Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.1% → 98.3%

Time: 8.7s

Alternatives: 12

Speedup: 8.5×

• 61.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% 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: 98.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $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. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   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. fma-def 0.0%

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

      3. fma-def 3.4%

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

      4. +-commutative 3.4%

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

      5. metadata-eval 3.4%

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

   3. Simplified 3.4%

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

   4. Taylor expanded in a around inf 98.4%

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

4. Final simplification 99.6%

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

Alternative 2: 98.3% 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. fma-def 0.0%

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

      3. fma-def 3.4%

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

      4. +-commutative 3.4%

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

      5. metadata-eval 3.4%

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

   3. Simplified 3.4%

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

   4. Taylor expanded in a around inf 98.4%

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

4. Final simplification 99.5%

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

Alternative 3: 94.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00285:\\ \;\;\;\;-1 + \left({a}^{4} + \left(1 - a\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+55}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 12 + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.00285)
   (+ -1.0 (+ (pow a 4.0) (* (- 1.0 a) (* (* a a) 4.0))))
   (if (<= a 4.5e+55)
     (+ -1.0 (+ (* (* b b) 12.0) (pow b 4.0)))
     (+ -1.0 (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.00285) {
		tmp = -1.0 + (pow(a, 4.0) + ((1.0 - a) * ((a * a) * 4.0)));
	} else if (a <= 4.5e+55) {
		tmp = -1.0 + (((b * b) * 12.0) + pow(b, 4.0));
	} else {
		tmp = -1.0 + pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.00285d0)) then
        tmp = (-1.0d0) + ((a ** 4.0d0) + ((1.0d0 - a) * ((a * a) * 4.0d0)))
    else if (a <= 4.5d+55) then
        tmp = (-1.0d0) + (((b * b) * 12.0d0) + (b ** 4.0d0))
    else
        tmp = (-1.0d0) + (a ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.00285) {
		tmp = -1.0 + (Math.pow(a, 4.0) + ((1.0 - a) * ((a * a) * 4.0)));
	} else if (a <= 4.5e+55) {
		tmp = -1.0 + (((b * b) * 12.0) + Math.pow(b, 4.0));
	} else {
		tmp = -1.0 + Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.00285:
		tmp = -1.0 + (math.pow(a, 4.0) + ((1.0 - a) * ((a * a) * 4.0)))
	elif a <= 4.5e+55:
		tmp = -1.0 + (((b * b) * 12.0) + math.pow(b, 4.0))
	else:
		tmp = -1.0 + math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.00285)
		tmp = Float64(-1.0 + Float64((a ^ 4.0) + Float64(Float64(1.0 - a) * Float64(Float64(a * a) * 4.0))));
	elseif (a <= 4.5e+55)
		tmp = Float64(-1.0 + Float64(Float64(Float64(b * b) * 12.0) + (b ^ 4.0)));
	else
		tmp = Float64(-1.0 + (a ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.00285)
		tmp = -1.0 + ((a ^ 4.0) + ((1.0 - a) * ((a * a) * 4.0)));
	elseif (a <= 4.5e+55)
		tmp = -1.0 + (((b * b) * 12.0) + (b ^ 4.0));
	else
		tmp = -1.0 + (a ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.00285], N[(-1.0 + N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(1.0 - a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.5e+55], N[(-1.0 + N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0028500000000000001

   1. Initial program 64.5%

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

      1. sub-neg 64.5%

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

      2. fma-def 64.5%

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

      3. fma-def 64.5%

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

      4. +-commutative 64.5%

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

      5. metadata-eval 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in b around 0 96.1%

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

      1. associate-*r* 96.1%

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

      2. unpow2 96.1%

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

   6. Simplified 96.1%

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

   if -0.0028500000000000001 < a < 4.49999999999999998e55

   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. fma-def 99.9%

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

      3. fma-def 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 84.7%

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

      1. associate-+r+ 84.7%

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

      2. associate-*r* 84.7%

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

      3. distribute-rgt-out 95.7%

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

      4. metadata-eval 95.7%

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

      5. distribute-lft-in 95.7%

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

      6. unpow2 95.7%

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

      7. distribute-rgt-in 95.7%

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

      8. metadata-eval 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in a around 0 95.7%

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

      1. unpow2 95.7%

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

   9. Simplified 95.7%

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

   if 4.49999999999999998e55 < a

   1. Initial program 22.2%

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

      1. sub-neg 22.2%

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

      2. fma-def 22.2%

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

      3. fma-def 25.9%

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

      4. +-commutative 25.9%

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

      5. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   4. Taylor expanded in a around inf 100.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00285:\\ \;\;\;\;-1 + \left({a}^{4} + \left(1 - a\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+55}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 12 + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \]

Alternative 4: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 12 + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.019)
   (+ -1.0 (* (pow a 3.0) (+ a -4.0)))
   (if (<= a 9.5e+54)
     (+ -1.0 (+ (* (* b b) 12.0) (pow b 4.0)))
     (+ -1.0 (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.019) {
		tmp = -1.0 + (pow(a, 3.0) * (a + -4.0));
	} else if (a <= 9.5e+54) {
		tmp = -1.0 + (((b * b) * 12.0) + pow(b, 4.0));
	} else {
		tmp = -1.0 + pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.019d0)) then
        tmp = (-1.0d0) + ((a ** 3.0d0) * (a + (-4.0d0)))
    else if (a <= 9.5d+54) then
        tmp = (-1.0d0) + (((b * b) * 12.0d0) + (b ** 4.0d0))
    else
        tmp = (-1.0d0) + (a ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.019) {
		tmp = -1.0 + (Math.pow(a, 3.0) * (a + -4.0));
	} else if (a <= 9.5e+54) {
		tmp = -1.0 + (((b * b) * 12.0) + Math.pow(b, 4.0));
	} else {
		tmp = -1.0 + Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.019:
		tmp = -1.0 + (math.pow(a, 3.0) * (a + -4.0))
	elif a <= 9.5e+54:
		tmp = -1.0 + (((b * b) * 12.0) + math.pow(b, 4.0))
	else:
		tmp = -1.0 + math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.019)
		tmp = Float64(-1.0 + Float64((a ^ 3.0) * Float64(a + -4.0)));
	elseif (a <= 9.5e+54)
		tmp = Float64(-1.0 + Float64(Float64(Float64(b * b) * 12.0) + (b ^ 4.0)));
	else
		tmp = Float64(-1.0 + (a ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.019)
		tmp = -1.0 + ((a ^ 3.0) * (a + -4.0));
	elseif (a <= 9.5e+54)
		tmp = -1.0 + (((b * b) * 12.0) + (b ^ 4.0));
	else
		tmp = -1.0 + (a ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.019], N[(-1.0 + N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.5e+54], N[(-1.0 + N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0189999999999999995

   1. Initial program 64.5%

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

      1. sub-neg 64.5%

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

      2. fma-def 64.5%

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

      3. fma-def 64.5%

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

      4. +-commutative 64.5%

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

      5. metadata-eval 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in b around 0 96.1%

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

      1. associate-*r* 96.1%

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

      2. unpow2 96.1%

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

   6. Simplified 96.1%

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

      1. flip-+ 8.6%

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

      2. pow-prod-up 8.5%

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

      3. metadata-eval 8.5%

         \[\leadsto \frac{{a}^{\color{blue}{8}} - \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right) \cdot \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right)}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      4. pow2 8.5%

         \[\leadsto \frac{{a}^{8} - \color{blue}{{\left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right)}^{2}}}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      5. *-commutative 8.5%

         \[\leadsto \frac{{a}^{8} - {\color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)}}^{2}}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      6. *-commutative 8.5%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)}^{2}}{{a}^{4} - \color{blue}{\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)}} + -1 \]

   8. Applied egg-rr 8.5%

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

      1. associate-*r* 8.5%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \color{blue}{\left(\left(4 \cdot a\right) \cdot a\right)}\right)}^{2}}{{a}^{4} - \left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)} + -1 \]

      2. associate-*r* 8.5%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}^{2}}{{a}^{4} - \left(1 - a\right) \cdot \color{blue}{\left(\left(4 \cdot a\right) \cdot a\right)}} + -1 \]

   10. Simplified 8.5%

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

       1. unpow2 8.5%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right) \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       2. associate-*r* 8.5%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot a\right)} \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       3. associate-*l* 8.5%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       4. *-commutative 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{\left(\left(\left(4 \cdot a\right) \cdot a\right) \cdot \left(1 - a\right)\right)}\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       5. associate-*l* 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\color{blue}{\left(4 \cdot \left(a \cdot a\right)\right)} \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       6. pow2 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(4 \cdot \color{blue}{{a}^{2}}\right) \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       7. metadata-eval 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(4 \cdot {a}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       8. associate-*l* 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{\left(4 \cdot \left({a}^{\left(\frac{4}{2}\right)} \cdot \left(1 - a\right)\right)\right)}\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       9. metadata-eval 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left({a}^{\color{blue}{2}} \cdot \left(1 - a\right)\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       10. pow2 8.5%

           \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(1 - a\right)\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

   12. Applied egg-rr 8.5%

       \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

   13. Taylor expanded in a around inf 94.6%

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

       1. *-commutative 94.6%

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

       2. metadata-eval 94.6%

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

       3. pow-plus 94.6%

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

       4. distribute-lft-out 94.6%

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

   15. Simplified 94.6%

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

   if -0.0189999999999999995 < a < 9.4999999999999999e54

   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. fma-def 99.9%

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

      3. fma-def 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 84.7%

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

      1. associate-+r+ 84.7%

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

      2. associate-*r* 84.7%

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

      3. distribute-rgt-out 95.7%

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

      4. metadata-eval 95.7%

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

      5. distribute-lft-in 95.7%

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

      6. unpow2 95.7%

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

      7. distribute-rgt-in 95.7%

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

      8. metadata-eval 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in a around 0 95.7%

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

      1. unpow2 95.7%

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

   9. Simplified 95.7%

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

   if 9.4999999999999999e54 < a

   1. Initial program 22.2%

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

      1. sub-neg 22.2%

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

      2. fma-def 22.2%

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

      3. fma-def 25.9%

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

      4. +-commutative 25.9%

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

      5. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   4. Taylor expanded in a around inf 100.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 12 + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \]

Alternative 5: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{elif}\;a \leq 5.1 \cdot 10^{+54}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.019)
   (+ -1.0 (* (pow a 3.0) (+ a -4.0)))
   (if (<= a 5.1e+54)
     (+ -1.0 (* b (* b (fma b b 12.0))))
     (+ -1.0 (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.019) {
		tmp = -1.0 + (pow(a, 3.0) * (a + -4.0));
	} else if (a <= 5.1e+54) {
		tmp = -1.0 + (b * (b * fma(b, b, 12.0)));
	} else {
		tmp = -1.0 + pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.019)
		tmp = Float64(-1.0 + Float64((a ^ 3.0) * Float64(a + -4.0)));
	elseif (a <= 5.1e+54)
		tmp = Float64(-1.0 + Float64(b * Float64(b * fma(b, b, 12.0))));
	else
		tmp = Float64(-1.0 + (a ^ 4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.019], N[(-1.0 + N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.1e+54], N[(-1.0 + N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0189999999999999995

   1. Initial program 64.5%

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

      1. sub-neg 64.5%

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

      2. fma-def 64.5%

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

      3. fma-def 64.5%

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

      4. +-commutative 64.5%

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

      5. metadata-eval 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in b around 0 96.1%

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

      1. associate-*r* 96.1%

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

      2. unpow2 96.1%

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

   6. Simplified 96.1%

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

      1. flip-+ 8.6%

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

      2. pow-prod-up 8.5%

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

      3. metadata-eval 8.5%

         \[\leadsto \frac{{a}^{\color{blue}{8}} - \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right) \cdot \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right)}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      4. pow2 8.5%

         \[\leadsto \frac{{a}^{8} - \color{blue}{{\left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right)}^{2}}}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      5. *-commutative 8.5%

         \[\leadsto \frac{{a}^{8} - {\color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)}}^{2}}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      6. *-commutative 8.5%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)}^{2}}{{a}^{4} - \color{blue}{\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)}} + -1 \]

   8. Applied egg-rr 8.5%

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

      1. associate-*r* 8.5%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \color{blue}{\left(\left(4 \cdot a\right) \cdot a\right)}\right)}^{2}}{{a}^{4} - \left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)} + -1 \]

      2. associate-*r* 8.5%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}^{2}}{{a}^{4} - \left(1 - a\right) \cdot \color{blue}{\left(\left(4 \cdot a\right) \cdot a\right)}} + -1 \]

   10. Simplified 8.5%

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

       1. unpow2 8.5%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right) \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       2. associate-*r* 8.5%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot a\right)} \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       3. associate-*l* 8.5%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       4. *-commutative 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{\left(\left(\left(4 \cdot a\right) \cdot a\right) \cdot \left(1 - a\right)\right)}\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       5. associate-*l* 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\color{blue}{\left(4 \cdot \left(a \cdot a\right)\right)} \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       6. pow2 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(4 \cdot \color{blue}{{a}^{2}}\right) \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       7. metadata-eval 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(4 \cdot {a}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       8. associate-*l* 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{\left(4 \cdot \left({a}^{\left(\frac{4}{2}\right)} \cdot \left(1 - a\right)\right)\right)}\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       9. metadata-eval 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left({a}^{\color{blue}{2}} \cdot \left(1 - a\right)\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       10. pow2 8.5%

           \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(1 - a\right)\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

   12. Applied egg-rr 8.5%

       \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

   13. Taylor expanded in a around inf 94.6%

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

       1. *-commutative 94.6%

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

       2. metadata-eval 94.6%

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

       3. pow-plus 94.6%

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

       4. distribute-lft-out 94.6%

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

   15. Simplified 94.6%

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

   if -0.0189999999999999995 < a < 5.10000000000000009e54

   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. fma-def 99.9%

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

      3. fma-def 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 84.7%

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

      1. associate-+r+ 84.7%

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

      2. associate-*r* 84.7%

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

      3. distribute-rgt-out 95.7%

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

      4. metadata-eval 95.7%

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

      5. distribute-lft-in 95.7%

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

      6. unpow2 95.7%

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

      7. distribute-rgt-in 95.7%

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

      8. metadata-eval 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in a around 0 95.7%

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

      1. unpow2 95.7%

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

   9. Simplified 95.7%

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

   10. Taylor expanded in b around 0 95.7%

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

       1. unpow2 95.7%

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

       2. metadata-eval 95.7%

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

       3. pow-sqr 95.7%

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

       4. unpow2 95.7%

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

       5. unpow2 95.7%

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

       6. distribute-rgt-in 95.7%

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

       7. associate-*l* 95.7%

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

       8. +-commutative 95.7%

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

       9. fma-def 95.7%

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

   12. Simplified 95.7%

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

   if 5.10000000000000009e54 < a

   1. Initial program 22.2%

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

      1. sub-neg 22.2%

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

      2. fma-def 22.2%

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

      3. fma-def 25.9%

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

      4. +-commutative 25.9%

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

      5. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   4. Taylor expanded in a around inf 100.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{elif}\;a \leq 5.1 \cdot 10^{+54}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \]

Alternative 6: 93.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.019)
   (+ -1.0 (* (pow a 3.0) (+ a -4.0)))
   (if (<= a 1.8e+54)
     (+ -1.0 (* (* b b) (+ (* b b) 12.0)))
     (+ -1.0 (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.019) {
		tmp = -1.0 + (pow(a, 3.0) * (a + -4.0));
	} else if (a <= 1.8e+54) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.019d0)) then
        tmp = (-1.0d0) + ((a ** 3.0d0) * (a + (-4.0d0)))
    else if (a <= 1.8d+54) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    else
        tmp = (-1.0d0) + (a ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.019) {
		tmp = -1.0 + (Math.pow(a, 3.0) * (a + -4.0));
	} else if (a <= 1.8e+54) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.019:
		tmp = -1.0 + (math.pow(a, 3.0) * (a + -4.0))
	elif a <= 1.8e+54:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	else:
		tmp = -1.0 + math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.019)
		tmp = Float64(-1.0 + Float64((a ^ 3.0) * Float64(a + -4.0)));
	elseif (a <= 1.8e+54)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	else
		tmp = Float64(-1.0 + (a ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.019)
		tmp = -1.0 + ((a ^ 3.0) * (a + -4.0));
	elseif (a <= 1.8e+54)
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	else
		tmp = -1.0 + (a ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.019], N[(-1.0 + N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e+54], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0189999999999999995

   1. Initial program 64.5%

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

      1. sub-neg 64.5%

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

      2. fma-def 64.5%

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

      3. fma-def 64.5%

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

      4. +-commutative 64.5%

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

      5. metadata-eval 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in b around 0 96.1%

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

      1. associate-*r* 96.1%

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

      2. unpow2 96.1%

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

   6. Simplified 96.1%

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

      1. flip-+ 8.6%

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

      2. pow-prod-up 8.5%

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

      3. metadata-eval 8.5%

         \[\leadsto \frac{{a}^{\color{blue}{8}} - \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right) \cdot \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right)}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      4. pow2 8.5%

         \[\leadsto \frac{{a}^{8} - \color{blue}{{\left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right)}^{2}}}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      5. *-commutative 8.5%

         \[\leadsto \frac{{a}^{8} - {\color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)}}^{2}}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      6. *-commutative 8.5%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)}^{2}}{{a}^{4} - \color{blue}{\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)}} + -1 \]

   8. Applied egg-rr 8.5%

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

      1. associate-*r* 8.5%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \color{blue}{\left(\left(4 \cdot a\right) \cdot a\right)}\right)}^{2}}{{a}^{4} - \left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)} + -1 \]

      2. associate-*r* 8.5%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}^{2}}{{a}^{4} - \left(1 - a\right) \cdot \color{blue}{\left(\left(4 \cdot a\right) \cdot a\right)}} + -1 \]

   10. Simplified 8.5%

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

       1. unpow2 8.5%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right) \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       2. associate-*r* 8.5%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot a\right)} \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       3. associate-*l* 8.5%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       4. *-commutative 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{\left(\left(\left(4 \cdot a\right) \cdot a\right) \cdot \left(1 - a\right)\right)}\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       5. associate-*l* 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\color{blue}{\left(4 \cdot \left(a \cdot a\right)\right)} \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       6. pow2 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(4 \cdot \color{blue}{{a}^{2}}\right) \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       7. metadata-eval 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(4 \cdot {a}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       8. associate-*l* 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{\left(4 \cdot \left({a}^{\left(\frac{4}{2}\right)} \cdot \left(1 - a\right)\right)\right)}\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       9. metadata-eval 8.5%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left({a}^{\color{blue}{2}} \cdot \left(1 - a\right)\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       10. pow2 8.5%

           \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(1 - a\right)\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

   12. Applied egg-rr 8.5%

       \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

   13. Taylor expanded in a around inf 94.6%

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

       1. *-commutative 94.6%

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

       2. metadata-eval 94.6%

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

       3. pow-plus 94.6%

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

       4. distribute-lft-out 94.6%

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

   15. Simplified 94.6%

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

   if -0.0189999999999999995 < a < 1.8000000000000001e54

   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. fma-def 99.9%

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

      3. fma-def 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 84.7%

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

      1. associate-+r+ 84.7%

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

      2. associate-*r* 84.7%

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

      3. distribute-rgt-out 95.7%

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

      4. metadata-eval 95.7%

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

      5. distribute-lft-in 95.7%

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

      6. unpow2 95.7%

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

      7. distribute-rgt-in 95.7%

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

      8. metadata-eval 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in a around 0 95.7%

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

      1. unpow2 95.7%

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

   9. Simplified 95.7%

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

       1. +-commutative 95.7%

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

       2. metadata-eval 95.7%

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

       3. pow-sqr 95.7%

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

       4. pow-prod-down 95.7%

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

       5. pow2 95.7%

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

       6. distribute-rgt-out 95.7%

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

   11. Applied egg-rr 95.7%

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

   if 1.8000000000000001e54 < a

   1. Initial program 22.2%

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

      1. sub-neg 22.2%

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

      2. fma-def 22.2%

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

      3. fma-def 25.9%

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

      4. +-commutative 25.9%

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

      5. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   4. Taylor expanded in a around inf 100.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.019:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+54}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \]

Alternative 7: 93.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.019 \lor \neg \left(a \leq 3.95 \cdot 10^{+62}\right):\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.019) (not (<= a 3.95e+62)))
   (+ -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 <= -0.019) || !(a <= 3.95e+62)) {
		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 <= (-0.019d0)) .or. (.not. (a <= 3.95d+62))) 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 <= -0.019) || !(a <= 3.95e+62)) {
		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 <= -0.019) or not (a <= 3.95e+62):
		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 <= -0.019) || !(a <= 3.95e+62))
		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 <= -0.019) || ~((a <= 3.95e+62)))
		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, -0.019], N[Not[LessEqual[a, 3.95e+62]], $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 < -0.0189999999999999995 or 3.9499999999999998e62 < a

   1. Initial program 42.1%

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

      1. sub-neg 42.1%

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

      2. fma-def 42.1%

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

      3. fma-def 44.1%

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

      4. +-commutative 44.1%

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

      5. metadata-eval 44.1%

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

   3. Simplified 44.1%

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

   4. Taylor expanded in a around inf 97.3%

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

   if -0.0189999999999999995 < a < 3.9499999999999998e62

   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. fma-def 99.9%

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

      3. fma-def 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 84.7%

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

      1. associate-+r+ 84.7%

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

      2. associate-*r* 84.7%

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

      3. distribute-rgt-out 95.7%

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

      4. metadata-eval 95.7%

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

      5. distribute-lft-in 95.7%

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

      6. unpow2 95.7%

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

      7. distribute-rgt-in 95.7%

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

      8. metadata-eval 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in a around 0 95.7%

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

      1. unpow2 95.7%

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

   9. Simplified 95.7%

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

       1. +-commutative 95.7%

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

       2. metadata-eval 95.7%

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

       3. pow-sqr 95.7%

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

       4. pow-prod-down 95.7%

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

       5. pow2 95.7%

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

       6. distribute-rgt-out 95.7%

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

   11. Applied egg-rr 95.7%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.019 \lor \neg \left(a \leq 3.95 \cdot 10^{+62}\right):\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \]

Alternative 8: 86.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+57}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+153}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7e+57)
   (+ -1.0 (* 4.0 (* a (* a (- 1.0 a)))))
   (if (<= a 5.8e+153)
     (+ -1.0 (* (* b b) (+ (* b b) 12.0)))
     (+ -1.0 (* a (* a 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7e+57) {
		tmp = -1.0 + (4.0 * (a * (a * (1.0 - a))));
	} else if (a <= 5.8e+153) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + (a * (a * 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-7d+57)) then
        tmp = (-1.0d0) + (4.0d0 * (a * (a * (1.0d0 - a))))
    else if (a <= 5.8d+153) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    else
        tmp = (-1.0d0) + (a * (a * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7e+57) {
		tmp = -1.0 + (4.0 * (a * (a * (1.0 - a))));
	} else if (a <= 5.8e+153) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + (a * (a * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7e+57:
		tmp = -1.0 + (4.0 * (a * (a * (1.0 - a))))
	elif a <= 5.8e+153:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	else:
		tmp = -1.0 + (a * (a * 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7e+57)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * Float64(a * Float64(1.0 - a)))));
	elseif (a <= 5.8e+153)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	else
		tmp = Float64(-1.0 + Float64(a * Float64(a * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7e+57)
		tmp = -1.0 + (4.0 * (a * (a * (1.0 - a))));
	elseif (a <= 5.8e+153)
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	else
		tmp = -1.0 + (a * (a * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7e+57], N[(-1.0 + N[(4.0 * N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.8e+153], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.9999999999999995e57

   1. Initial program 60.9%

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

      1. sub-neg 60.9%

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

      2. fma-def 60.9%

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

      3. fma-def 60.9%

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

      4. +-commutative 60.9%

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

      5. metadata-eval 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in b around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. flip-+ 0.0%

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

      2. pow-prod-up 0.0%

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

      3. metadata-eval 0.0%

         \[\leadsto \frac{{a}^{\color{blue}{8}} - \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right) \cdot \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right)}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      4. pow2 0.0%

         \[\leadsto \frac{{a}^{8} - \color{blue}{{\left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right)}^{2}}}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      5. *-commutative 0.0%

         \[\leadsto \frac{{a}^{8} - {\color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)}}^{2}}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      6. *-commutative 0.0%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)}^{2}}{{a}^{4} - \color{blue}{\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)}} + -1 \]

   8. Applied egg-rr 0.0%

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

      1. associate-*r* 0.0%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \color{blue}{\left(\left(4 \cdot a\right) \cdot a\right)}\right)}^{2}}{{a}^{4} - \left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)} + -1 \]

      2. associate-*r* 0.0%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}^{2}}{{a}^{4} - \left(1 - a\right) \cdot \color{blue}{\left(\left(4 \cdot a\right) \cdot a\right)}} + -1 \]

   10. Simplified 0.0%

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

       1. unpow2 0.0%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right) \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       2. associate-*r* 0.0%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot a\right)} \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       3. associate-*l* 0.0%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       4. *-commutative 0.0%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{\left(\left(\left(4 \cdot a\right) \cdot a\right) \cdot \left(1 - a\right)\right)}\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       5. associate-*l* 0.0%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\color{blue}{\left(4 \cdot \left(a \cdot a\right)\right)} \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       6. pow2 0.0%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(4 \cdot \color{blue}{{a}^{2}}\right) \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       7. metadata-eval 0.0%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(4 \cdot {a}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       8. associate-*l* 0.0%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{\left(4 \cdot \left({a}^{\left(\frac{4}{2}\right)} \cdot \left(1 - a\right)\right)\right)}\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       9. metadata-eval 0.0%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left({a}^{\color{blue}{2}} \cdot \left(1 - a\right)\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       10. pow2 0.0%

           \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(1 - a\right)\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

   12. Applied egg-rr 0.0%

       \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

   13. Taylor expanded in a around 0 86.5%

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

       1. *-commutative 86.5%

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

       2. unpow2 86.5%

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

       3. associate-*r* 86.5%

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

       4. *-commutative 86.5%

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

       5. cube-mult 86.5%

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

       6. unpow2 86.5%

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

       7. associate-*l* 86.5%

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

       8. metadata-eval 86.5%

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

       9. distribute-rgt-neg-in 86.5%

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

       10. unpow2 86.5%

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

       11. associate-*r* 86.5%

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

       12. distribute-lft-out 86.5%

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

       13. *-lft-identity 86.5%

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

       14. distribute-lft-neg-in 86.5%

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

       15. distribute-rgt-in 86.5%

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

       16. sub-neg 86.5%

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

       17. associate-*l* 86.5%

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

       18. *-commutative 86.5%

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

       19. associate-*l* 86.5%

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

       20. *-commutative 86.5%

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

       21. associate-*r* 86.5%

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

   15. Simplified 86.5%

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

   if -6.9999999999999995e57 < a < 5.80000000000000004e153

   1. Initial program 94.4%

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

      1. sub-neg 94.4%

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

      2. fma-def 94.4%

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

      3. fma-def 95.5%

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

      4. +-commutative 95.5%

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

      5. metadata-eval 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in a around 0 76.0%

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

      1. associate-+r+ 76.0%

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

      2. associate-*r* 76.0%

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

      3. distribute-rgt-out 85.3%

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

      4. metadata-eval 85.3%

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

      5. distribute-lft-in 85.3%

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

      6. unpow2 85.3%

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

      7. distribute-rgt-in 85.3%

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

      8. metadata-eval 85.3%

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

   6. Simplified 85.3%

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

   7. Taylor expanded in a around 0 85.8%

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

      1. unpow2 85.8%

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

   9. Simplified 85.8%

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

       1. +-commutative 85.8%

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

       2. metadata-eval 85.8%

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

       3. pow-sqr 85.8%

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

       4. pow-prod-down 85.8%

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

       5. pow2 85.8%

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

       6. distribute-rgt-out 85.8%

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

   11. Applied egg-rr 85.8%

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

   if 5.80000000000000004e153 < 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. fma-def 0.0%

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

      3. fma-def 0.0%

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

      4. +-commutative 0.0%

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

      5. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in b around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. unpow2 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+57}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+153}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 9: 71.6% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+40}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.5e+40)
   (+ -1.0 (* 4.0 (* a (* a (- 1.0 a)))))
   (if (<= a 6.5e+153) (+ -1.0 (* (* b b) 12.0)) (+ -1.0 (* a (* a 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.5e+40) {
		tmp = -1.0 + (4.0 * (a * (a * (1.0 - a))));
	} else if (a <= 6.5e+153) {
		tmp = -1.0 + ((b * b) * 12.0);
	} else {
		tmp = -1.0 + (a * (a * 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-9.5d+40)) then
        tmp = (-1.0d0) + (4.0d0 * (a * (a * (1.0d0 - a))))
    else if (a <= 6.5d+153) then
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    else
        tmp = (-1.0d0) + (a * (a * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9.5e+40) {
		tmp = -1.0 + (4.0 * (a * (a * (1.0 - a))));
	} else if (a <= 6.5e+153) {
		tmp = -1.0 + ((b * b) * 12.0);
	} else {
		tmp = -1.0 + (a * (a * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9.5e+40:
		tmp = -1.0 + (4.0 * (a * (a * (1.0 - a))))
	elif a <= 6.5e+153:
		tmp = -1.0 + ((b * b) * 12.0)
	else:
		tmp = -1.0 + (a * (a * 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.5e+40)
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * Float64(a * Float64(1.0 - a)))));
	elseif (a <= 6.5e+153)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 12.0));
	else
		tmp = Float64(-1.0 + Float64(a * Float64(a * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9.5e+40)
		tmp = -1.0 + (4.0 * (a * (a * (1.0 - a))));
	elseif (a <= 6.5e+153)
		tmp = -1.0 + ((b * b) * 12.0);
	else
		tmp = -1.0 + (a * (a * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9.5e+40], N[(-1.0 + N[(4.0 * N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.5e+153], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.5000000000000003e40

   1. Initial program 62.7%

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

      1. sub-neg 62.7%

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

      2. fma-def 62.7%

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

      3. fma-def 62.7%

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

      4. +-commutative 62.7%

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

      5. metadata-eval 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in b around 0 97.8%

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

      1. associate-*r* 97.8%

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

      2. unpow2 97.8%

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

   6. Simplified 97.8%

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

      1. flip-+ 0.1%

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

      2. pow-prod-up 0.1%

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

      3. metadata-eval 0.1%

         \[\leadsto \frac{{a}^{\color{blue}{8}} - \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right) \cdot \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right)}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      4. pow2 0.1%

         \[\leadsto \frac{{a}^{8} - \color{blue}{{\left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)\right)}^{2}}}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      5. *-commutative 0.1%

         \[\leadsto \frac{{a}^{8} - {\color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)}}^{2}}{{a}^{4} - \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 - a\right)} + -1 \]

      6. *-commutative 0.1%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)}^{2}}{{a}^{4} - \color{blue}{\left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)}} + -1 \]

   8. Applied egg-rr 0.1%

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

      1. associate-*r* 0.1%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \color{blue}{\left(\left(4 \cdot a\right) \cdot a\right)}\right)}^{2}}{{a}^{4} - \left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)} + -1 \]

      2. associate-*r* 0.1%

         \[\leadsto \frac{{a}^{8} - {\left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}^{2}}{{a}^{4} - \left(1 - a\right) \cdot \color{blue}{\left(\left(4 \cdot a\right) \cdot a\right)}} + -1 \]

   10. Simplified 0.1%

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

       1. unpow2 0.1%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right) \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       2. associate-*r* 0.1%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot a\right)} \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       3. associate-*l* 0.1%

          \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       4. *-commutative 0.1%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{\left(\left(\left(4 \cdot a\right) \cdot a\right) \cdot \left(1 - a\right)\right)}\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       5. associate-*l* 0.1%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\color{blue}{\left(4 \cdot \left(a \cdot a\right)\right)} \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       6. pow2 0.1%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(4 \cdot \color{blue}{{a}^{2}}\right) \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       7. metadata-eval 0.1%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(\left(4 \cdot {a}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) \cdot \left(1 - a\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       8. associate-*l* 0.1%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \color{blue}{\left(4 \cdot \left({a}^{\left(\frac{4}{2}\right)} \cdot \left(1 - a\right)\right)\right)}\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       9. metadata-eval 0.1%

          \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left({a}^{\color{blue}{2}} \cdot \left(1 - a\right)\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

       10. pow2 0.1%

           \[\leadsto \frac{{a}^{8} - \left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(1 - a\right)\right)\right)\right)}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

   12. Applied egg-rr 0.1%

       \[\leadsto \frac{{a}^{8} - \color{blue}{\left(\left(1 - a\right) \cdot \left(4 \cdot a\right)\right) \cdot \left(a \cdot \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\right)}}{{a}^{4} - \left(1 - a\right) \cdot \left(\left(4 \cdot a\right) \cdot a\right)} + -1 \]

   13. Taylor expanded in a around 0 82.7%

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

       1. *-commutative 82.7%

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

       2. unpow2 82.7%

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

       3. associate-*r* 82.7%

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

       4. *-commutative 82.7%

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

       5. cube-mult 82.7%

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

       6. unpow2 82.7%

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

       7. associate-*l* 82.7%

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

       8. metadata-eval 82.7%

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

       9. distribute-rgt-neg-in 82.7%

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

       10. unpow2 82.7%

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

       11. associate-*r* 82.7%

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

       12. distribute-lft-out 82.7%

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

       13. *-lft-identity 82.7%

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

       14. distribute-lft-neg-in 82.7%

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

       15. distribute-rgt-in 82.7%

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

       16. sub-neg 82.7%

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

       17. associate-*l* 82.7%

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

       18. *-commutative 82.7%

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

       19. associate-*l* 82.7%

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

       20. *-commutative 82.7%

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

       21. associate-*r* 82.7%

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

   15. Simplified 82.7%

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

   if -9.5000000000000003e40 < a < 6.49999999999999972e153

   1. Initial program 94.3%

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

      1. sub-neg 94.3%

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

      2. fma-def 94.3%

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

      3. fma-def 95.4%

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

      4. +-commutative 95.4%

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

      5. metadata-eval 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in a around 0 76.2%

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

      1. associate-+r+ 76.2%

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

      2. associate-*r* 76.2%

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

      3. distribute-rgt-out 85.7%

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

      4. metadata-eval 85.7%

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

      5. distribute-lft-in 85.7%

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

      6. unpow2 85.7%

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

      7. distribute-rgt-in 85.7%

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

      8. metadata-eval 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in a around 0 86.2%

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

      1. unpow2 86.2%

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

   9. Simplified 86.2%

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

   10. Taylor expanded in b around 0 86.2%

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

       1. unpow2 86.2%

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

       2. metadata-eval 86.2%

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

       3. pow-sqr 86.1%

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

       4. unpow2 86.1%

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

       5. unpow2 86.1%

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

       6. distribute-rgt-in 86.1%

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

       7. associate-*l* 86.1%

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

       8. +-commutative 86.1%

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

       9. fma-def 86.1%

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

   12. Simplified 86.1%

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

   13. Taylor expanded in b around 0 60.8%

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

       1. unpow2 60.8%

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

   15. Simplified 60.8%

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

   if 6.49999999999999972e153 < 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. fma-def 0.0%

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

      3. fma-def 0.0%

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

      4. +-commutative 0.0%

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

      5. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in b around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. unpow2 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+40}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 10: 69.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+295}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+295) (+ -1.0 (* a (* a 4.0))) (+ -1.0 (* (* b b) 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+295) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = -1.0 + ((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 ((b * b) <= 2d+295) then
        tmp = (-1.0d0) + (a * (a * 4.0d0))
    else
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+295) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+295:
		tmp = -1.0 + (a * (a * 4.0))
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+295)
		tmp = Float64(-1.0 + Float64(a * Float64(a * 4.0)));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+295)
		tmp = -1.0 + (a * (a * 4.0));
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+295], N[(-1.0 + N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e295

   1. Initial program 80.7%

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

      1. sub-neg 80.7%

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

      2. fma-def 80.7%

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

      3. fma-def 80.7%

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

      4. +-commutative 80.7%

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

      5. metadata-eval 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in b around 0 61.2%

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

      1. associate-*r* 61.2%

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

      2. unpow2 61.2%

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

   6. Simplified 61.2%

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

   7. Taylor expanded in a around 0 58.3%

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

      1. unpow2 58.3%

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

      2. associate-*r* 58.3%

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

   9. Simplified 58.3%

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

   if 2e295 < (*.f64 b b)

   1. Initial program 62.3%

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

      1. sub-neg 62.3%

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

      2. fma-def 62.3%

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

      3. fma-def 66.0%

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

      4. +-commutative 66.0%

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

      5. metadata-eval 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in a around 0 50.9%

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

      1. associate-+r+ 50.9%

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

      2. associate-*r* 50.9%

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

      3. distribute-rgt-out 83.0%

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

      4. metadata-eval 83.0%

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

      5. distribute-lft-in 83.0%

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

      6. unpow2 83.0%

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

      7. distribute-rgt-in 83.0%

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

      8. metadata-eval 83.0%

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

   6. Simplified 83.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

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

       2. metadata-eval 100.0%

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

       3. pow-sqr 100.0%

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

       4. unpow2 100.0%

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

       5. unpow2 100.0%

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

       6. distribute-rgt-in 100.0%

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

       7. associate-*l* 100.0%

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

       8. +-commutative 100.0%

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

       9. fma-def 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

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

   15. Simplified 100.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+295}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 11: 50.7% 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 76.9%

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

   1. sub-neg 76.9%

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

   2. fma-def 76.9%

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

   3. fma-def 77.6%

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

   4. +-commutative 77.6%

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

   5. metadata-eval 77.6%

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

3. Simplified 77.6%

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

4. Taylor expanded in a around 0 60.9%

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

   1. associate-+r+ 60.9%

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

   2. associate-*r* 60.9%

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

   3. distribute-rgt-out 67.5%

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

   4. metadata-eval 67.5%

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

   5. distribute-lft-in 67.5%

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

   6. unpow2 67.5%

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

   7. distribute-rgt-in 67.5%

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

   8. metadata-eval 67.5%

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

6. Simplified 67.5%

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

7. Taylor expanded in a around 0 73.0%

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

   1. unpow2 73.0%

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

9. Simplified 73.0%

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

10. Taylor expanded in b around 0 73.0%

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

    1. unpow2 73.0%

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

    2. metadata-eval 73.0%

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

    3. pow-sqr 73.0%

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

    4. unpow2 73.0%

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

    5. unpow2 73.0%

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

    6. distribute-rgt-in 73.0%

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

    7. associate-*l* 73.0%

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

    8. +-commutative 73.0%

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

    9. fma-def 73.0%

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

12. Simplified 73.0%

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

13. Taylor expanded in b around 0 50.0%

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

    1. unpow2 50.0%

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

15. Simplified 50.0%

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

16. Final simplification 50.0%

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

Alternative 12: 24.8% accurate, 128.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 76.9%

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

   1. sub-neg 76.9%

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

   2. fma-def 76.9%

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

   3. fma-def 77.6%

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

   4. +-commutative 77.6%

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

   5. metadata-eval 77.6%

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

3. Simplified 77.6%

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

4. Taylor expanded in a around inf 66.9%

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

5. Taylor expanded in a around 0 27.4%

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

6. Final simplification 27.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023261 
(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))