Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.9% → 98.6%

Time: 10.7s

Alternatives: 10

Speedup: 1.2×

• 60.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\\ \mathbf{if}\;a \leq -5 \cdot 10^{+48}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, {a}^{2} \cdot \mathsf{fma}\left(4, a, 4\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (pow (hypot a b) 2.0)))
   (if (<= a -5e+48)
     (pow a 4.0)
     (+ (fma t_0 t_0 (* (pow a 2.0) (fma 4.0 a 4.0))) -1.0))))

C

double code(double a, double b) {
	double t_0 = pow(hypot(a, b), 2.0);
	double tmp;
	if (a <= -5e+48) {
		tmp = pow(a, 4.0);
	} else {
		tmp = fma(t_0, t_0, (pow(a, 2.0) * fma(4.0, a, 4.0))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = hypot(a, b) ^ 2.0
	tmp = 0.0
	if (a <= -5e+48)
		tmp = a ^ 4.0;
	else
		tmp = Float64(fma(t_0, t_0, Float64((a ^ 2.0) * fma(4.0, a, 4.0))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[a, -5e+48], N[Power[a, 4.0], $MachinePrecision], N[(N[(t$95$0 * t$95$0 + N[(N[Power[a, 2.0], $MachinePrecision] * N[(4.0 * a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.99999999999999973e48

   1. Initial program 15.5%

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

      1. associate--l+ 15.5%

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

      2. +-commutative 15.5%

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

      3. +-commutative 15.5%

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

      4. sub-neg 15.5%

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

      5. associate-+l+ 15.5%

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

      6. +-commutative 15.5%

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

      7. associate-+l+ 15.5%

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

   3. Simplified 15.5%

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

   5. Taylor expanded in a around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around inf 100.0%

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

   if -4.99999999999999973e48 < a

   1. Initial program 87.1%

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

      1. sub-neg 87.1%

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

   3. Simplified 87.1%

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

   6. Applied egg-rr 91.8%

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

   7. Taylor expanded in b around 0 99.2%

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

      1. distribute-lft-in 99.2%

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

      2. unpow2 99.2%

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

      3. associate-*r* 99.2%

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

      4. unpow3 99.2%

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

      5. unpow2 99.2%

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

      6. associate-*r* 99.2%

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

      7. unpow2 99.2%

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

      8. associate-*r* 99.2%

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

      9. distribute-rgt-in 99.2%

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

      10. distribute-rgt-in 99.2%

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

      11. associate-*r* 99.2%

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

      12. unpow2 99.2%

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

      13. +-commutative 99.2%

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

      14. fma-define 99.2%

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

   9. Simplified 99.2%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #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. 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. associate-+l+ 0.0%

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

   3. Simplified 15.4%

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

   5. Taylor expanded in a around inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. metadata-eval 94.1%

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

   7. Simplified 94.1%

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

   8. Taylor expanded in a around inf 94.1%

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

4. Final simplification 98.4%

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

Alternative 3: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -63:\\ \;\;\;\;{a}^{4} \cdot \frac{a + 4}{a}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -63.0)
   (* (pow a 4.0) (/ (+ a 4.0) a))
   (if (<= a 3.1e+34) (+ (+ (* 4.0 (* b b)) (pow b 4.0)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -63.0) {
		tmp = pow(a, 4.0) * ((a + 4.0) / a);
	} else if (a <= 3.1e+34) {
		tmp = ((4.0 * (b * b)) + pow(b, 4.0)) + -1.0;
	} 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 (a <= (-63.0d0)) then
        tmp = (a ** 4.0d0) * ((a + 4.0d0) / a)
    else if (a <= 3.1d+34) then
        tmp = ((4.0d0 * (b * b)) + (b ** 4.0d0)) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -63.0) {
		tmp = Math.pow(a, 4.0) * ((a + 4.0) / a);
	} else if (a <= 3.1e+34) {
		tmp = ((4.0 * (b * b)) + Math.pow(b, 4.0)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -63.0:
		tmp = math.pow(a, 4.0) * ((a + 4.0) / a)
	elif a <= 3.1e+34:
		tmp = ((4.0 * (b * b)) + math.pow(b, 4.0)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -63.0)
		tmp = Float64((a ^ 4.0) * Float64(Float64(a + 4.0) / a));
	elseif (a <= 3.1e+34)
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + (b ^ 4.0)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -63.0)
		tmp = (a ^ 4.0) * ((a + 4.0) / a);
	elseif (a <= 3.1e+34)
		tmp = ((4.0 * (b * b)) + (b ^ 4.0)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -63.0], N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[(a + 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.1e+34], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -63

   1. Initial program 30.8%

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

      1. associate--l+ 30.8%

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

      2. +-commutative 30.8%

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

      3. +-commutative 30.8%

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

      4. sub-neg 30.8%

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

      5. associate-+l+ 30.8%

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

      6. +-commutative 30.8%

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

      7. associate-+l+ 30.8%

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

   3. Simplified 30.8%

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

   5. Taylor expanded in a around inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. metadata-eval 94.1%

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

   7. Simplified 94.1%

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

   8. Taylor expanded in a around 0 94.1%

      \[\leadsto {a}^{4} \cdot \color{blue}{\frac{4 + a}{a}} \]

   if -63 < a < 3.09999999999999977e34

   1. Initial program 97.8%

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

      1. associate--l+ 97.8%

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

      2. +-commutative 97.8%

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

      3. +-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. associate-+l+ 97.8%

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

      6. +-commutative 97.8%

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

      7. associate-+l+ 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in a around 0 97.4%

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

      1. unpow2 97.4%

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

   7. Applied egg-rr 97.4%

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

   if 3.09999999999999977e34 < a

   1. Initial program 54.6%

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

      1. associate--l+ 54.6%

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

      2. +-commutative 54.6%

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

      3. +-commutative 54.6%

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

      4. sub-neg 54.6%

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

      5. associate-+l+ 54.6%

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

      6. +-commutative 54.6%

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

      7. associate-+l+ 54.6%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in a around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around inf 98.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -63:\\ \;\;\;\;{a}^{4} \cdot \frac{a + 4}{a}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -63:\\ \;\;\;\;{a}^{4} \cdot \frac{a + 4}{a}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -63.0)
   (* (pow a 4.0) (/ (+ a 4.0) a))
   (if (<= a 3.1e+34) (+ (* 4.0 (* b b)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -63.0) {
		tmp = pow(a, 4.0) * ((a + 4.0) / a);
	} else if (a <= 3.1e+34) {
		tmp = (4.0 * (b * b)) + -1.0;
	} 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 (a <= (-63.0d0)) then
        tmp = (a ** 4.0d0) * ((a + 4.0d0) / a)
    else if (a <= 3.1d+34) then
        tmp = (4.0d0 * (b * b)) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -63.0) {
		tmp = Math.pow(a, 4.0) * ((a + 4.0) / a);
	} else if (a <= 3.1e+34) {
		tmp = (4.0 * (b * b)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -63.0:
		tmp = math.pow(a, 4.0) * ((a + 4.0) / a)
	elif a <= 3.1e+34:
		tmp = (4.0 * (b * b)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -63.0)
		tmp = Float64((a ^ 4.0) * Float64(Float64(a + 4.0) / a));
	elseif (a <= 3.1e+34)
		tmp = Float64(Float64(4.0 * Float64(b * b)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -63.0)
		tmp = (a ^ 4.0) * ((a + 4.0) / a);
	elseif (a <= 3.1e+34)
		tmp = (4.0 * (b * b)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -63.0], N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[(a + 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.1e+34], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -63

   1. Initial program 30.8%

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

      1. associate--l+ 30.8%

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

      2. +-commutative 30.8%

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

      3. +-commutative 30.8%

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

      4. sub-neg 30.8%

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

      5. associate-+l+ 30.8%

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

      6. +-commutative 30.8%

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

      7. associate-+l+ 30.8%

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

   3. Simplified 30.8%

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

   5. Taylor expanded in a around inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. metadata-eval 94.1%

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

   7. Simplified 94.1%

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

   8. Taylor expanded in a around 0 94.1%

      \[\leadsto {a}^{4} \cdot \color{blue}{\frac{4 + a}{a}} \]

   if -63 < a < 3.09999999999999977e34

   1. Initial program 97.8%

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

      1. associate--l+ 97.8%

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

      2. +-commutative 97.8%

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

      3. +-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. associate-+l+ 97.8%

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

      6. +-commutative 97.8%

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

      7. associate-+l+ 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in a around 0 97.4%

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

   6. Taylor expanded in b around 0 73.0%

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

      1. unpow2 97.4%

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

   8. Applied egg-rr 73.0%

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

   if 3.09999999999999977e34 < a

   1. Initial program 54.6%

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

      1. associate--l+ 54.6%

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

      2. +-commutative 54.6%

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

      3. +-commutative 54.6%

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

      4. sub-neg 54.6%

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

      5. associate-+l+ 54.6%

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

      6. +-commutative 54.6%

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

      7. associate-+l+ 54.6%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in a around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around inf 98.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -63:\\ \;\;\;\;{a}^{4} \cdot \frac{a + 4}{a}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -63:\\ \;\;\;\;{a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -63.0)
   (* (pow a 4.0) (+ 1.0 (/ 4.0 a)))
   (if (<= a 3.1e+34) (+ (* 4.0 (* b b)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -63.0) {
		tmp = pow(a, 4.0) * (1.0 + (4.0 / a));
	} else if (a <= 3.1e+34) {
		tmp = (4.0 * (b * b)) + -1.0;
	} 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 (a <= (-63.0d0)) then
        tmp = (a ** 4.0d0) * (1.0d0 + (4.0d0 / a))
    else if (a <= 3.1d+34) then
        tmp = (4.0d0 * (b * b)) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -63.0) {
		tmp = Math.pow(a, 4.0) * (1.0 + (4.0 / a));
	} else if (a <= 3.1e+34) {
		tmp = (4.0 * (b * b)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -63.0:
		tmp = math.pow(a, 4.0) * (1.0 + (4.0 / a))
	elif a <= 3.1e+34:
		tmp = (4.0 * (b * b)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -63.0)
		tmp = Float64((a ^ 4.0) * Float64(1.0 + Float64(4.0 / a)));
	elseif (a <= 3.1e+34)
		tmp = Float64(Float64(4.0 * Float64(b * b)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -63.0)
		tmp = (a ^ 4.0) * (1.0 + (4.0 / a));
	elseif (a <= 3.1e+34)
		tmp = (4.0 * (b * b)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -63.0], N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 + N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.1e+34], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -63

   1. Initial program 30.8%

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

      1. associate--l+ 30.8%

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

      2. +-commutative 30.8%

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

      3. +-commutative 30.8%

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

      4. sub-neg 30.8%

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

      5. associate-+l+ 30.8%

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

      6. +-commutative 30.8%

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

      7. associate-+l+ 30.8%

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

   3. Simplified 30.8%

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

   5. Taylor expanded in a around inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. metadata-eval 94.1%

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

   7. Simplified 94.1%

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

   if -63 < a < 3.09999999999999977e34

   1. Initial program 97.8%

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

      1. associate--l+ 97.8%

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

      2. +-commutative 97.8%

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

      3. +-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. associate-+l+ 97.8%

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

      6. +-commutative 97.8%

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

      7. associate-+l+ 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in a around 0 97.4%

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

   6. Taylor expanded in b around 0 73.0%

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

      1. unpow2 97.4%

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

   8. Applied egg-rr 73.0%

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

   if 3.09999999999999977e34 < a

   1. Initial program 54.6%

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

      1. associate--l+ 54.6%

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

      2. +-commutative 54.6%

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

      3. +-commutative 54.6%

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

      4. sub-neg 54.6%

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

      5. associate-+l+ 54.6%

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

      6. +-commutative 54.6%

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

      7. associate-+l+ 54.6%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in a around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around inf 98.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -63:\\ \;\;\;\;{a}^{4} \cdot \left(1 + \frac{4}{a}\right)\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -58 \lor \neg \left(a \leq 3.1 \cdot 10^{+34}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -58.0) (not (<= a 3.1e+34)))
   (pow a 4.0)
   (+ (* 4.0 (* b b)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -58.0) || !(a <= 3.1e+34)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (4.0 * (b * b)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-58.0d0)) .or. (.not. (a <= 3.1d+34))) then
        tmp = a ** 4.0d0
    else
        tmp = (4.0d0 * (b * b)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -58.0) || !(a <= 3.1e+34)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (4.0 * (b * b)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -58.0) or not (a <= 3.1e+34):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (4.0 * (b * b)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -58.0) || !(a <= 3.1e+34))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(4.0 * Float64(b * b)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -58.0) || ~((a <= 3.1e+34)))
		tmp = a ^ 4.0;
	else
		tmp = (4.0 * (b * b)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -58.0], N[Not[LessEqual[a, 3.1e+34]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -58 or 3.09999999999999977e34 < a

   1. Initial program 42.4%

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

      1. associate--l+ 42.4%

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

      2. +-commutative 42.4%

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

      3. +-commutative 42.4%

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

      4. sub-neg 42.4%

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

      5. associate-+l+ 42.4%

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

      6. +-commutative 42.4%

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

      7. associate-+l+ 42.4%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in a around inf 96.1%

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

      1. associate-*r/ 96.1%

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

      2. metadata-eval 96.1%

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

   7. Simplified 96.1%

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

   8. Taylor expanded in a around inf 95.3%

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

   if -58 < a < 3.09999999999999977e34

   1. Initial program 97.8%

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

      1. associate--l+ 97.8%

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

      2. +-commutative 97.8%

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

      3. +-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. associate-+l+ 97.8%

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

      6. +-commutative 97.8%

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

      7. associate-+l+ 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in a around 0 97.4%

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

   6. Taylor expanded in b around 0 73.0%

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

      1. unpow2 97.4%

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

   8. Applied egg-rr 73.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -58 \lor \neg \left(a \leq 3.1 \cdot 10^{+34}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8:\\ \;\;\;\;\left(a + 4\right) \cdot {a}^{3}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.8)
   (* (+ a 4.0) (pow a 3.0))
   (if (<= a 3.1e+34) (+ (* 4.0 (* b b)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.8) {
		tmp = (a + 4.0) * pow(a, 3.0);
	} else if (a <= 3.1e+34) {
		tmp = (4.0 * (b * b)) + -1.0;
	} 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 (a <= (-7.8d0)) then
        tmp = (a + 4.0d0) * (a ** 3.0d0)
    else if (a <= 3.1d+34) then
        tmp = (4.0d0 * (b * b)) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7.8) {
		tmp = (a + 4.0) * Math.pow(a, 3.0);
	} else if (a <= 3.1e+34) {
		tmp = (4.0 * (b * b)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7.8:
		tmp = (a + 4.0) * math.pow(a, 3.0)
	elif a <= 3.1e+34:
		tmp = (4.0 * (b * b)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.8)
		tmp = Float64(Float64(a + 4.0) * (a ^ 3.0));
	elseif (a <= 3.1e+34)
		tmp = Float64(Float64(4.0 * Float64(b * b)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7.8)
		tmp = (a + 4.0) * (a ^ 3.0);
	elseif (a <= 3.1e+34)
		tmp = (4.0 * (b * b)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.8], N[(N[(a + 4.0), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.1e+34], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.79999999999999982

   1. Initial program 30.8%

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

      1. associate--l+ 30.8%

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

      2. +-commutative 30.8%

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

      3. +-commutative 30.8%

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

      4. sub-neg 30.8%

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

      5. associate-+l+ 30.8%

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

      6. +-commutative 30.8%

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

      7. associate-+l+ 30.8%

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

   3. Simplified 30.8%

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

   5. Taylor expanded in a around inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. metadata-eval 94.1%

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

   7. Simplified 94.1%

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

   8. Taylor expanded in a around 0 94.0%

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

   if -7.79999999999999982 < a < 3.09999999999999977e34

   1. Initial program 97.8%

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

      1. associate--l+ 97.8%

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

      2. +-commutative 97.8%

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

      3. +-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. associate-+l+ 97.8%

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

      6. +-commutative 97.8%

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

      7. associate-+l+ 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in a around 0 97.4%

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

   6. Taylor expanded in b around 0 73.0%

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

      1. unpow2 97.4%

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

   8. Applied egg-rr 73.0%

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

   if 3.09999999999999977e34 < a

   1. Initial program 54.6%

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

      1. associate--l+ 54.6%

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

      2. +-commutative 54.6%

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

      3. +-commutative 54.6%

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

      4. sub-neg 54.6%

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

      5. associate-+l+ 54.6%

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

      6. +-commutative 54.6%

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

      7. associate-+l+ 54.6%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in a around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around inf 98.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8:\\ \;\;\;\;\left(a + 4\right) \cdot {a}^{3}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.7% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return (4.0 * (b * b)) + -1.0;
}

Python

def code(a, b):
	return (4.0 * (b * b)) + -1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate--l+ 74.5%

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

   2. +-commutative 74.5%

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

   3. +-commutative 74.5%

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

   4. sub-neg 74.5%

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

   5. associate-+l+ 74.5%

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

   6. +-commutative 74.5%

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

   7. associate-+l+ 74.5%

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

3. Simplified 78.4%

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

5. Taylor expanded in a around 0 70.8%

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

6. Taylor expanded in b around 0 50.8%

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

   1. unpow2 70.8%

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

8. Applied egg-rr 50.8%

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

9. Final simplification 50.8%

   \[\leadsto 4 \cdot \left(b \cdot b\right) + -1 \]
10. Add Preprocessing

Alternative 9: 26.4% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ b \cdot 2 + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate--l+ 74.5%

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

   2. +-commutative 74.5%

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

   3. +-commutative 74.5%

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

   4. sub-neg 74.5%

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

   5. associate-+l+ 74.5%

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

   6. +-commutative 74.5%

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

   7. associate-+l+ 74.5%

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

3. Simplified 78.4%

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

5. Taylor expanded in a around 0 70.8%

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

6. Taylor expanded in b around 0 50.8%

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

   1. pow2 50.8%

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

   2. add-sqr-sqrt 50.8%

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

   3. difference-of-sqr-1 50.8%

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

   4. *-commutative 50.8%

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

   5. sqrt-prod 50.8%

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

   6. sqrt-prod 29.4%

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

   7. add-sqr-sqrt 39.4%

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

   8. metadata-eval 39.4%

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

   9. *-commutative 39.4%

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

   10. sqrt-prod 39.4%

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

   11. sqrt-prod 29.4%

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

   12. add-sqr-sqrt 50.8%

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

   13. metadata-eval 50.8%

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

8. Applied egg-rr 50.8%

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

9. Taylor expanded in b around 0 25.3%

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

10. Final simplification 25.3%

    \[\leadsto b \cdot 2 + -1 \]
11. Add Preprocessing

Alternative 10: 25.5% 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 74.5%

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

   1. associate--l+ 74.5%

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

   2. +-commutative 74.5%

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

   3. +-commutative 74.5%

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

   4. sub-neg 74.5%

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

   5. associate-+l+ 74.5%

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

   6. +-commutative 74.5%

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

   7. associate-+l+ 74.5%

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

3. Simplified 78.4%

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

5. Taylor expanded in a around 0 70.8%

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

6. Taylor expanded in b around 0 24.5%

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

Reproduce

herbie shell --seed 2024165 
(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))