Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.8% → 99.4%

Time: 10.6s

Alternatives: 8

Speedup: 1.1×

• 61.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+103}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left({a}^{3} \cdot \left(1 + \frac{1}{a}\right)\right) + -1\right) + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+103)
   (pow a 4.0)
   (+
    (+ (* 4.0 (* (pow a 3.0) (+ 1.0 (/ 1.0 a)))) -1.0)
    (pow (hypot a b) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+103) {
		tmp = pow(a, 4.0);
	} else {
		tmp = ((4.0 * (pow(a, 3.0) * (1.0 + (1.0 / a)))) + -1.0) + pow(hypot(a, b), 4.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+103) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = ((4.0 * (Math.pow(a, 3.0) * (1.0 + (1.0 / a)))) + -1.0) + Math.pow(Math.hypot(a, b), 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+103:
		tmp = math.pow(a, 4.0)
	else:
		tmp = ((4.0 * (math.pow(a, 3.0) * (1.0 + (1.0 / a)))) + -1.0) + math.pow(math.hypot(a, b), 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+103)
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(Float64(4.0 * Float64((a ^ 3.0) * Float64(1.0 + Float64(1.0 / a)))) + -1.0) + (hypot(a, b) ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+103)
		tmp = a ^ 4.0;
	else
		tmp = ((4.0 * ((a ^ 3.0) * (1.0 + (1.0 / a)))) + -1.0) + (hypot(a, b) ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+103], N[Power[a, 4.0], $MachinePrecision], N[(N[(N[(4.0 * N[(N[Power[a, 3.0], $MachinePrecision] * N[(1.0 + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5e103

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. +-commutative 0.0%

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

      3. +-commutative 0.0%

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

      4. sub-neg 0.0%

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

      5. associate-+l+ 0.0%

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

      6. +-commutative 0.0%

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

      7. fma-define 0.0%

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

   3. Simplified 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in a around inf 100.0%

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

   if -5e103 < a

   1. Initial program 89.6%

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

      1. associate--l+ 89.7%

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

      2. +-commutative 89.7%

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

      3. +-commutative 89.7%

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

      4. sub-neg 89.7%

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

      5. associate-+l+ 89.6%

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

      6. +-commutative 89.6%

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

      7. fma-define 89.6%

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

   3. Simplified 91.6%

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

   6. Applied egg-rr 91.6%

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

   7. Taylor expanded in a around inf 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. pow-pow 99.4%

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

      3. metadata-eval 99.4%

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

   9. Applied egg-rr 99.4%

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

       1. *-lft-identity 99.4%

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

   11. Simplified 99.4%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = -1.0 + t_0
	else:
		tmp = -1.0 + (math.pow(a, 2.0) * (4.0 + (a * (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(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(-1.0 + t_0);
	else
		tmp = Float64(-1.0 + Float64((a ^ 2.0) * Float64(4.0 + Float64(a * Float64(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) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = -1.0 + t_0;
	else
		tmp = -1.0 + ((a ^ 2.0) * (4.0 + (a * (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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(-1.0 + t$95$0), $MachinePrecision], N[(-1.0 + N[(N[Power[a, 2.0], $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < +inf.0

   1. Initial program 99.8%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. fma-define 2.8%

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

      4. +-commutative 2.8%

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

      5. associate-*l* 2.8%

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

      6. cancel-sign-sub-inv 2.8%

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

      7. metadata-eval 2.8%

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

      8. fma-define 2.8%

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

      9. metadata-eval 2.8%

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

   3. Simplified 2.8%

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

   5. Taylor expanded in b around 0 24.2%

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

   6. Taylor expanded in a around 0 94.6%

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

4. Final simplification 98.4%

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

Alternative 3: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+34} \lor \neg \left(b \leq 4.9 \cdot 10^{+64}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8000.0)
   (+ (pow a 4.0) -1.0)
   (if (or (<= b 2.1e+34) (not (<= b 4.9e+64))) (pow b 4.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else if ((b <= 2.1e+34) || !(b <= 4.9e+64)) {
		tmp = pow(b, 4.0);
	} else {
		tmp = 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 (b <= 8000.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else if ((b <= 2.1d+34) .or. (.not. (b <= 4.9d+64))) then
        tmp = b ** 4.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 8000.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else if ((b <= 2.1e+34) || !(b <= 4.9e+64)) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8000.0:
		tmp = math.pow(a, 4.0) + -1.0
	elif (b <= 2.1e+34) or not (b <= 4.9e+64):
		tmp = math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	elseif ((b <= 2.1e+34) || !(b <= 4.9e+64))
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8000.0)
		tmp = (a ^ 4.0) + -1.0;
	elseif ((b <= 2.1e+34) || ~((b <= 4.9e+64)))
		tmp = b ^ 4.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], If[Or[LessEqual[b, 2.1e+34], N[Not[LessEqual[b, 4.9e+64]], $MachinePrecision]], N[Power[b, 4.0], $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 8e3

   1. Initial program 74.7%

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

      1. sub-neg 74.7%

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

      2. +-commutative 74.7%

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

      3. fma-define 75.2%

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

      4. +-commutative 75.2%

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

      5. associate-*l* 75.2%

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

      6. cancel-sign-sub-inv 75.2%

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

      7. metadata-eval 75.2%

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

      8. fma-define 75.2%

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

      9. metadata-eval 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in a around inf 79.3%

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

   if 8e3 < b < 2.10000000000000017e34 or 4.9000000000000003e64 < b

   1. Initial program 64.1%

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

      1. associate--l+ 64.1%

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

      2. +-commutative 64.1%

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

      3. +-commutative 64.1%

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

      4. sub-neg 64.1%

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

      5. associate-+l+ 64.1%

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

      6. +-commutative 64.1%

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

      7. fma-define 64.1%

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

   3. Simplified 67.9%

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

   6. Applied egg-rr 67.9%

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

   7. Taylor expanded in b around inf 99.3%

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

   if 2.10000000000000017e34 < b < 4.9000000000000003e64

   1. Initial program 50.0%

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

      1. associate--l+ 50.0%

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

      2. +-commutative 50.0%

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

      3. +-commutative 50.0%

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

      4. sub-neg 50.0%

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

      5. associate-+l+ 50.0%

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

      6. +-commutative 50.0%

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

      7. fma-define 50.0%

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

   3. Simplified 50.0%

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in a around inf 75.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+34} \lor \neg \left(b \leq 4.9 \cdot 10^{+64}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{+57} \lor \neg \left(a \leq 3.2 \cdot 10^{+40}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.6e+57) (not (<= a 3.2e+40)))
   (pow a 4.0)
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.6e+57) || !(a <= 3.2e+40)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.6d+57)) .or. (.not. (a <= 3.2d+40))) then
        tmp = a ** 4.0d0
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.6e+57) || !(a <= 3.2e+40)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.6e+57) or not (a <= 3.2e+40):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.6e+57) || !(a <= 3.2e+40))
		tmp = a ^ 4.0;
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.6e+57) || ~((a <= 3.2e+40)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.6e+57], N[Not[LessEqual[a, 3.2e+40]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.60000000000000015e57 or 3.19999999999999981e40 < a

   1. Initial program 41.1%

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

      1. associate--l+ 41.1%

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

      2. +-commutative 41.1%

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

      3. +-commutative 41.1%

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

      4. sub-neg 41.1%

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

      5. associate-+l+ 41.1%

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

      6. +-commutative 41.1%

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

      7. fma-define 41.1%

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

   3. Simplified 44.7%

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

   6. Applied egg-rr 44.7%

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

   7. Taylor expanded in a around inf 99.2%

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

   if -1.60000000000000015e57 < a < 3.19999999999999981e40

   1. Initial program 97.0%

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

      1. sub-neg 97.0%

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

      2. +-commutative 97.0%

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

      3. fma-define 97.0%

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

      4. +-commutative 97.0%

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

      5. associate-*l* 97.0%

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

      6. cancel-sign-sub-inv 97.0%

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

      7. metadata-eval 97.0%

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

      8. fma-define 97.0%

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

      9. metadata-eval 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in b around inf 93.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{+57} \lor \neg \left(a \leq 3.2 \cdot 10^{+40}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-8} \lor \neg \left(a \leq 1.95\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(-1 + a \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -9.8e-8) (not (<= a 1.95)))
   (pow a 4.0)
   (* (+ 1.0 (* a 2.0)) (+ -1.0 (* a 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -9.8e-8) || !(a <= 1.95)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (1.0 + (a * 2.0)) * (-1.0 + (a * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-9.8d-8)) .or. (.not. (a <= 1.95d0))) then
        tmp = a ** 4.0d0
    else
        tmp = (1.0d0 + (a * 2.0d0)) * ((-1.0d0) + (a * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -9.8e-8) || !(a <= 1.95)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (1.0 + (a * 2.0)) * (-1.0 + (a * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -9.8e-8) or not (a <= 1.95):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (1.0 + (a * 2.0)) * (-1.0 + (a * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -9.8e-8) || !(a <= 1.95))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(1.0 + Float64(a * 2.0)) * Float64(-1.0 + Float64(a * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -9.8e-8) || ~((a <= 1.95)))
		tmp = a ^ 4.0;
	else
		tmp = (1.0 + (a * 2.0)) * (-1.0 + (a * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -9.8e-8], N[Not[LessEqual[a, 1.95]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(1.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.8000000000000004e-8 or 1.94999999999999996 < a

   1. Initial program 47.6%

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

      1. associate--l+ 47.6%

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

      2. +-commutative 47.6%

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

      3. +-commutative 47.6%

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

      4. sub-neg 47.6%

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

      5. associate-+l+ 47.6%

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

      6. +-commutative 47.6%

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

      7. fma-define 47.6%

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

   3. Simplified 50.5%

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

   6. Applied egg-rr 50.5%

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

   7. Taylor expanded in a around inf 88.0%

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

   if -9.8000000000000004e-8 < a < 1.94999999999999996

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. fma-define 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-*l* 99.9%

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

      6. cancel-sign-sub-inv 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 52.7%

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

   6. Taylor expanded in a around 0 51.9%

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

      1. add-sqr-sqrt 51.9%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

      2. difference-of-sqr-1 51.9%

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

      3. *-commutative 51.9%

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

      4. sqrt-prod 51.9%

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

      5. sqrt-pow1 51.8%

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

      6. metadata-eval 51.8%

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

      7. pow1 51.8%

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

      8. metadata-eval 51.8%

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

      9. *-commutative 51.8%

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

      10. sqrt-prod 51.8%

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

      11. sqrt-pow1 51.9%

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

      12. metadata-eval 51.9%

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

      13. pow1 51.9%

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

      14. metadata-eval 51.9%

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

   8. Applied egg-rr 51.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-8} \lor \neg \left(a \leq 1.95\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(-1 + a \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7000:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(-1 + a \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7000.0) (* (+ 1.0 (* a 2.0)) (+ -1.0 (* a 2.0))) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7000.0) {
		tmp = (1.0 + (a * 2.0)) * (-1.0 + (a * 2.0));
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 7000.0d0) then
        tmp = (1.0d0 + (a * 2.0d0)) * ((-1.0d0) + (a * 2.0d0))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7000.0) {
		tmp = (1.0 + (a * 2.0)) * (-1.0 + (a * 2.0));
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7000.0:
		tmp = (1.0 + (a * 2.0)) * (-1.0 + (a * 2.0))
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7000.0)
		tmp = Float64(Float64(1.0 + Float64(a * 2.0)) * Float64(-1.0 + Float64(a * 2.0)));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7000.0)
		tmp = (1.0 + (a * 2.0)) * (-1.0 + (a * 2.0));
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7000.0], N[(N[(1.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7e3

   1. Initial program 74.7%

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

      1. sub-neg 74.7%

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

      2. +-commutative 74.7%

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

      3. fma-define 75.2%

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

      4. +-commutative 75.2%

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

      5. associate-*l* 75.2%

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

      6. cancel-sign-sub-inv 75.2%

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

      7. metadata-eval 75.2%

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

      8. fma-define 75.2%

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

      9. metadata-eval 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in b around 0 61.4%

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

   6. Taylor expanded in a around 0 58.6%

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

      1. add-sqr-sqrt 58.6%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

      2. difference-of-sqr-1 58.6%

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

      3. *-commutative 58.6%

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

      4. sqrt-prod 58.6%

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

      5. sqrt-pow1 42.4%

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

      6. metadata-eval 42.4%

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

      7. pow1 42.4%

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

      8. metadata-eval 42.4%

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

      9. *-commutative 42.4%

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

      10. sqrt-prod 42.4%

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

      11. sqrt-pow1 58.6%

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

      12. metadata-eval 58.6%

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

      13. pow1 58.6%

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

      14. metadata-eval 58.6%

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

   8. Applied egg-rr 58.6%

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

   if 7e3 < b

   1. Initial program 63.1%

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

      1. associate--l+ 63.1%

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

      2. +-commutative 63.1%

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

      3. +-commutative 63.1%

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

      4. sub-neg 63.1%

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

      5. associate-+l+ 63.1%

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

      6. +-commutative 63.1%

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

      7. fma-define 63.1%

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

   3. Simplified 66.6%

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

   6. Applied egg-rr 66.6%

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

   7. Taylor expanded in b around inf 94.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7000:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(-1 + a \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.8% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return (1.0 + (a * 2.0)) * (-1.0 + (a * 2.0));
}

Python

def code(a, b):
	return (1.0 + (a * 2.0)) * (-1.0 + (a * 2.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. sub-neg 72.1%

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

   2. +-commutative 72.1%

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

   3. fma-define 72.9%

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

   4. +-commutative 72.9%

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

   5. associate-*l* 72.9%

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

   6. cancel-sign-sub-inv 72.9%

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

   7. metadata-eval 72.9%

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

   8. fma-define 72.9%

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

   9. metadata-eval 72.9%

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

3. Simplified 72.9%

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

5. Taylor expanded in b around 0 52.6%

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

6. Taylor expanded in a around 0 51.4%

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

   1. add-sqr-sqrt 51.4%

      \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

   2. difference-of-sqr-1 51.4%

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

   3. *-commutative 51.4%

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

   4. sqrt-prod 51.4%

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

   5. sqrt-pow1 34.8%

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

   6. metadata-eval 34.8%

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

   7. pow1 34.8%

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

   8. metadata-eval 34.8%

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

   9. *-commutative 34.8%

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

   10. sqrt-prod 34.8%

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

   11. sqrt-pow1 51.4%

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

   12. metadata-eval 51.4%

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

   13. pow1 51.4%

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

   14. metadata-eval 51.4%

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

8. Applied egg-rr 51.4%

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

9. Final simplification 51.4%

   \[\leadsto \left(1 + a \cdot 2\right) \cdot \left(-1 + a \cdot 2\right) \]
10. Add Preprocessing

Alternative 8: 24.5% accurate, 130.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 72.1%

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

   1. sub-neg 72.1%

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

   2. +-commutative 72.1%

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

   3. fma-define 72.9%

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

   4. +-commutative 72.9%

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

   5. associate-*l* 72.9%

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

   6. cancel-sign-sub-inv 72.9%

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

   7. metadata-eval 72.9%

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

   8. fma-define 72.9%

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

   9. metadata-eval 72.9%

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

3. Simplified 72.9%

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

5. Taylor expanded in a around inf 70.9%

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

6. Taylor expanded in a around 0 24.6%

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

Reproduce

herbie shell --seed 2024083 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))