.0%

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

      9. unpow2 0.0%

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

      10. pow-sqr 0.0%

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

      11. metadata-eval 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in a around inf 0.0%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} + 4 \cdot {a}^{3}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 0.0%

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

      2. *-lft-identity 0.0%

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

      3. cube-mult 0.0%

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

      4. unpow2 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. distribute-rgt-in 0.0%

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

      7. *-lft-identity 0.0%

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

      8. unpow2 0.0%

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

      9. unpow2 0.0%

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

      10. cube-mult 0.0%

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

   10. Simplified 0.0%

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

   11. Taylor expanded in a around 0 100.0%

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

       1. unpow2 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*r* 100.0%

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

       4. *-commutative 100.0%

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

   13. Simplified 100.0%

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

   if -6.2e153 < a < 3.4e102

   1. Initial program 87.8%

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

      1. associate--l+ 87.8%

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

      2. fma-def 87.8%

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

   3. Simplified 92.2%

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

      1. fma-def 92.2%

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

      2. add-sqr-sqrt 92.2%

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

      3. hypot-udef 92.2%

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

      4. hypot-udef 92.2%

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

      5. pow-prod-down 92.2%

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

      6. pow-prod-up 92.3%

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

      7. metadata-eval 92.3%

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

      8. expm1-log1p-u 90.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

      9. expm1-udef 90.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 97.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{{b}^{2}} - 1\right) \]
   7. Step-by-step derivation

      1. unpow2 97.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 97.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right) + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   9. Taylor expanded in a around 0 78.9%

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

       1. associate-+r- 78.9%

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

       2. sqr-pow 78.8%

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

       3. metadata-eval 78.8%

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

       4. pow2 78.8%

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

       5. metadata-eval 78.8%

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

       6. pow2 78.8%

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

       7. distribute-rgt-out 78.8%

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

   11. Applied egg-rr 78.8%

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

   if 3.4e102 < a

   1. Initial program 70.3%

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

      1. associate--l+ 70.3%

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

      2. fma-def 70.3%

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

   3. Simplified 70.3%

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

      1. fma-def 70.3%

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

      2. add-sqr-sqrt 70.3%

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

      3. hypot-udef 70.3%

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

      4. hypot-udef 70.3%

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

      5. pow-prod-down 70.3%

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

      6. pow-prod-up 70.3%

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

      7. metadata-eval 70.3%

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

      8. add-cube-cbrt 70.3%

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

      9. unpow-prod-down 70.3%

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

      10. pow2 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. *-commutative 70.3%

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

      2. metadata-eval 70.3%

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

      3. pow-sqr 70.3%

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

      4. sqr-pow 70.3%

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

      5. unpow2 70.3%

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

      6. metadata-eval 70.3%

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

      7. cube-unmult 70.3%

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

      8. metadata-eval 70.3%

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

      9. unpow2 70.3%

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

      10. pow-sqr 70.3%

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

      11. metadata-eval 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in a around inf 100.0%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} + 4 \cdot {a}^{3}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

      2. *-lft-identity 100.0%

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

      3. cube-mult 100.0%

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

      4. unpow2 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-lft-identity 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. cube-mult 100.0%

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

   10. Simplified 100.0%

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

       1. cube-mult 100.0%

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

       2. distribute-rgt1-in 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+102}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\\ \end{array} \]

Alternative 11: 49.8% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{-20} \lor \neg \left(a \leq 3 \cdot 10^{-9}\right):\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -5.1e-20) (not (<= a 3e-9))) (* a (* a 4.0)) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -5.1e-20) || !(a <= 3e-9)) {
		tmp = a * (a * 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-5.1d-20)) .or. (.not. (a <= 3d-9))) then
        tmp = a * (a * 4.0d0)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -5.1e-20) || !(a <= 3e-9)) {
		tmp = a * (a * 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -5.1e-20) or not (a <= 3e-9):
		tmp = a * (a * 4.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -5.1e-20) || !(a <= 3e-9))
		tmp = Float64(a * Float64(a * 4.0));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -5.1e-20) || ~((a <= 3e-9)))
		tmp = a * (a * 4.0);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -5.1e-20], N[Not[LessEqual[a, 3e-9]], $MachinePrecision]], N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -5.10000000000000019e-20 or 2.99999999999999998e-9 < a

   1. Initial program 52.0%

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

      1. associate--l+ 52.0%

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

      2. fma-def 52.0%

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

   3. Simplified 57.5%

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

      1. fma-def 57.5%

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

      2. add-sqr-sqrt 57.5%

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

      3. hypot-udef 57.5%

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

      4. hypot-udef 57.5%

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

      5. pow-prod-down 57.5%

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

      6. pow-prod-up 57.6%

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

      7. metadata-eval 57.6%

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

      8. add-cube-cbrt 56.9%

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

      9. unpow-prod-down 56.9%

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

      10. pow2 56.9%

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

   5. Applied egg-rr 56.9%

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

      1. *-commutative 56.9%

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

      2. metadata-eval 56.9%

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

      3. pow-sqr 56.8%

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

      4. sqr-pow 56.8%

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

      5. unpow2 56.8%

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

      6. metadata-eval 56.8%

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

      7. cube-unmult 56.8%

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

      8. metadata-eval 56.8%

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

      9. unpow2 56.8%

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

      10. pow-sqr 56.9%

          \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\mathsf{hypot}\left(a, b\right)}\right)}^{\left(2 \cdot 2\right)}\right)}}^{3} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

      11. metadata-eval 56.9%

          \[\leadsto {\left({\left(\sqrt[3]{\mathsf{hypot}\left(a, b\right)}\right)}^{\color{blue}{4}}\right)}^{3} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   7. Simplified 56.9%

      \[\leadsto \color{blue}{{\left({\left(\sqrt[3]{\mathsf{hypot}\left(a, b\right)}\right)}^{4}\right)}^{3}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   8. Taylor expanded in a around inf 27.3%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} + 4 \cdot {a}^{3}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 27.3%

         \[\leadsto \color{blue}{4 \cdot \left({a}^{2} + {a}^{3}\right)} \]

      2. *-lft-identity 27.3%

         \[\leadsto 4 \cdot \left(\color{blue}{1 \cdot {a}^{2}} + {a}^{3}\right) \]

      3. cube-mult 27.3%

         \[\leadsto 4 \cdot \left(1 \cdot {a}^{2} + \color{blue}{a \cdot \left(a \cdot a\right)}\right) \]

      4. unpow2 27.3%

         \[\leadsto 4 \cdot \left(1 \cdot {a}^{2} + a \cdot \color{blue}{{a}^{2}}\right) \]

      5. distribute-rgt-in 27.3%

         \[\leadsto 4 \cdot \color{blue}{\left({a}^{2} \cdot \left(1 + a\right)\right)} \]

      6. distribute-rgt-in 27.3%

         \[\leadsto 4 \cdot \color{blue}{\left(1 \cdot {a}^{2} + a \cdot {a}^{2}\right)} \]

      7. *-lft-identity 27.3%

         \[\leadsto 4 \cdot \left(\color{blue}{{a}^{2}} + a \cdot {a}^{2}\right) \]

      8. unpow2 27.3%

         \[\leadsto 4 \cdot \left(\color{blue}{a \cdot a} + a \cdot {a}^{2}\right) \]

      9. unpow2 27.3%

         \[\leadsto 4 \cdot \left(a \cdot a + a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

      10. cube-mult 27.3%

          \[\leadsto 4 \cdot \left(a \cdot a + \color{blue}{{a}^{3}}\right) \]

   10. Simplified 27.3%

       \[\leadsto \color{blue}{4 \cdot \left(a \cdot a + {a}^{3}\right)} \]

   11. Taylor expanded in a around 0 47.1%

       \[\leadsto \color{blue}{4 \cdot {a}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 47.1%

          \[\leadsto 4 \cdot \color{blue}{\left(a \cdot a\right)} \]

       2. *-commutative 47.1%

          \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot 4} \]

       3. associate-*r* 47.1%

          \[\leadsto \color{blue}{a \cdot \left(a \cdot 4\right)} \]

       4. *-commutative 47.1%

          \[\leadsto a \cdot \color{blue}{\left(4 \cdot a\right)} \]

   13. Simplified 47.1%

       \[\leadsto \color{blue}{a \cdot \left(4 \cdot a\right)} \]

   if -5.10000000000000019e-20 < a < 2.99999999999999998e-9

   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. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. fma-def 99.9%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   4. Taylor expanded in b around 0 55.4%

      \[\leadsto \color{blue}{\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1} \]
   5. Step-by-step derivation

      1. associate--l+ 55.4%

         \[\leadsto \color{blue}{{a}^{4} + \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) - 1\right)} \]

      2. associate-*r* 55.4%

         \[\leadsto {a}^{4} + \left(\color{blue}{\left(4 \cdot {a}^{2}\right) \cdot \left(1 + a\right)} - 1\right) \]

      3. unpow2 55.4%

         \[\leadsto {a}^{4} + \left(\left(4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(1 + a\right) - 1\right) \]

   6. Simplified 55.4%

      \[\leadsto \color{blue}{{a}^{4} + \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 + a\right) - 1\right)} \]

   7. Taylor expanded in a around 0 55.4%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{-20} \lor \neg \left(a \leq 3 \cdot 10^{-9}\right):\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 12: 24.4% accurate, 130.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

double code(double a, double b) {
	return -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -1.0d0
end function

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0;
end

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 72.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Step-by-step derivation

   1. associate--l+ 72.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

   2. fma-def 72.9%

      \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

3. Simplified 76.1%

   \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

4. Taylor expanded in b around 0 52.6%

   \[\leadsto \color{blue}{\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) - 1} \]
5. Step-by-step derivation

   1. associate--l+ 52.6%

      \[\leadsto \color{blue}{{a}^{4} + \left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) - 1\right)} \]

   2. associate-*r* 52.6%

      \[\leadsto {a}^{4} + \left(\color{blue}{\left(4 \cdot {a}^{2}\right) \cdot \left(1 + a\right)} - 1\right) \]

   3. unpow2 52.6%

      \[\leadsto {a}^{4} + \left(\left(4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(1 + a\right) - 1\right) \]

6. Simplified 52.6%

   \[\leadsto \color{blue}{{a}^{4} + \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 + a\right) - 1\right)} \]

7. Taylor expanded in a around 0 24.6%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 24.6%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023172 
(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))