Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.4% → 98.2%

Time: 9.7s

Alternatives: 11

Speedup: 7.5×

• 60.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(4.0 * N[(a * N[(a - N[(a * a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $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 3 a))))) < +inf.0

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. fma-def 99.9%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

      7. distribute-lft-out-- 99.9%

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

      8. *-rgt-identity 99.9%

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

      9. +-commutative 99.9%

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

   3. Simplified 100.0%

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

   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 3 a)))))

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. sqr-pow 0.0%

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

      3. sqr-pow 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-rgt-in 0.0%

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

   3. Simplified 1.3%

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

   4. Taylor expanded in a around inf 96.3%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))))
   (if (<= t_0 INFINITY) (+ t_0 -1.0) (+ -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 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = -1.0 + 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 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = -1.0 + Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = -1.0 + 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(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(-1.0 + (a ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = -1.0 + (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[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $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 3 a))))) < +inf.0

   1. Initial program 99.9%

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

   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 3 a)))))

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. sqr-pow 0.0%

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

      3. sqr-pow 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-rgt-in 0.0%

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

   3. Simplified 1.3%

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

   4. Taylor expanded in a around inf 96.3%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

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

Alternative 3: 93.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+18} \lor \neg \left(a \leq 1.15 \cdot 10^{+68}\right):\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.8e+18) (not (<= a 1.15e+68)))
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (+ (pow b 4.0) (* b (* b 12.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.8e+18) || !(a <= 1.15e+68)) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + (pow(b, 4.0) + (b * (b * 12.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-2.8d+18)) .or. (.not. (a <= 1.15d+68))) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b ** 4.0d0) + (b * (b * 12.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.8e+18) || !(a <= 1.15e+68)) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + (Math.pow(b, 4.0) + (b * (b * 12.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.8e+18) or not (a <= 1.15e+68):
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + (math.pow(b, 4.0) + (b * (b * 12.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.8e+18) || !(a <= 1.15e+68))
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(b * Float64(b * 12.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.8e+18) || ~((a <= 1.15e+68)))
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b ^ 4.0) + (b * (b * 12.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.8e+18], N[Not[LessEqual[a, 1.15e+68]], $MachinePrecision]], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.8e18 or 1.15e68 < a

   1. Initial program 34.7%

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

      1. sub-neg 34.7%

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

      2. sqr-pow 34.7%

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

      3. sqr-pow 34.7%

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

      4. sqr-neg 34.7%

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

      5. distribute-rgt-in 34.7%

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

      6. sqr-neg 34.7%

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

      7. distribute-rgt-in 34.7%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in a around inf 98.4%

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

   if -2.8e18 < a < 1.15e68

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

      2. sqr-pow 99.1%

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

      3. sqr-pow 99.1%

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

      4. sqr-neg 99.1%

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

      5. distribute-rgt-in 99.1%

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

      6. sqr-neg 99.1%

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

      7. distribute-rgt-in 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 83.1%

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

      1. +-commutative 83.1%

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

      2. +-commutative 83.1%

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

      3. associate-+l+ 83.1%

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

      4. unpow2 83.1%

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

      5. unpow2 83.1%

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

      6. associate-*r* 83.1%

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

      7. distribute-rgt-in 94.0%

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

      8. metadata-eval 94.0%

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

      9. distribute-lft-in 94.0%

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

      10. associate-*l* 94.0%

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

      11. +-commutative 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in a around 0 94.7%

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

      1. unpow2 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*l* 94.7%

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

   9. Simplified 94.7%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+18} \lor \neg \left(a \leq 1.15 \cdot 10^{+68}\right):\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 12\right)\right)\\ \end{array} \]

Alternative 4: 93.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+18} \lor \neg \left(a \leq 6.2 \cdot 10^{+69}\right):\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.2e+18) (not (<= a 6.2e+69)))
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.2e+18) || !(a <= 6.2e+69)) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-3.2d+18)) .or. (.not. (a <= 6.2d+69))) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -3.2e+18) || !(a <= 6.2e+69)) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -3.2e+18) or not (a <= 6.2e+69):
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.2e+18) || !(a <= 6.2e+69))
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -3.2e+18) || ~((a <= 6.2e+69)))
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.2e+18], N[Not[LessEqual[a, 6.2e+69]], $MachinePrecision]], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.2e18 or 6.1999999999999997e69 < a

   1. Initial program 34.7%

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

      1. sub-neg 34.7%

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

      2. sqr-pow 34.7%

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

      3. sqr-pow 34.7%

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

      4. sqr-neg 34.7%

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

      5. distribute-rgt-in 34.7%

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

      6. sqr-neg 34.7%

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

      7. distribute-rgt-in 34.7%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in a around inf 98.4%

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

   if -3.2e18 < a < 6.1999999999999997e69

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

      2. sqr-pow 99.1%

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

      3. sqr-pow 99.1%

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

      4. sqr-neg 99.1%

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

      5. distribute-rgt-in 99.1%

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

      6. sqr-neg 99.1%

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

      7. distribute-rgt-in 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 83.1%

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

      1. +-commutative 83.1%

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

      2. +-commutative 83.1%

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

      3. associate-+l+ 83.1%

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

      4. unpow2 83.1%

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

      5. unpow2 83.1%

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

      6. associate-*r* 83.1%

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

      7. distribute-rgt-in 94.0%

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

      8. metadata-eval 94.0%

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

      9. distribute-lft-in 94.0%

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

      10. associate-*l* 94.0%

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

      11. +-commutative 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in a around 0 94.7%

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

      1. unpow2 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*l* 94.7%

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

   9. Simplified 94.7%

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

       1. sqr-pow 94.6%

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

       2. metadata-eval 94.6%

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

       3. pow2 94.6%

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

       4. metadata-eval 94.6%

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

       5. pow2 94.6%

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

       6. associate-*r* 94.6%

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

       7. distribute-lft-out 94.6%

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

   11. Applied egg-rr 94.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+18} \lor \neg \left(a \leq 6.2 \cdot 10^{+69}\right):\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \]

Alternative 5: 79.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+77}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+147}:\\ \;\;\;\;-1 + \frac{\left(b \cdot \left(b \cdot 4\right)\right) \cdot \frac{81 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{a \cdot a + 9}}{3 - a}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot 4\right) \cdot \frac{b \cdot \left(9 - a \cdot a\right)}{3 - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.5e+77)
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))
   (if (<= a 3.4e+147)
     (+
      -1.0
      (/
       (* (* b (* b 4.0)) (/ (- 81.0 (* (* a a) (* a a))) (+ (* a a) 9.0)))
       (- 3.0 a)))
     (+ -1.0 (* (* b 4.0) (/ (* b (- 9.0 (* a a))) (- 3.0 a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.5e+77) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else if (a <= 3.4e+147) {
		tmp = -1.0 + (((b * (b * 4.0)) * ((81.0 - ((a * a) * (a * a))) / ((a * a) + 9.0))) / (3.0 - a));
	} else {
		tmp = -1.0 + ((b * 4.0) * ((b * (9.0 - (a * a))) / (3.0 - a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 1.5d+77) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    else if (a <= 3.4d+147) then
        tmp = (-1.0d0) + (((b * (b * 4.0d0)) * ((81.0d0 - ((a * a) * (a * a))) / ((a * a) + 9.0d0))) / (3.0d0 - a))
    else
        tmp = (-1.0d0) + ((b * 4.0d0) * ((b * (9.0d0 - (a * a))) / (3.0d0 - a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1.5e+77) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else if (a <= 3.4e+147) {
		tmp = -1.0 + (((b * (b * 4.0)) * ((81.0 - ((a * a) * (a * a))) / ((a * a) + 9.0))) / (3.0 - a));
	} else {
		tmp = -1.0 + ((b * 4.0) * ((b * (9.0 - (a * a))) / (3.0 - a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.5e+77:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	elif a <= 3.4e+147:
		tmp = -1.0 + (((b * (b * 4.0)) * ((81.0 - ((a * a) * (a * a))) / ((a * a) + 9.0))) / (3.0 - a))
	else:
		tmp = -1.0 + ((b * 4.0) * ((b * (9.0 - (a * a))) / (3.0 - a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.5e+77)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	elseif (a <= 3.4e+147)
		tmp = Float64(-1.0 + Float64(Float64(Float64(b * Float64(b * 4.0)) * Float64(Float64(81.0 - Float64(Float64(a * a) * Float64(a * a))) / Float64(Float64(a * a) + 9.0))) / Float64(3.0 - a)));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * 4.0) * Float64(Float64(b * Float64(9.0 - Float64(a * a))) / Float64(3.0 - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.5e+77)
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	elseif (a <= 3.4e+147)
		tmp = -1.0 + (((b * (b * 4.0)) * ((81.0 - ((a * a) * (a * a))) / ((a * a) + 9.0))) / (3.0 - a));
	else
		tmp = -1.0 + ((b * 4.0) * ((b * (9.0 - (a * a))) / (3.0 - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.5e+77], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.4e+147], N[(-1.0 + N[(N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(81.0 - N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * a), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * 4.0), $MachinePrecision] * N[(N[(b * N[(9.0 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.4999999999999999e77

   1. Initial program 85.0%

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

      1. sub-neg 85.0%

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

      2. sqr-pow 85.0%

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

      3. sqr-pow 85.0%

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

      4. sqr-neg 85.0%

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

      5. distribute-rgt-in 85.0%

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

      6. sqr-neg 85.0%

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

      7. distribute-rgt-in 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in a around 0 56.9%

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

      1. +-commutative 56.9%

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

      2. +-commutative 56.9%

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

      3. associate-+l+ 56.9%

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

      4. unpow2 56.9%

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

      5. unpow2 56.9%

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

      6. associate-*r* 56.9%

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

      7. distribute-rgt-in 64.4%

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

      8. metadata-eval 64.4%

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

      9. distribute-lft-in 64.4%

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

      10. associate-*l* 64.4%

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

      11. +-commutative 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in a around 0 77.9%

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

      1. unpow2 77.9%

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

      2. *-commutative 77.9%

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

      3. associate-*l* 77.9%

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

   9. Simplified 77.9%

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

       1. sqr-pow 77.8%

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

       2. metadata-eval 77.8%

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

       3. pow2 77.8%

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

       4. metadata-eval 77.8%

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

       5. pow2 77.8%

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

       6. associate-*r* 77.8%

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

       7. distribute-lft-out 77.8%

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

   11. Applied egg-rr 77.8%

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

   if 1.4999999999999999e77 < a < 3.4e147

   1. Initial program 35.3%

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

      1. sub-neg 35.3%

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

      2. sqr-pow 35.3%

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

      3. sqr-pow 35.3%

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

      4. sqr-neg 35.3%

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

      5. distribute-rgt-in 35.3%

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

      6. sqr-neg 35.3%

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

      7. distribute-rgt-in 35.3%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in a around 0 19.3%

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

      1. +-commutative 19.3%

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

      2. +-commutative 19.3%

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

      3. associate-+l+ 19.3%

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

      4. unpow2 19.3%

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

      5. unpow2 19.3%

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

      6. associate-*r* 19.3%

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

      7. distribute-rgt-in 19.3%

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

      8. metadata-eval 19.3%

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

      9. distribute-lft-in 19.3%

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

      10. associate-*l* 19.3%

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

      11. +-commutative 19.3%

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

   6. Simplified 19.3%

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

   7. Taylor expanded in b around 0 19.3%

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

      1. unpow2 19.3%

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

   9. Simplified 19.3%

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

       1. associate-*r* 19.3%

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

       2. flip-+ 19.3%

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

       3. associate-*r/ 30.5%

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

       4. associate-*r* 30.5%

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

       5. *-commutative 30.5%

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

       6. metadata-eval 30.5%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \left(\color{blue}{9} - a \cdot a\right)}{3 - a} + -1 \]

   11. Applied egg-rr 30.5%

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

       1. sub-neg 30.5%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \color{blue}{\left(9 + \left(-a \cdot a\right)\right)}}{3 - a} + -1 \]

       2. flip-+ 70.6%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \color{blue}{\frac{9 \cdot 9 - \left(-a \cdot a\right) \cdot \left(-a \cdot a\right)}{9 - \left(-a \cdot a\right)}}}{3 - a} + -1 \]

       3. metadata-eval 70.6%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \frac{\color{blue}{81} - \left(-a \cdot a\right) \cdot \left(-a \cdot a\right)}{9 - \left(-a \cdot a\right)}}{3 - a} + -1 \]

       4. distribute-rgt-neg-in 70.6%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \frac{81 - \color{blue}{\left(a \cdot \left(-a\right)\right)} \cdot \left(-a \cdot a\right)}{9 - \left(-a \cdot a\right)}}{3 - a} + -1 \]

       5. distribute-rgt-neg-in 70.6%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \frac{81 - \left(a \cdot \left(-a\right)\right) \cdot \color{blue}{\left(a \cdot \left(-a\right)\right)}}{9 - \left(-a \cdot a\right)}}{3 - a} + -1 \]

       6. distribute-rgt-neg-in 70.6%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \frac{81 - \left(a \cdot \left(-a\right)\right) \cdot \left(a \cdot \left(-a\right)\right)}{9 - \color{blue}{a \cdot \left(-a\right)}}}{3 - a} + -1 \]

   13. Applied egg-rr 70.6%

       \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \color{blue}{\frac{81 - \left(a \cdot \left(-a\right)\right) \cdot \left(a \cdot \left(-a\right)\right)}{9 - a \cdot \left(-a\right)}}}{3 - a} + -1 \]

   if 3.4e147 < a

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. sqr-pow 0.0%

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

      3. sqr-pow 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-rgt-in 0.0%

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

   3. Simplified 2.7%

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

   4. Taylor expanded in a around 0 37.3%

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

      1. +-commutative 37.3%

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

      2. +-commutative 37.3%

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

      3. associate-+l+ 37.3%

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

      4. unpow2 37.3%

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

      5. unpow2 37.3%

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

      6. associate-*r* 37.3%

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

      7. distribute-rgt-in 37.3%

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

      8. metadata-eval 37.3%

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

      9. distribute-lft-in 37.3%

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

      10. associate-*l* 37.3%

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

      11. +-commutative 37.3%

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

   6. Simplified 37.3%

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

   7. Taylor expanded in b around 0 37.3%

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

      1. unpow2 37.3%

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

   9. Simplified 37.3%

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

       1. associate-*r* 37.3%

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

       2. flip-+ 78.5%

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

       3. associate-*r/ 81.1%

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

       4. associate-*r* 81.1%

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

       5. *-commutative 81.1%

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

       6. metadata-eval 81.1%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \left(\color{blue}{9} - a \cdot a\right)}{3 - a} + -1 \]

   11. Applied egg-rr 81.1%

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

       1. associate-*l* 100.0%

          \[\leadsto \frac{\color{blue}{\left(b \cdot 4\right) \cdot \left(b \cdot \left(9 - a \cdot a\right)\right)}}{3 - a} + -1 \]

       2. *-un-lft-identity 100.0%

          \[\leadsto \frac{\left(b \cdot 4\right) \cdot \left(b \cdot \left(9 - a \cdot a\right)\right)}{\color{blue}{1 \cdot \left(3 - a\right)}} + -1 \]

       3. times-frac 100.0%

          \[\leadsto \color{blue}{\frac{b \cdot 4}{1} \cdot \frac{b \cdot \left(9 - a \cdot a\right)}{3 - a}} + -1 \]

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{b \cdot 4}{1} \cdot \frac{b \cdot \left(9 - a \cdot a\right)}{3 - a}} + -1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+77}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+147}:\\ \;\;\;\;-1 + \frac{\left(b \cdot \left(b \cdot 4\right)\right) \cdot \frac{81 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{a \cdot a + 9}}{3 - a}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot 4\right) \cdot \frac{b \cdot \left(9 - a \cdot a\right)}{3 - a}\\ \end{array} \]

Alternative 6: 74.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+50}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\left(b \cdot \left(b \cdot 4\right)\right) \cdot \left(9 - a \cdot a\right)}{3 - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 9.5e+50)
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))
   (+ -1.0 (/ (* (* b (* b 4.0)) (- 9.0 (* a a))) (- 3.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 9.5e+50) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + (((b * (b * 4.0)) * (9.0 - (a * a))) / (3.0 - a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 9.5d+50) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    else
        tmp = (-1.0d0) + (((b * (b * 4.0d0)) * (9.0d0 - (a * a))) / (3.0d0 - a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 9.5e+50) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + (((b * (b * 4.0)) * (9.0 - (a * a))) / (3.0 - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 9.5e+50:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	else:
		tmp = -1.0 + (((b * (b * 4.0)) * (9.0 - (a * a))) / (3.0 - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 9.5e+50)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(b * Float64(b * 4.0)) * Float64(9.0 - Float64(a * a))) / Float64(3.0 - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 9.5e+50)
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	else
		tmp = -1.0 + (((b * (b * 4.0)) * (9.0 - (a * a))) / (3.0 - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 9.5e+50], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] * N[(9.0 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.4999999999999993e50

   1. Initial program 84.4%

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

      1. sub-neg 84.4%

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

      2. sqr-pow 84.4%

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

      3. sqr-pow 84.4%

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

      4. sqr-neg 84.4%

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

      5. distribute-rgt-in 84.4%

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

      6. sqr-neg 84.4%

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

      7. distribute-rgt-in 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in a around 0 56.7%

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

      1. +-commutative 56.7%

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

      2. +-commutative 56.7%

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

      3. associate-+l+ 56.7%

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

      4. unpow2 56.7%

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

      5. unpow2 56.7%

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

      6. associate-*r* 56.7%

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

      7. distribute-rgt-in 64.4%

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

      8. metadata-eval 64.4%

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

      9. distribute-lft-in 64.4%

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

      10. associate-*l* 64.4%

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

      11. +-commutative 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in a around 0 78.5%

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

      1. unpow2 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-*l* 78.5%

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

   9. Simplified 78.5%

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

       1. sqr-pow 78.4%

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

       2. metadata-eval 78.4%

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

       3. pow2 78.4%

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

       4. metadata-eval 78.4%

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

       5. pow2 78.4%

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

       6. associate-*r* 78.4%

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

       7. distribute-lft-out 78.4%

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

   11. Applied egg-rr 78.4%

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

   if 9.4999999999999993e50 < a

   1. Initial program 22.6%

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

      1. sub-neg 22.6%

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

      2. sqr-pow 22.6%

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

      3. sqr-pow 22.6%

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

      4. sqr-neg 22.6%

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

      5. distribute-rgt-in 22.6%

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

      6. sqr-neg 22.6%

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

      7. distribute-rgt-in 22.6%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in a around 0 35.7%

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

      1. +-commutative 35.7%

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

      2. +-commutative 35.7%

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

      3. associate-+l+ 35.7%

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

      4. unpow2 35.7%

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

      5. unpow2 35.7%

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

      6. associate-*r* 35.7%

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

      7. distribute-rgt-in 35.7%

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

      8. metadata-eval 35.7%

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

      9. distribute-lft-in 35.7%

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

      10. associate-*l* 35.7%

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

      11. +-commutative 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in b around 0 34.2%

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

      1. unpow2 34.2%

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

   9. Simplified 34.2%

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

       1. associate-*r* 34.2%

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

       2. flip-+ 58.8%

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

       3. associate-*r/ 64.9%

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

       4. associate-*r* 64.9%

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

       5. *-commutative 64.9%

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

       6. metadata-eval 64.9%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \left(\color{blue}{9} - a \cdot a\right)}{3 - a} + -1 \]

   11. Applied egg-rr 64.9%

       \[\leadsto \color{blue}{\frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \left(9 - a \cdot a\right)}{3 - a}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+50}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\left(b \cdot \left(b \cdot 4\right)\right) \cdot \left(9 - a \cdot a\right)}{3 - a}\\ \end{array} \]

Alternative 7: 77.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{+62}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot 4\right) \cdot \frac{b \cdot \left(9 - a \cdot a\right)}{3 - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.4e+62)
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))
   (+ -1.0 (* (* b 4.0) (/ (* b (- 9.0 (* a a))) (- 3.0 a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.4e+62) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + ((b * 4.0) * ((b * (9.0 - (a * a))) / (3.0 - a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 2.4d+62) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    else
        tmp = (-1.0d0) + ((b * 4.0d0) * ((b * (9.0d0 - (a * a))) / (3.0d0 - a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 2.4e+62) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + ((b * 4.0) * ((b * (9.0 - (a * a))) / (3.0 - a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 2.4e+62:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	else:
		tmp = -1.0 + ((b * 4.0) * ((b * (9.0 - (a * a))) / (3.0 - a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.4e+62)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * 4.0) * Float64(Float64(b * Float64(9.0 - Float64(a * a))) / Float64(3.0 - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 2.4e+62)
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	else
		tmp = -1.0 + ((b * 4.0) * ((b * (9.0 - (a * a))) / (3.0 - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.4e+62], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * 4.0), $MachinePrecision] * N[(N[(b * N[(9.0 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.4e62

   1. Initial program 84.8%

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

      1. sub-neg 84.8%

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

      2. sqr-pow 84.8%

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

      3. sqr-pow 84.8%

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

      4. sqr-neg 84.8%

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

      5. distribute-rgt-in 84.8%

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

      6. sqr-neg 84.8%

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

      7. distribute-rgt-in 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in a around 0 56.8%

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

      1. +-commutative 56.8%

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

      2. +-commutative 56.8%

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

      3. associate-+l+ 56.8%

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

      4. unpow2 56.8%

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

      5. unpow2 56.8%

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

      6. associate-*r* 56.8%

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

      7. distribute-rgt-in 64.3%

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

      8. metadata-eval 64.3%

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

      9. distribute-lft-in 64.3%

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

      10. associate-*l* 64.3%

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

      11. +-commutative 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in a around 0 78.1%

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

      1. unpow2 78.1%

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

      2. *-commutative 78.1%

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

      3. associate-*l* 78.1%

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

   9. Simplified 78.1%

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

       1. sqr-pow 78.0%

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

       2. metadata-eval 78.0%

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

       3. pow2 78.0%

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

       4. metadata-eval 78.0%

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

       5. pow2 78.0%

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

       6. associate-*r* 78.0%

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

       7. distribute-lft-out 78.0%

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

   11. Applied egg-rr 78.0%

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

   if 2.4e62 < a

   1. Initial program 15.8%

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

      1. sub-neg 15.8%

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

      2. sqr-pow 15.8%

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

      3. sqr-pow 15.8%

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

      4. sqr-neg 15.8%

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

      5. distribute-rgt-in 15.8%

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

      6. sqr-neg 15.8%

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

      7. distribute-rgt-in 15.8%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in a around 0 33.5%

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

      1. +-commutative 33.5%

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

      2. +-commutative 33.5%

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

      3. associate-+l+ 33.5%

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

      4. unpow2 33.5%

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

      5. unpow2 33.5%

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

      6. associate-*r* 33.5%

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

      7. distribute-rgt-in 33.5%

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

      8. metadata-eval 33.5%

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

      9. distribute-lft-in 33.5%

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

      10. associate-*l* 33.5%

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

      11. +-commutative 33.5%

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

   6. Simplified 33.5%

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

   7. Taylor expanded in b around 0 33.5%

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

      1. unpow2 33.5%

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

   9. Simplified 33.5%

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

       1. associate-*r* 33.5%

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

       2. flip-+ 60.3%

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

       3. associate-*r/ 65.3%

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

       4. associate-*r* 65.3%

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

       5. *-commutative 65.3%

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

       6. metadata-eval 65.3%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \left(\color{blue}{9} - a \cdot a\right)}{3 - a} + -1 \]

   11. Applied egg-rr 65.3%

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

       1. associate-*l* 77.5%

          \[\leadsto \frac{\color{blue}{\left(b \cdot 4\right) \cdot \left(b \cdot \left(9 - a \cdot a\right)\right)}}{3 - a} + -1 \]

       2. *-un-lft-identity 77.5%

          \[\leadsto \frac{\left(b \cdot 4\right) \cdot \left(b \cdot \left(9 - a \cdot a\right)\right)}{\color{blue}{1 \cdot \left(3 - a\right)}} + -1 \]

       3. times-frac 74.2%

          \[\leadsto \color{blue}{\frac{b \cdot 4}{1} \cdot \frac{b \cdot \left(9 - a \cdot a\right)}{3 - a}} + -1 \]

   13. Applied egg-rr 74.2%

       \[\leadsto \color{blue}{\frac{b \cdot 4}{1} \cdot \frac{b \cdot \left(9 - a \cdot a\right)}{3 - a}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{+62}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot 4\right) \cdot \frac{b \cdot \left(9 - a \cdot a\right)}{3 - a}\\ \end{array} \]

Alternative 8: 74.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+87}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot -4\right)}{3 - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 5e+87)
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))
   (+ -1.0 (/ (* (* a a) (* (* b b) -4.0)) (- 3.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 5e+87) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + (((a * a) * ((b * b) * -4.0)) / (3.0 - a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 5d+87) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    else
        tmp = (-1.0d0) + (((a * a) * ((b * b) * (-4.0d0))) / (3.0d0 - a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 5e+87) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + (((a * a) * ((b * b) * -4.0)) / (3.0 - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 5e+87:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	else:
		tmp = -1.0 + (((a * a) * ((b * b) * -4.0)) / (3.0 - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 5e+87)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(a * a) * Float64(Float64(b * b) * -4.0)) / Float64(3.0 - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 5e+87)
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	else
		tmp = -1.0 + (((a * a) * ((b * b) * -4.0)) / (3.0 - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 5e+87], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(a * a), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.9999999999999998e87

   1. Initial program 85.3%

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

      1. sub-neg 85.3%

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

      2. sqr-pow 85.3%

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

      3. sqr-pow 85.3%

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

      4. sqr-neg 85.3%

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

      5. distribute-rgt-in 85.3%

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

      6. sqr-neg 85.3%

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

      7. distribute-rgt-in 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in a around 0 56.8%

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

      1. +-commutative 56.8%

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

      2. +-commutative 56.8%

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

      3. associate-+l+ 56.8%

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

      4. unpow2 56.8%

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

      5. unpow2 56.8%

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

      6. associate-*r* 56.8%

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

      7. distribute-rgt-in 64.1%

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

      8. metadata-eval 64.1%

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

      9. distribute-lft-in 64.1%

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

      10. associate-*l* 64.1%

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

      11. +-commutative 64.1%

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

   6. Simplified 64.1%

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

   7. Taylor expanded in a around 0 77.4%

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

      1. unpow2 77.4%

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

      2. *-commutative 77.4%

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

      3. associate-*l* 77.4%

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

   9. Simplified 77.4%

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

       1. sqr-pow 77.3%

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

       2. metadata-eval 77.3%

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

       3. pow2 77.3%

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

       4. metadata-eval 77.3%

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

       5. pow2 77.3%

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

       6. associate-*r* 77.3%

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

       7. distribute-lft-out 77.3%

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

   11. Applied egg-rr 77.3%

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

   if 4.9999999999999998e87 < a

   1. Initial program 4.0%

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

      1. sub-neg 4.0%

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

      2. sqr-pow 4.0%

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

      3. sqr-pow 4.0%

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

      4. sqr-neg 4.0%

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

      5. distribute-rgt-in 4.0%

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

      6. sqr-neg 4.0%

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

      7. distribute-rgt-in 4.0%

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

   3. Simplified 6.0%

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

   4. Taylor expanded in a around 0 30.2%

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

      1. +-commutative 30.2%

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

      2. +-commutative 30.2%

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

      3. associate-+l+ 30.2%

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

      4. unpow2 30.2%

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

      5. unpow2 30.2%

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

      6. associate-*r* 30.2%

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

      7. distribute-rgt-in 30.2%

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

      8. metadata-eval 30.2%

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

      9. distribute-lft-in 30.2%

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

      10. associate-*l* 30.2%

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

      11. +-commutative 30.2%

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

   6. Simplified 30.2%

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

   7. Taylor expanded in b around 0 30.2%

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

      1. unpow2 30.2%

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

   9. Simplified 30.2%

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

       1. associate-*r* 30.2%

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

       2. flip-+ 60.7%

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

       3. associate-*r/ 66.4%

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

       4. associate-*r* 66.4%

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

       5. *-commutative 66.4%

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

       6. metadata-eval 66.4%

          \[\leadsto \frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \left(\color{blue}{9} - a \cdot a\right)}{3 - a} + -1 \]

   11. Applied egg-rr 66.4%

       \[\leadsto \color{blue}{\frac{\left(\left(b \cdot 4\right) \cdot b\right) \cdot \left(9 - a \cdot a\right)}{3 - a}} + -1 \]

   12. Taylor expanded in a around inf 66.4%

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

       1. *-commutative 66.4%

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

       2. unpow2 66.4%

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

       3. unpow2 66.4%

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

       4. associate-*l* 66.4%

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

   14. Simplified 66.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+87}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot -4\right)}{3 - a}\\ \end{array} \]

Alternative 9: 69.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ -1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ -1.0 (* (* b b) (+ (* b b) 12.0))))

C

double code(double a, double b) {
	return -1.0 + ((b * b) * ((b * b) + 12.0));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
end function

Java

public static double code(double a, double b) {
	return -1.0 + ((b * b) * ((b * b) + 12.0));
}

Python

def code(a, b):
	return -1.0 + ((b * b) * ((b * b) + 12.0))

Julia

function code(a, b)
	return Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)))
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
end

Wolfram

code[a_, b_] := N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

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

   1. sub-neg 69.4%

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

   2. sqr-pow 69.4%

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

   3. sqr-pow 69.4%

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

   4. sqr-neg 69.4%

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

   5. distribute-rgt-in 69.4%

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

   6. sqr-neg 69.4%

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

   7. distribute-rgt-in 69.4%

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

3. Simplified 69.8%

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

4. Taylor expanded in a around 0 51.6%

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

   1. +-commutative 51.6%

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

   2. +-commutative 51.6%

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

   3. associate-+l+ 51.6%

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

   4. unpow2 51.6%

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

   5. unpow2 51.6%

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

   6. associate-*r* 51.6%

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

   7. distribute-rgt-in 57.5%

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

   8. metadata-eval 57.5%

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

   9. distribute-lft-in 57.5%

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

   10. associate-*l* 57.5%

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

   11. +-commutative 57.5%

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

6. Simplified 57.5%

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

7. Taylor expanded in a around 0 67.9%

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

   1. unpow2 67.9%

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

   2. *-commutative 67.9%

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

   3. associate-*l* 67.9%

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

9. Simplified 67.9%

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

    1. sqr-pow 67.9%

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

    2. metadata-eval 67.9%

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

    3. pow2 67.9%

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

    4. metadata-eval 67.9%

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

    5. pow2 67.9%

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

    6. associate-*r* 67.9%

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

    7. distribute-lft-out 67.9%

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

11. Applied egg-rr 67.9%

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

12. Final simplification 67.9%

    \[\leadsto -1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) \]

Alternative 10: 69.0% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ -1 + \left(b \cdot b\right) \cdot \left(b \cdot b\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0 + ((b * b) * (b * b))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

   1. sub-neg 69.4%

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

   2. sqr-pow 69.4%

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

   3. sqr-pow 69.4%

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

   4. sqr-neg 69.4%

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

   5. distribute-rgt-in 69.4%

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

   6. sqr-neg 69.4%

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

   7. distribute-rgt-in 69.4%

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

3. Simplified 69.8%

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

4. Taylor expanded in b around inf 67.2%

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

   1. sqr-pow 67.1%

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

   2. metadata-eval 67.1%

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

   3. pow2 67.1%

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

   4. metadata-eval 67.1%

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

   5. pow2 67.1%

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

6. Applied egg-rr 67.1%

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

7. Final simplification 67.1%

   \[\leadsto -1 + \left(b \cdot b\right) \cdot \left(b \cdot b\right) \]

Alternative 11: 51.2% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ -1 + \left(b \cdot b\right) \cdot 12 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ -1.0 (* (* b b) 12.0)))

C

double code(double a, double b) {
	return -1.0 + ((b * b) * 12.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + ((b * b) * 12.0d0)
end function

Java

public static double code(double a, double b) {
	return -1.0 + ((b * b) * 12.0);
}

Python

def code(a, b):
	return -1.0 + ((b * b) * 12.0)

Julia

function code(a, b)
	return Float64(-1.0 + Float64(Float64(b * b) * 12.0))
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + ((b * b) * 12.0);
end

Wolfram

code[a_, b_] := N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

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

   1. sub-neg 69.4%

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

   2. sqr-pow 69.4%

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

   3. sqr-pow 69.4%

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

   4. sqr-neg 69.4%

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

   5. distribute-rgt-in 69.4%

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

   6. sqr-neg 69.4%

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

   7. distribute-rgt-in 69.4%

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

3. Simplified 69.8%

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

4. Taylor expanded in a around 0 51.6%

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

   1. +-commutative 51.6%

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

   2. +-commutative 51.6%

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

   3. associate-+l+ 51.6%

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

   4. unpow2 51.6%

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

   5. unpow2 51.6%

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

   6. associate-*r* 51.6%

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

   7. distribute-rgt-in 57.5%

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

   8. metadata-eval 57.5%

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

   9. distribute-lft-in 57.5%

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

   10. associate-*l* 57.5%

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

   11. +-commutative 57.5%

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

6. Simplified 57.5%

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

7. Taylor expanded in a around 0 67.9%

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

   1. unpow2 67.9%

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

   2. *-commutative 67.9%

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

   3. associate-*l* 67.9%

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

9. Simplified 67.9%

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

    1. sqr-pow 67.9%

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

    2. metadata-eval 67.9%

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

    3. pow2 67.9%

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

    4. metadata-eval 67.9%

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

    5. pow2 67.9%

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

    6. associate-*r* 67.9%

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

    7. distribute-lft-out 67.9%

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

11. Applied egg-rr 67.9%

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

12. Taylor expanded in b around 0 53.0%

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

    1. unpow2 53.0%

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

14. Simplified 53.0%

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

15. Final simplification 53.0%

    \[\leadsto -1 + \left(b \cdot b\right) \cdot 12 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))