Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.4% → 100.0%

Time: 8.0s

Alternatives: 12

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(2.0 * N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision] + N[(b * N[(b * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(a * N[(N[(a * N[(4.0 + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * N[(4.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 3 binary64) a))))) < +inf.0

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. fma-define 99.7%

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

      3. distribute-rgt-in 99.7%

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

      4. sqr-neg 99.7%

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

      5. distribute-rgt-in 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around 0 94.2%

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

      1. +-commutative 94.2%

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

      2. distribute-rgt-in 90.5%

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

      3. pow-sqr 90.6%

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

      4. metadata-eval 90.6%

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

      5. associate-*r* 90.6%

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

      6. *-commutative 90.6%

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

      7. fma-define 90.6%

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

      8. unpow2 90.6%

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

      9. unpow2 90.6%

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

      10. swap-sqr 100.0%

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

      11. unpow2 100.0%

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

      12. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   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 3 binary64) a)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0 71.9%

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

   4. Taylor expanded in a around inf 71.9%

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

   5. Taylor expanded in a around 0 79.7%

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

   6. Taylor expanded in b around 0 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(a * N[(N[(a * N[(4.0 + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * N[(4.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 3 binary64) a))))) < +inf.0

   1. Initial program 99.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 3 binary64) a)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0 71.9%

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

   4. Taylor expanded in a around inf 71.9%

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

   5. Taylor expanded in a around 0 79.7%

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

   6. Taylor expanded in b around 0 100.0%

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

4. Final simplification 99.8%

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

Alternative 3: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(4.0 + N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 3 binary64) a))))) < +inf.0

   1. Initial program 99.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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 3 binary64) a)))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-define 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

   3. Simplified 9.4%

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

   5. Taylor expanded in b around 0 25.1%

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

   6. Taylor expanded in a around 0 97.0%

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

4. Final simplification 99.0%

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

Alternative 4: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+59}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.6e+59)
   (pow a 4.0)
   (+
    (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* a a) (* (* b b) (+ a 3.0)))))
    -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.6e+59) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a * a) + ((b * b) * (a + 3.0))))) + -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 <= (-2.6d+59)) then
        tmp = a ** 4.0d0
    else
        tmp = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * ((a * a) + ((b * b) * (a + 3.0d0))))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.6e+59) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a * a) + ((b * b) * (a + 3.0))))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.6e+59:
		tmp = math.pow(a, 4.0)
	else:
		tmp = (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a * a) + ((b * b) * (a + 3.0))))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.6e+59)
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(a * a) + Float64(Float64(b * b) * Float64(a + 3.0))))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.6e+59)
		tmp = a ^ 4.0;
	else
		tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * ((a * a) + ((b * b) * (a + 3.0))))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.6e+59], N[Power[a, 4.0], $MachinePrecision], N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.59999999999999999e59

   1. Initial program 63.5%

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

      1. associate--l+ 63.5%

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

      2. fma-define 63.5%

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

      3. distribute-rgt-in 63.5%

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

      4. sqr-neg 63.5%

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

      5. distribute-rgt-in 63.5%

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

   3. Simplified 63.5%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, 1 - a, b \cdot \left(b \cdot \left(a + 3\right)\right)\right) - 1\right)} \]
   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 -2.59999999999999999e59 < a

   1. Initial program 77.1%

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

   3. Taylor expanded in a around 0 97.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+59}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a + a \cdot \left(b \cdot b\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+140)
   (+ (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* a a) (* a (* b b))))) -1.0)
   (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+140) {
		tmp = (pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a * a) + (a * (b * b))))) + -1.0;
	} 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 * b) <= 5d+140) then
        tmp = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * ((a * a) + (a * (b * b))))) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+140) {
		tmp = (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a * a) + (a * (b * b))))) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+140:
		tmp = (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * ((a * a) + (a * (b * b))))) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+140)
		tmp = Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(a * a) + Float64(a * Float64(b * b))))) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+140)
		tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * ((a * a) + (a * (b * b))))) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+140], N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(a * a), $MachinePrecision] + N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.00000000000000008e140

   1. Initial program 82.3%

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

   3. Taylor expanded in a around 0 97.4%

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

   4. Taylor expanded in a around inf 96.8%

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

   if 5.00000000000000008e140 < (*.f64 b b)

   1. Initial program 59.0%

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

      1. associate--l+ 59.0%

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

      2. fma-define 59.0%

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

      3. distribute-rgt-in 59.0%

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

      4. sqr-neg 59.0%

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

      5. distribute-rgt-in 59.0%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a + a \cdot \left(b \cdot b\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -75000000000 \lor \neg \left(a \leq 152000000\right):\\ \;\;\;\;{a}^{4} \cdot \left(1 - \frac{4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 12\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -75000000000.0) (not (<= a 152000000.0)))
   (* (pow a 4.0) (- 1.0 (/ 4.0 a)))
   (+ (+ (pow b 4.0) (* (* b b) 12.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -75000000000.0) || !(a <= 152000000.0)) {
		tmp = pow(a, 4.0) * (1.0 - (4.0 / a));
	} else {
		tmp = (pow(b, 4.0) + ((b * b) * 12.0)) + -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 <= (-75000000000.0d0)) .or. (.not. (a <= 152000000.0d0))) then
        tmp = (a ** 4.0d0) * (1.0d0 - (4.0d0 / a))
    else
        tmp = ((b ** 4.0d0) + ((b * b) * 12.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -75000000000.0) || !(a <= 152000000.0)) {
		tmp = Math.pow(a, 4.0) * (1.0 - (4.0 / a));
	} else {
		tmp = (Math.pow(b, 4.0) + ((b * b) * 12.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -75000000000.0) or not (a <= 152000000.0):
		tmp = math.pow(a, 4.0) * (1.0 - (4.0 / a))
	else:
		tmp = (math.pow(b, 4.0) + ((b * b) * 12.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -75000000000.0) || !(a <= 152000000.0))
		tmp = Float64((a ^ 4.0) * Float64(1.0 - Float64(4.0 / a)));
	else
		tmp = Float64(Float64((b ^ 4.0) + Float64(Float64(b * b) * 12.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -75000000000.0) || ~((a <= 152000000.0)))
		tmp = (a ^ 4.0) * (1.0 - (4.0 / a));
	else
		tmp = ((b ^ 4.0) + ((b * b) * 12.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -75000000000.0], N[Not[LessEqual[a, 152000000.0]], $MachinePrecision]], N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 - N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5e10 or 1.52e8 < a

   1. Initial program 45.9%

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

      1. associate--l+ 45.9%

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

      2. fma-define 45.9%

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

      3. distribute-rgt-in 45.9%

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

      4. sqr-neg 45.9%

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

      5. distribute-rgt-in 45.9%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in a around inf 94.3%

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

      1. associate-*r/ 94.3%

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

      2. metadata-eval 94.3%

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

   7. Simplified 94.3%

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

   if -7.5e10 < a < 1.52e8

   1. Initial program 98.4%

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

      1. associate--l+ 98.4%

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

      2. fma-define 98.4%

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

      3. distribute-rgt-in 98.4%

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

      4. sqr-neg 98.4%

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

      5. distribute-rgt-in 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in a around 0 99.0%

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

      1. pow2 99.0%

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

   7. Applied egg-rr 99.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -75000000000 \lor \neg \left(a \leq 152000000\right):\\ \;\;\;\;{a}^{4} \cdot \left(1 - \frac{4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 12\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -75000000000 \lor \neg \left(a \leq 15500000\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -75000000000.0) (not (<= a 15500000.0)))
   (pow a 4.0)
   (+ (* (* b b) 12.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -75000000000.0) || !(a <= 15500000.0)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = ((b * b) * 12.0) + -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 <= (-75000000000.0d0)) .or. (.not. (a <= 15500000.0d0))) then
        tmp = a ** 4.0d0
    else
        tmp = ((b * b) * 12.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -75000000000.0) || !(a <= 15500000.0)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = ((b * b) * 12.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -75000000000.0) or not (a <= 15500000.0):
		tmp = math.pow(a, 4.0)
	else:
		tmp = ((b * b) * 12.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -75000000000.0) || !(a <= 15500000.0))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(Float64(b * b) * 12.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -75000000000.0) || ~((a <= 15500000.0)))
		tmp = a ^ 4.0;
	else
		tmp = ((b * b) * 12.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -75000000000.0], N[Not[LessEqual[a, 15500000.0]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5e10 or 1.55e7 < a

   1. Initial program 45.9%

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

      1. associate--l+ 45.9%

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

      2. fma-define 45.9%

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

      3. distribute-rgt-in 45.9%

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

      4. sqr-neg 45.9%

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

      5. distribute-rgt-in 45.9%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in a around inf 94.3%

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

      1. associate-*r/ 94.3%

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

      2. metadata-eval 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in a around inf 93.5%

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

   if -7.5e10 < a < 1.55e7

   1. Initial program 98.4%

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

      1. associate--l+ 98.4%

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

      2. fma-define 98.4%

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

      3. distribute-rgt-in 98.4%

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

      4. sqr-neg 98.4%

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

      5. distribute-rgt-in 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in a around 0 99.0%

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

   6. Taylor expanded in b around 0 69.7%

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

      1. pow2 99.0%

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

   8. Applied egg-rr 69.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -75000000000 \lor \neg \left(a \leq 15500000\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.5) (+ (* (* a a) 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = ((a * a) * 4.0) + -1.0;
	} 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 * b) <= 0.5d0) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.5) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.5:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.5)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.5)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.5

   1. Initial program 83.1%

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

      1. associate--l+ 83.1%

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

      2. fma-define 83.1%

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

      3. distribute-rgt-in 83.1%

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

      4. sqr-neg 83.1%

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

      5. distribute-rgt-in 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in b around 0 82.4%

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

   6. Taylor expanded in a around 0 67.2%

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

      1. *-commutative 67.2%

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

   8. Simplified 67.2%

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

      1. pow2 67.2%

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

   10. Applied egg-rr 67.2%

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

   11. Taylor expanded in a around 0 80.6%

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

   if 0.5 < (*.f64 b b)

   1. Initial program 66.1%

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

      1. associate--l+ 66.1%

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

      2. fma-define 66.1%

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

      3. distribute-rgt-in 66.1%

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

      4. sqr-neg 66.1%

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

      5. distribute-rgt-in 66.1%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in a around 0 90.4%

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

   6. Taylor expanded in b around inf 90.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.5:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+86}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot -4\right) + -1\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+138}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3e+86)
   (+ (* (* a a) (+ 4.0 (* a -4.0))) -1.0)
   (if (<= a 2.6e+138) (+ (* (* b b) 12.0) -1.0) (+ (* (* a a) 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3e+86) {
		tmp = ((a * a) * (4.0 + (a * -4.0))) + -1.0;
	} else if (a <= 2.6e+138) {
		tmp = ((b * b) * 12.0) + -1.0;
	} else {
		tmp = ((a * a) * 4.0) + -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 <= (-3d+86)) then
        tmp = ((a * a) * (4.0d0 + (a * (-4.0d0)))) + (-1.0d0)
    else if (a <= 2.6d+138) then
        tmp = ((b * b) * 12.0d0) + (-1.0d0)
    else
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3e+86) {
		tmp = ((a * a) * (4.0 + (a * -4.0))) + -1.0;
	} else if (a <= 2.6e+138) {
		tmp = ((b * b) * 12.0) + -1.0;
	} else {
		tmp = ((a * a) * 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3e+86:
		tmp = ((a * a) * (4.0 + (a * -4.0))) + -1.0
	elif a <= 2.6e+138:
		tmp = ((b * b) * 12.0) + -1.0
	else:
		tmp = ((a * a) * 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3e+86)
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * -4.0))) + -1.0);
	elseif (a <= 2.6e+138)
		tmp = Float64(Float64(Float64(b * b) * 12.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3e+86)
		tmp = ((a * a) * (4.0 + (a * -4.0))) + -1.0;
	elseif (a <= 2.6e+138)
		tmp = ((b * b) * 12.0) + -1.0;
	else
		tmp = ((a * a) * 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3e+86], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 2.6e+138], N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.99999999999999977e86

   1. Initial program 62.5%

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

      1. associate--l+ 62.5%

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

      2. fma-define 62.5%

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

      3. distribute-rgt-in 62.5%

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

      4. sqr-neg 62.5%

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

      5. distribute-rgt-in 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in b around 0 100.0%

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

   6. Taylor expanded in a around 0 90.8%

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

      1. *-commutative 90.8%

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

   8. Simplified 90.8%

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

      1. pow2 90.8%

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

   10. Applied egg-rr 90.8%

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

   if -2.99999999999999977e86 < a < 2.6000000000000001e138

   1. Initial program 95.1%

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

      1. associate--l+ 95.1%

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

      2. fma-define 95.1%

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

      3. distribute-rgt-in 95.1%

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

      4. sqr-neg 95.1%

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

      5. distribute-rgt-in 95.1%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in a around 0 87.6%

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

   6. Taylor expanded in b around 0 62.7%

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

      1. pow2 87.6%

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

   8. Applied egg-rr 62.7%

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

   if 2.6000000000000001e138 < a

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-define 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

   3. Simplified 2.4%

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

   5. Taylor expanded in b around 0 0.0%

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

   6. Taylor expanded in a around 0 0.0%

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

      1. *-commutative 0.0%

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

   8. Simplified 0.0%

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

      1. pow2 0.0%

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

   10. Applied egg-rr 0.0%

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

   11. Taylor expanded in a around 0 95.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+86}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot -4\right) + -1\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+138}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 8.6 \cdot 10^{+303}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 8.6e+303) (+ (* (* a a) 4.0) -1.0) (+ (* (* b b) 12.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 8.6e+303) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = ((b * b) * 12.0) + -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 ((b * b) <= 8.6d+303) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else
        tmp = ((b * b) * 12.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 8.6e+303) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = ((b * b) * 12.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 8.6e+303:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = ((b * b) * 12.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 8.6e+303)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 12.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 8.6e+303)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = ((b * b) * 12.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 8.6e+303], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 8.59999999999999982e303

   1. Initial program 80.8%

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

      1. associate--l+ 80.8%

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

      2. fma-define 80.8%

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

      3. distribute-rgt-in 80.8%

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

      4. sqr-neg 80.8%

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

      5. distribute-rgt-in 80.8%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in b around 0 60.0%

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

   6. Taylor expanded in a around 0 47.8%

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

      1. *-commutative 47.8%

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

   8. Simplified 47.8%

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

      1. pow2 47.8%

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

   10. Applied egg-rr 47.8%

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

   11. Taylor expanded in a around 0 61.0%

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

   if 8.59999999999999982e303 < (*.f64 b b)

   1. Initial program 55.0%

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

      1. associate--l+ 55.0%

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

      2. fma-define 55.0%

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

      3. distribute-rgt-in 55.0%

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

      4. sqr-neg 55.0%

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

      5. distribute-rgt-in 55.0%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 98.6%

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

      1. pow2 100.0%

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

   8. Applied egg-rr 98.6%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 8.6 \cdot 10^{+303}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.6% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

   1. associate--l+ 74.8%

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

   2. fma-define 74.8%

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

   3. distribute-rgt-in 74.8%

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

   4. sqr-neg 74.8%

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

   5. distribute-rgt-in 74.8%

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

3. Simplified 77.1%

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

5. Taylor expanded in a around 0 70.4%

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

6. Taylor expanded in b around 0 50.8%

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

   1. pow2 70.4%

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

8. Applied egg-rr 50.8%

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

9. Final simplification 50.8%

   \[\leadsto \left(b \cdot b\right) \cdot 12 + -1 \]
10. Add Preprocessing

Alternative 12: 25.1% accurate, 128.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.8%

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

   1. associate--l+ 74.8%

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

   2. fma-define 74.8%

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

   3. distribute-rgt-in 74.8%

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

   4. sqr-neg 74.8%

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

   5. distribute-rgt-in 74.8%

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

3. Simplified 77.1%

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

5. Taylor expanded in a around 0 70.4%

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

6. Taylor expanded in b around 0 26.1%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))