Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.8% → 100.0%

Time: 8.2s

Alternatives: 14

Speedup: 1.2×

• 60.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 86.9%

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

      1. fma-def 86.9%

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

      2. +-commutative 86.9%

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

      3. *-commutative 86.9%

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

      4. unpow2 86.9%

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

      5. associate-*l* 100.0%

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

      6. unpow2 100.0%

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

   6. Simplified 100.0%

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

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

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. fma-def 0.0%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in a around 0 7.7%

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

      1. fma-def 7.7%

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

      2. +-commutative 7.7%

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

      3. *-commutative 7.7%

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

      4. unpow2 7.7%

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

      5. associate-*l* 7.7%

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

      6. unpow2 7.7%

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

   6. Simplified 7.7%

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

   7. Taylor expanded in a around inf 36.9%

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

      1. +-commutative 36.9%

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

      2. *-commutative 36.9%

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

      3. cube-mult 36.9%

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

      4. associate-*r* 36.9%

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

      5. associate-+l+ 36.9%

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

      6. unpow2 36.9%

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

      7. metadata-eval 36.9%

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

      8. pow-plus 36.9%

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

      9. associate-*r* 36.9%

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

      10. cube-mult 36.9%

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

      11. distribute-lft-in 100.0%

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

      12. *-commutative 100.0%

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

      13. cube-mult 100.0%

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

      14. associate-*r* 100.0%

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

      15. distribute-rgt-out 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 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. Step-by-step derivation

      1. sub-neg 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. fma-def 0.0%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in a around 0 7.7%

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

      1. fma-def 7.7%

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

      2. +-commutative 7.7%

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

      3. *-commutative 7.7%

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

      4. unpow2 7.7%

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

      5. associate-*l* 7.7%

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

      6. unpow2 7.7%

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

   6. Simplified 7.7%

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

   7. Taylor expanded in a around inf 36.9%

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

      1. +-commutative 36.9%

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

      2. *-commutative 36.9%

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

      3. cube-mult 36.9%

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

      4. associate-*r* 36.9%

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

      5. associate-+l+ 36.9%

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

      6. unpow2 36.9%

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

      7. metadata-eval 36.9%

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

      8. pow-plus 36.9%

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

      9. associate-*r* 36.9%

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

      10. cube-mult 36.9%

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

      11. distribute-lft-in 100.0%

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

      12. *-commutative 100.0%

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

      13. cube-mult 100.0%

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

      14. associate-*r* 100.0%

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

      15. distribute-rgt-out 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   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. fma-def 0.0%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in a around 0 7.7%

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

      1. fma-def 7.7%

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

      2. +-commutative 7.7%

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

      3. *-commutative 7.7%

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

      4. unpow2 7.7%

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

      5. associate-*l* 7.7%

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

      6. unpow2 7.7%

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

   6. Simplified 7.7%

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

   7. Taylor expanded in a around inf 36.9%

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

      1. +-commutative 36.9%

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

      2. *-commutative 36.9%

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

      3. cube-mult 36.9%

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

      4. associate-*r* 36.9%

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

      5. associate-+l+ 36.9%

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

      6. unpow2 36.9%

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

      7. metadata-eval 36.9%

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

      8. pow-plus 36.9%

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

      9. associate-*r* 36.9%

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

      10. cube-mult 36.9%

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

      11. distribute-lft-in 100.0%

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

      12. *-commutative 100.0%

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

      13. cube-mult 100.0%

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

      14. associate-*r* 100.0%

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

      15. distribute-rgt-out 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.9%

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

Alternative 4: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8200000000000 \lor \neg \left(a \leq 0.46\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\mathsf{fma}\left(2, b \cdot b, 4\right) + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -8200000000000.0) (not (<= a 0.46)))
   (* (* a a) (+ (fma 2.0 (* b b) 4.0) (* a (+ a 4.0))))
   (+ -1.0 (+ (pow b 4.0) (* (* b b) (+ 4.0 (* a -12.0)))))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -8200000000000.0) || !(a <= 0.46)) {
		tmp = (a * a) * (fma(2.0, (b * b), 4.0) + (a * (a + 4.0)));
	} else {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * (4.0 + (a * -12.0))));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -8200000000000.0) || !(a <= 0.46))
		tmp = Float64(Float64(a * a) * Float64(fma(2.0, Float64(b * b), 4.0) + Float64(a * Float64(a + 4.0))));
	else
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * Float64(4.0 + Float64(a * -12.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -8200000000000.0], N[Not[LessEqual[a, 0.46]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(N[(2.0 * N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision] + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(a * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.2e12 or 0.46000000000000002 < a

   1. Initial program 43.3%

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

      1. associate--l+ 43.3%

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

      2. fma-def 43.3%

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

   3. Simplified 47.7%

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

   4. Taylor expanded in a around 0 39.1%

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

      1. fma-def 39.1%

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

      2. +-commutative 39.1%

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

      3. *-commutative 39.1%

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

      4. unpow2 39.1%

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

      5. associate-*l* 47.8%

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

      6. unpow2 47.8%

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

   6. Simplified 47.8%

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

   7. Taylor expanded in a around inf 61.4%

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

      1. +-commutative 61.4%

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

      2. *-commutative 61.4%

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

      3. cube-mult 61.4%

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

      4. associate-*r* 61.4%

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

      5. associate-+l+ 61.4%

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

      6. unpow2 61.4%

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

      7. metadata-eval 61.4%

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

      8. pow-plus 61.3%

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

      9. associate-*r* 61.3%

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

      10. cube-mult 61.3%

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

      11. distribute-lft-in 97.0%

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

      12. *-commutative 97.0%

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

      13. cube-mult 97.0%

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

      14. associate-*r* 97.0%

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

      15. distribute-rgt-out 97.0%

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

   9. Simplified 97.0%

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

   if -8.2e12 < a < 0.46000000000000002

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. associate-+l+ 88.6%

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

      3. *-commutative 88.6%

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

      4. associate-*r* 88.6%

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

      5. *-commutative 88.6%

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

      6. distribute-lft-out 98.5%

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

      7. unpow2 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8200000000000 \lor \neg \left(a \leq 0.46\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\mathsf{fma}\left(2, b \cdot b, 4\right) + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right)\\ \end{array} \]

Alternative 5: 94.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.8:\\ \;\;\;\;{b}^{4} + \left(-1 + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e+17)
   (pow a 4.0)
   (if (<= a 0.8)
     (+ (pow b 4.0) (+ -1.0 (* (* b b) 4.0)))
     (+ (pow a 4.0) (+ -1.0 (* (+ a 1.0) (* (* a a) 4.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.7e+17) {
		tmp = pow(a, 4.0);
	} else if (a <= 0.8) {
		tmp = pow(b, 4.0) + (-1.0 + ((b * b) * 4.0));
	} else {
		tmp = pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.7d+17)) then
        tmp = a ** 4.0d0
    else if (a <= 0.8d0) then
        tmp = (b ** 4.0d0) + ((-1.0d0) + ((b * b) * 4.0d0))
    else
        tmp = (a ** 4.0d0) + ((-1.0d0) + ((a + 1.0d0) * ((a * a) * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.7e+17) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 0.8) {
		tmp = Math.pow(b, 4.0) + (-1.0 + ((b * b) * 4.0));
	} else {
		tmp = Math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.7e+17:
		tmp = math.pow(a, 4.0)
	elif a <= 0.8:
		tmp = math.pow(b, 4.0) + (-1.0 + ((b * b) * 4.0))
	else:
		tmp = math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.7e+17)
		tmp = a ^ 4.0;
	elseif (a <= 0.8)
		tmp = Float64((b ^ 4.0) + Float64(-1.0 + Float64(Float64(b * b) * 4.0)));
	else
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + Float64(Float64(a + 1.0) * Float64(Float64(a * a) * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.7e+17)
		tmp = a ^ 4.0;
	elseif (a <= 0.8)
		tmp = (b ^ 4.0) + (-1.0 + ((b * b) * 4.0));
	else
		tmp = (a ^ 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.7e+17], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 0.8], N[(N[Power[b, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(a + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.7e17

   1. Initial program 27.9%

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

      1. associate--l+ 27.9%

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

      2. fma-def 27.9%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in a around inf 96.6%

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

   if -2.7e17 < a < 0.80000000000000004

   1. Initial program 99.2%

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

      1. associate--l+ 99.2%

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

      2. fma-def 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in a around 0 88.9%

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

      1. fma-def 88.9%

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

      2. +-commutative 88.9%

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

      3. *-commutative 88.9%

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

      4. unpow2 88.9%

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

      5. associate-*l* 99.3%

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

      6. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in a around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. associate-+l+ 87.5%

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

      3. associate--l+ 87.5%

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

      4. associate-*r* 87.5%

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

      5. distribute-rgt-out 97.3%

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

      6. unpow2 97.3%

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

      7. metadata-eval 97.3%

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

      8. associate-*r* 97.3%

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

      9. *-commutative 97.3%

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

      10. *-commutative 97.3%

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

      11. associate-*l* 97.3%

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

      12. metadata-eval 97.3%

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

   9. Simplified 97.3%

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

   10. Taylor expanded in a around 0 97.9%

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

   if 0.80000000000000004 < a

   1. Initial program 58.0%

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

      1. associate--l+ 58.0%

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

      2. fma-def 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in b around 0 91.4%

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

      1. associate--l+ 91.4%

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

      2. associate-*r* 91.4%

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

      3. unpow2 91.4%

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

   6. Simplified 91.4%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+17}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.8:\\ \;\;\;\;{b}^{4} + \left(-1 + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \end{array} \]

Alternative 6: 94.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+20}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.85:\\ \;\;\;\;{b}^{4} + \left(-1 + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e+20)
   (pow a 4.0)
   (if (<= a 0.85)
     (+ (pow b 4.0) (+ -1.0 (* (* b b) 4.0)))
     (* (pow a 3.0) (+ a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.4e+20) {
		tmp = pow(a, 4.0);
	} else if (a <= 0.85) {
		tmp = pow(b, 4.0) + (-1.0 + ((b * b) * 4.0));
	} else {
		tmp = pow(a, 3.0) * (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 <= (-5.4d+20)) then
        tmp = a ** 4.0d0
    else if (a <= 0.85d0) then
        tmp = (b ** 4.0d0) + ((-1.0d0) + ((b * b) * 4.0d0))
    else
        tmp = (a ** 3.0d0) * (a + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.4e+20) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 0.85) {
		tmp = Math.pow(b, 4.0) + (-1.0 + ((b * b) * 4.0));
	} else {
		tmp = Math.pow(a, 3.0) * (a + 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.4e+20:
		tmp = math.pow(a, 4.0)
	elif a <= 0.85:
		tmp = math.pow(b, 4.0) + (-1.0 + ((b * b) * 4.0))
	else:
		tmp = math.pow(a, 3.0) * (a + 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.4e+20)
		tmp = a ^ 4.0;
	elseif (a <= 0.85)
		tmp = Float64((b ^ 4.0) + Float64(-1.0 + Float64(Float64(b * b) * 4.0)));
	else
		tmp = Float64((a ^ 3.0) * Float64(a + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.4e+20)
		tmp = a ^ 4.0;
	elseif (a <= 0.85)
		tmp = (b ^ 4.0) + (-1.0 + ((b * b) * 4.0));
	else
		tmp = (a ^ 3.0) * (a + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.4e+20], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 0.85], N[(N[Power[b, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.4e20

   1. Initial program 27.9%

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

      1. associate--l+ 27.9%

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

      2. fma-def 27.9%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in a around inf 96.6%

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

   if -5.4e20 < a < 0.849999999999999978

   1. Initial program 99.2%

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

      1. associate--l+ 99.2%

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

      2. fma-def 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in a around 0 88.9%

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

      1. fma-def 88.9%

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

      2. +-commutative 88.9%

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

      3. *-commutative 88.9%

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

      4. unpow2 88.9%

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

      5. associate-*l* 99.3%

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

      6. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in a around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. associate-+l+ 87.5%

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

      3. associate--l+ 87.5%

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

      4. associate-*r* 87.5%

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

      5. distribute-rgt-out 97.3%

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

      6. unpow2 97.3%

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

      7. metadata-eval 97.3%

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

      8. associate-*r* 97.3%

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

      9. *-commutative 97.3%

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

      10. *-commutative 97.3%

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

      11. associate-*l* 97.3%

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

      12. metadata-eval 97.3%

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

   9. Simplified 97.3%

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

   10. Taylor expanded in a around 0 97.9%

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

   if 0.849999999999999978 < a

   1. Initial program 58.0%

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

      1. associate--l+ 58.0%

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

      2. fma-def 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in a around inf 90.0%

      \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   5. Step-by-step derivation

      1. *-commutative 90.0%

         \[\leadsto \color{blue}{{a}^{3} \cdot 4} + {a}^{4} \]

      2. metadata-eval 90.0%

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

      3. pow-plus 89.9%

         \[\leadsto {a}^{3} \cdot 4 + \color{blue}{{a}^{3} \cdot a} \]

      4. distribute-lft-out 90.0%

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

   6. Simplified 90.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+20}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.85:\\ \;\;\;\;{b}^{4} + \left(-1 + \left(b \cdot b\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \end{array} \]

Alternative 7: 81.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+50}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+50) (+ -1.0 (* a (* a 4.0))) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+50) {
		tmp = -1.0 + (a * (a * 4.0));
	} 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) <= 2d+50) then
        tmp = (-1.0d0) + (a * (a * 4.0d0))
    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) <= 2e+50) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+50:
		tmp = -1.0 + (a * (a * 4.0))
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+50)
		tmp = Float64(-1.0 + Float64(a * Float64(a * 4.0)));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+50)
		tmp = -1.0 + (a * (a * 4.0));
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+50], N[(-1.0 + N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.0000000000000002e50

   1. Initial program 84.6%

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

      1. associate--l+ 84.6%

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

      2. fma-def 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in b around 0 83.4%

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

      1. associate--l+ 83.4%

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

      2. associate-*r* 83.4%

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

      3. unpow2 83.4%

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

   6. Simplified 83.4%

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

   7. Taylor expanded in a around 0 78.7%

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

      1. fma-neg 78.7%

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

      2. unpow2 78.7%

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

      3. metadata-eval 78.7%

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

   9. Simplified 78.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, a \cdot a, -1\right)} \]
   10. Step-by-step derivation

       1. fma-udef 78.7%

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

       2. *-commutative 78.7%

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

       3. associate-*l* 78.7%

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

   11. Applied egg-rr 78.7%

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

   if 2.0000000000000002e50 < (*.f64 b b)

   1. Initial program 62.6%

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

      1. associate--l+ 62.6%

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

      2. fma-def 62.6%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in b around inf 95.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+50}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 8: 68.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \mathbf{if}\;a \leq -4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -1.5 \cdot 10^{-226}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-266}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{elif}\;a \leq 0.0045:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a (+ a 4.0)))))
   (if (<= a -4.0)
     t_0
     (if (<= a -1.5e-226)
       -1.0
       (if (<= a -4.5e-266)
         (* a (* (* b b) -12.0))
         (if (<= a 0.0045) (+ -1.0 (* a (* a 4.0))) t_0))))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * (a + 4.0));
	double tmp;
	if (a <= -4.0) {
		tmp = t_0;
	} else if (a <= -1.5e-226) {
		tmp = -1.0;
	} else if (a <= -4.5e-266) {
		tmp = a * ((b * b) * -12.0);
	} else if (a <= 0.0045) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (a * (a + 4.0d0))
    if (a <= (-4.0d0)) then
        tmp = t_0
    else if (a <= (-1.5d-226)) then
        tmp = -1.0d0
    else if (a <= (-4.5d-266)) then
        tmp = a * ((b * b) * (-12.0d0))
    else if (a <= 0.0045d0) then
        tmp = (-1.0d0) + (a * (a * 4.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) * (a * (a + 4.0));
	double tmp;
	if (a <= -4.0) {
		tmp = t_0;
	} else if (a <= -1.5e-226) {
		tmp = -1.0;
	} else if (a <= -4.5e-266) {
		tmp = a * ((b * b) * -12.0);
	} else if (a <= 0.0045) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * (a + 4.0))
	tmp = 0
	if a <= -4.0:
		tmp = t_0
	elif a <= -1.5e-226:
		tmp = -1.0
	elif a <= -4.5e-266:
		tmp = a * ((b * b) * -12.0)
	elif a <= 0.0045:
		tmp = -1.0 + (a * (a * 4.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * Float64(a + 4.0)))
	tmp = 0.0
	if (a <= -4.0)
		tmp = t_0;
	elseif (a <= -1.5e-226)
		tmp = -1.0;
	elseif (a <= -4.5e-266)
		tmp = Float64(a * Float64(Float64(b * b) * -12.0));
	elseif (a <= 0.0045)
		tmp = Float64(-1.0 + Float64(a * Float64(a * 4.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * (a * (a + 4.0));
	tmp = 0.0;
	if (a <= -4.0)
		tmp = t_0;
	elseif (a <= -1.5e-226)
		tmp = -1.0;
	elseif (a <= -4.5e-266)
		tmp = a * ((b * b) * -12.0);
	elseif (a <= 0.0045)
		tmp = -1.0 + (a * (a * 4.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.0], t$95$0, If[LessEqual[a, -1.5e-226], -1.0, If[LessEqual[a, -4.5e-266], N[(a * N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.0045], N[(-1.0 + N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -4 or 0.00449999999999999966 < a

   1. Initial program 44.3%

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

      1. associate--l+ 44.3%

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

      2. fma-def 44.3%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in a around 0 40.1%

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

      1. fma-def 40.1%

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

      2. +-commutative 40.1%

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

      3. *-commutative 40.1%

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

      4. unpow2 40.1%

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

      5. associate-*l* 48.7%

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

      6. unpow2 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in a around inf 60.5%

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

      1. +-commutative 60.5%

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

      2. *-commutative 60.5%

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

      3. cube-mult 60.5%

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

      4. associate-*r* 60.5%

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

      5. associate-+l+ 60.5%

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

      6. unpow2 60.5%

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

      7. metadata-eval 60.5%

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

      8. pow-plus 60.4%

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

      9. associate-*r* 60.4%

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

      10. cube-mult 60.4%

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

      11. distribute-lft-in 95.5%

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

      12. *-commutative 95.5%

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

      13. cube-mult 95.5%

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

      14. associate-*r* 95.4%

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

      15. distribute-rgt-out 95.4%

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

   9. Simplified 95.4%

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

   10. Taylor expanded in a around inf 88.8%

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

       1. unpow2 88.8%

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

       2. distribute-rgt-in 89.6%

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

   12. Simplified 89.6%

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

   if -4 < a < -1.49999999999999998e-226

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 52.9%

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

      1. associate--l+ 52.9%

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

      2. associate-*r* 52.9%

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

      3. unpow2 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in a around 0 51.5%

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

   if -1.49999999999999998e-226 < a < -4.5000000000000003e-266

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 53.8%

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

      1. fma-def 53.8%

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

      2. +-commutative 53.8%

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

      3. *-commutative 53.8%

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

      4. unpow2 53.8%

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

      5. associate-*l* 100.0%

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

      6. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. associate--l+ 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. unpow2 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in a around inf 47.8%

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

       1. *-commutative 47.8%

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

       2. associate-*l* 47.8%

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

       3. *-commutative 47.8%

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

       4. unpow2 47.8%

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

   12. Simplified 47.8%

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

   if -4.5000000000000003e-266 < a < 0.00449999999999999966

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 59.4%

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

      1. associate--l+ 59.4%

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

      2. associate-*r* 59.4%

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

      3. unpow2 59.4%

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

   6. Simplified 59.4%

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

   7. Taylor expanded in a around 0 58.8%

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

      1. fma-neg 58.8%

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

      2. unpow2 58.8%

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

      3. metadata-eval 58.8%

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

   9. Simplified 58.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, a \cdot a, -1\right)} \]
   10. Step-by-step derivation

       1. fma-udef 58.8%

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

       2. *-commutative 58.8%

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

       3. associate-*l* 58.8%

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

   11. Applied egg-rr 58.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \mathbf{elif}\;a \leq -1.5 \cdot 10^{-226}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-266}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{elif}\;a \leq 0.0045:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + 4\right)\right)\\ \end{array} \]

Alternative 9: 52.4% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot 4\right)\\ t_1 := a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -7.6 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-227}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq -3.4 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 0.0045:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a 4.0))) (t_1 (* a (* (* b b) -12.0))))
   (if (<= a -6.5e+153)
     t_0
     (if (<= a -7.6e-50)
       t_1
       (if (<= a -7e-227)
         -1.0
         (if (<= a -3.4e-266) t_1 (if (<= a 0.0045) -1.0 t_0)))))))

C

double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double t_1 = a * ((b * b) * -12.0);
	double tmp;
	if (a <= -6.5e+153) {
		tmp = t_0;
	} else if (a <= -7.6e-50) {
		tmp = t_1;
	} else if (a <= -7e-227) {
		tmp = -1.0;
	} else if (a <= -3.4e-266) {
		tmp = t_1;
	} else if (a <= 0.0045) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (a * 4.0d0)
    t_1 = a * ((b * b) * (-12.0d0))
    if (a <= (-6.5d+153)) then
        tmp = t_0
    else if (a <= (-7.6d-50)) then
        tmp = t_1
    else if (a <= (-7d-227)) then
        tmp = -1.0d0
    else if (a <= (-3.4d-266)) then
        tmp = t_1
    else if (a <= 0.0045d0) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double t_1 = a * ((b * b) * -12.0);
	double tmp;
	if (a <= -6.5e+153) {
		tmp = t_0;
	} else if (a <= -7.6e-50) {
		tmp = t_1;
	} else if (a <= -7e-227) {
		tmp = -1.0;
	} else if (a <= -3.4e-266) {
		tmp = t_1;
	} else if (a <= 0.0045) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * 4.0)
	t_1 = a * ((b * b) * -12.0)
	tmp = 0
	if a <= -6.5e+153:
		tmp = t_0
	elif a <= -7.6e-50:
		tmp = t_1
	elif a <= -7e-227:
		tmp = -1.0
	elif a <= -3.4e-266:
		tmp = t_1
	elif a <= 0.0045:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * 4.0))
	t_1 = Float64(a * Float64(Float64(b * b) * -12.0))
	tmp = 0.0
	if (a <= -6.5e+153)
		tmp = t_0;
	elseif (a <= -7.6e-50)
		tmp = t_1;
	elseif (a <= -7e-227)
		tmp = -1.0;
	elseif (a <= -3.4e-266)
		tmp = t_1;
	elseif (a <= 0.0045)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * 4.0);
	t_1 = a * ((b * b) * -12.0);
	tmp = 0.0;
	if (a <= -6.5e+153)
		tmp = t_0;
	elseif (a <= -7.6e-50)
		tmp = t_1;
	elseif (a <= -7e-227)
		tmp = -1.0;
	elseif (a <= -3.4e-266)
		tmp = t_1;
	elseif (a <= 0.0045)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.5e+153], t$95$0, If[LessEqual[a, -7.6e-50], t$95$1, If[LessEqual[a, -7e-227], -1.0, If[LessEqual[a, -3.4e-266], t$95$1, If[LessEqual[a, 0.0045], -1.0, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.49999999999999972e153 or 0.00449999999999999966 < a

   1. Initial program 37.0%

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

      1. associate--l+ 37.0%

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

      2. fma-def 37.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in b around 0 56.5%

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

      1. associate--l+ 56.5%

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

      2. associate-*r* 56.5%

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

      3. unpow2 56.5%

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

   6. Simplified 56.5%

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

   7. Taylor expanded in a around 0 75.7%

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

      1. fma-neg 75.7%

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

      2. unpow2 75.7%

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

      3. metadata-eval 75.7%

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

   9. Simplified 75.7%

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

   10. Taylor expanded in a around inf 75.8%

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

       1. *-commutative 75.8%

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

       2. unpow2 75.8%

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

       3. associate-*r* 75.8%

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

   12. Simplified 75.8%

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

   if -6.49999999999999972e153 < a < -7.5999999999999998e-50 or -7.0000000000000002e-227 < a < -3.39999999999999995e-266

   1. Initial program 81.7%

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

      1. associate--l+ 81.7%

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

      2. fma-def 81.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in a around 0 79.9%

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

      1. fma-def 79.9%

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

      2. +-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. unpow2 79.9%

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

      5. associate-*l* 91.9%

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

      6. unpow2 91.9%

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

   6. Simplified 91.9%

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

   7. Taylor expanded in a around 0 63.3%

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

      1. +-commutative 63.3%

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

      2. associate-+l+ 63.3%

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

      3. associate--l+ 63.3%

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

      4. associate-*r* 63.3%

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

      5. distribute-rgt-out 63.3%

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

      6. unpow2 63.3%

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

      7. metadata-eval 63.3%

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

      8. associate-*r* 63.3%

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

      9. *-commutative 63.3%

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

      10. *-commutative 63.3%

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

      11. associate-*l* 63.3%

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

      12. metadata-eval 63.3%

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

   9. Simplified 63.3%

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

   10. Taylor expanded in a around inf 36.4%

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

       1. *-commutative 36.4%

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

       2. associate-*l* 36.4%

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

       3. *-commutative 36.4%

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

       4. unpow2 36.4%

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

   12. Simplified 36.4%

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

   if -7.5999999999999998e-50 < a < -7.0000000000000002e-227 or -3.39999999999999995e-266 < a < 0.00449999999999999966

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 59.2%

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

      1. associate--l+ 59.2%

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

      2. associate-*r* 59.2%

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

      3. unpow2 59.2%

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

   6. Simplified 59.2%

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

   7. Taylor expanded in a around 0 58.5%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -7.6 \cdot 10^{-50}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-227}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq -3.4 \cdot 10^{-266}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{elif}\;a \leq 0.0045:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 10: 52.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -7.6 \cdot 10^{-50}:\\ \;\;\;\;b \cdot \left(-12 \cdot \left(a \cdot b\right)\right)\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-226}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{-266}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{elif}\;a \leq 0.0045:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a 4.0))))
   (if (<= a -6.6e+153)
     t_0
     (if (<= a -7.6e-50)
       (* b (* -12.0 (* a b)))
       (if (<= a -1.45e-226)
         -1.0
         (if (<= a -2.6e-266)
           (* a (* (* b b) -12.0))
           (if (<= a 0.0045) -1.0 t_0)))))))

C

double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double tmp;
	if (a <= -6.6e+153) {
		tmp = t_0;
	} else if (a <= -7.6e-50) {
		tmp = b * (-12.0 * (a * b));
	} else if (a <= -1.45e-226) {
		tmp = -1.0;
	} else if (a <= -2.6e-266) {
		tmp = a * ((b * b) * -12.0);
	} else if (a <= 0.0045) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * 4.0d0)
    if (a <= (-6.6d+153)) then
        tmp = t_0
    else if (a <= (-7.6d-50)) then
        tmp = b * ((-12.0d0) * (a * b))
    else if (a <= (-1.45d-226)) then
        tmp = -1.0d0
    else if (a <= (-2.6d-266)) then
        tmp = a * ((b * b) * (-12.0d0))
    else if (a <= 0.0045d0) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double tmp;
	if (a <= -6.6e+153) {
		tmp = t_0;
	} else if (a <= -7.6e-50) {
		tmp = b * (-12.0 * (a * b));
	} else if (a <= -1.45e-226) {
		tmp = -1.0;
	} else if (a <= -2.6e-266) {
		tmp = a * ((b * b) * -12.0);
	} else if (a <= 0.0045) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * 4.0)
	tmp = 0
	if a <= -6.6e+153:
		tmp = t_0
	elif a <= -7.6e-50:
		tmp = b * (-12.0 * (a * b))
	elif a <= -1.45e-226:
		tmp = -1.0
	elif a <= -2.6e-266:
		tmp = a * ((b * b) * -12.0)
	elif a <= 0.0045:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * 4.0))
	tmp = 0.0
	if (a <= -6.6e+153)
		tmp = t_0;
	elseif (a <= -7.6e-50)
		tmp = Float64(b * Float64(-12.0 * Float64(a * b)));
	elseif (a <= -1.45e-226)
		tmp = -1.0;
	elseif (a <= -2.6e-266)
		tmp = Float64(a * Float64(Float64(b * b) * -12.0));
	elseif (a <= 0.0045)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * 4.0);
	tmp = 0.0;
	if (a <= -6.6e+153)
		tmp = t_0;
	elseif (a <= -7.6e-50)
		tmp = b * (-12.0 * (a * b));
	elseif (a <= -1.45e-226)
		tmp = -1.0;
	elseif (a <= -2.6e-266)
		tmp = a * ((b * b) * -12.0);
	elseif (a <= 0.0045)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.6e+153], t$95$0, If[LessEqual[a, -7.6e-50], N[(b * N[(-12.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.45e-226], -1.0, If[LessEqual[a, -2.6e-266], N[(a * N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.0045], -1.0, t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.59999999999999989e153 or 0.00449999999999999966 < a

   1. Initial program 37.0%

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

      1. associate--l+ 37.0%

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

      2. fma-def 37.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in b around 0 56.5%

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

      1. associate--l+ 56.5%

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

      2. associate-*r* 56.5%

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

      3. unpow2 56.5%

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

   6. Simplified 56.5%

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

   7. Taylor expanded in a around 0 75.7%

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

      1. fma-neg 75.7%

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

      2. unpow2 75.7%

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

      3. metadata-eval 75.7%

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

   9. Simplified 75.7%

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

   10. Taylor expanded in a around inf 75.8%

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

       1. *-commutative 75.8%

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

       2. unpow2 75.8%

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

       3. associate-*r* 75.8%

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

   12. Simplified 75.8%

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

   if -6.59999999999999989e153 < a < -7.5999999999999998e-50

   1. Initial program 75.3%

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

      1. associate--l+ 75.3%

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

      2. fma-def 75.3%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in a around 0 89.1%

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

      1. fma-def 89.1%

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

      2. +-commutative 89.1%

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

      3. *-commutative 89.1%

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

      4. unpow2 89.1%

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

      5. associate-*l* 89.1%

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

      6. unpow2 89.1%

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

   6. Simplified 89.1%

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

   7. Taylor expanded in a around 0 50.4%

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

      1. +-commutative 50.4%

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

      2. associate-+l+ 50.4%

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

      3. associate--l+ 50.4%

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

      4. associate-*r* 50.4%

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

      5. distribute-rgt-out 50.4%

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

      6. unpow2 50.4%

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

      7. metadata-eval 50.4%

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

      8. associate-*r* 50.4%

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

      9. *-commutative 50.4%

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

      10. *-commutative 50.4%

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

      11. associate-*l* 50.4%

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

      12. metadata-eval 50.4%

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

   9. Simplified 50.4%

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

   10. Taylor expanded in a around inf 32.4%

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

       1. associate-*r* 32.4%

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

       2. *-commutative 32.4%

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

       3. unpow2 32.4%

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

       4. associate-*r* 32.4%

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

   12. Simplified 32.4%

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

   13. Taylor expanded in a around 0 32.4%

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

   if -7.5999999999999998e-50 < a < -1.45000000000000001e-226 or -2.6e-266 < a < 0.00449999999999999966

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 59.2%

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

      1. associate--l+ 59.2%

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

      2. associate-*r* 59.2%

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

      3. unpow2 59.2%

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

   6. Simplified 59.2%

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

   7. Taylor expanded in a around 0 58.5%

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

   if -1.45000000000000001e-226 < a < -2.6e-266

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 53.8%

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

      1. fma-def 53.8%

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

      2. +-commutative 53.8%

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

      3. *-commutative 53.8%

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

      4. unpow2 53.8%

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

      5. associate-*l* 100.0%

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

      6. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. associate--l+ 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. unpow2 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in a around inf 47.8%

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

       1. *-commutative 47.8%

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

       2. associate-*l* 47.8%

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

       3. *-commutative 47.8%

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

       4. unpow2 47.8%

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

   12. Simplified 47.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -7.6 \cdot 10^{-50}:\\ \;\;\;\;b \cdot \left(-12 \cdot \left(a \cdot b\right)\right)\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-226}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{-266}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{elif}\;a \leq 0.0045:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 11: 66.5% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d-5) then
        tmp = (-1.0d0) + (a * (a * 4.0d0))
    else
        tmp = (b * b) * ((a * a) * 2.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.00000000000000024e-5

   1. Initial program 86.2%

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

      1. associate--l+ 86.2%

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

      2. fma-def 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in b around 0 86.3%

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

      1. associate--l+ 86.3%

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

      2. associate-*r* 86.3%

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

      3. unpow2 86.3%

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

   6. Simplified 86.3%

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

   7. Taylor expanded in a around 0 79.1%

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

      1. fma-neg 79.1%

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

      2. unpow2 79.1%

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

      3. metadata-eval 79.1%

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

   9. Simplified 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, a \cdot a, -1\right)} \]
   10. Step-by-step derivation

       1. fma-udef 79.1%

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

       2. *-commutative 79.1%

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

       3. associate-*l* 79.1%

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

   11. Applied egg-rr 79.1%

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

   if 5.00000000000000024e-5 < (*.f64 b b)

   1. Initial program 62.0%

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

      1. associate--l+ 62.0%

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

      2. fma-def 62.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in a around 0 54.0%

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

      1. fma-def 54.0%

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

      2. +-commutative 54.0%

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

      3. *-commutative 54.0%

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

      4. unpow2 54.0%

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

      5. associate-*l* 66.1%

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

      6. unpow2 66.1%

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

   6. Simplified 66.1%

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

   7. Taylor expanded in a around inf 43.0%

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

      1. +-commutative 43.0%

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

      2. *-commutative 43.0%

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

      3. cube-mult 43.0%

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

      4. associate-*r* 43.0%

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

      5. associate-+l+ 43.0%

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

      6. unpow2 43.0%

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

      7. metadata-eval 43.0%

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

      8. pow-plus 43.0%

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

      9. associate-*r* 43.0%

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

      10. cube-mult 43.0%

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

      11. distribute-lft-in 61.6%

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

      12. *-commutative 61.6%

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

      13. cube-mult 61.6%

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

      14. associate-*r* 61.6%

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

      15. distribute-rgt-out 61.6%

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

   9. Simplified 61.6%

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

   10. Taylor expanded in b around inf 60.8%

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

       1. unpow2 60.8%

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

       2. unpow2 60.8%

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

       3. associate-*r* 60.8%

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

       4. *-commutative 60.8%

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

   12. Simplified 60.8%

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

4. Final simplification 70.2%

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

Alternative 12: 50.6% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \lor \neg \left(a \leq 0.0045\right):\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.4) (not (<= a 0.0045))) (* a (* a 4.0)) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.4) || !(a <= 0.0045)) {
		tmp = a * (a * 4.0);
	} else {
		tmp = -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 <= (-2.4d0)) .or. (.not. (a <= 0.0045d0))) then
        tmp = a * (a * 4.0d0)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.4) || !(a <= 0.0045)) {
		tmp = a * (a * 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.4) or not (a <= 0.0045):
		tmp = a * (a * 4.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.4) || !(a <= 0.0045))
		tmp = Float64(a * Float64(a * 4.0));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.4) || ~((a <= 0.0045)))
		tmp = a * (a * 4.0);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.4], N[Not[LessEqual[a, 0.0045]], $MachinePrecision]], N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -2.39999999999999991 or 0.00449999999999999966 < a

   1. Initial program 44.3%

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

      1. associate--l+ 44.3%

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

      2. fma-def 44.3%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in b around 0 55.3%

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

      1. associate--l+ 55.3%

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

      2. associate-*r* 55.3%

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

      3. unpow2 55.3%

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

   6. Simplified 55.3%

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

   7. Taylor expanded in a around 0 59.1%

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

      1. fma-neg 59.1%

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

      2. unpow2 59.1%

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

      3. metadata-eval 59.1%

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

   9. Simplified 59.1%

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

   10. Taylor expanded in a around inf 59.2%

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

       1. *-commutative 59.2%

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

       2. unpow2 59.2%

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

       3. associate-*r* 59.2%

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

   12. Simplified 59.2%

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

   if -2.39999999999999991 < a < 0.00449999999999999966

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 52.8%

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

      1. associate--l+ 52.8%

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

      2. associate-*r* 52.8%

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

      3. unpow2 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in a around 0 51.6%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \lor \neg \left(a \leq 0.0045\right):\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 13: 53.1% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+117}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.2e+117) (+ -1.0 (* a (* a 4.0))) (* a (* (* b b) -12.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.2e+117) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = a * ((b * b) * -12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 9.2d+117) then
        tmp = (-1.0d0) + (a * (a * 4.0d0))
    else
        tmp = a * ((b * b) * (-12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.2e+117) {
		tmp = -1.0 + (a * (a * 4.0));
	} else {
		tmp = a * ((b * b) * -12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.2e+117:
		tmp = -1.0 + (a * (a * 4.0))
	else:
		tmp = a * ((b * b) * -12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.2e+117)
		tmp = Float64(-1.0 + Float64(a * Float64(a * 4.0)));
	else
		tmp = Float64(a * Float64(Float64(b * b) * -12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.2e+117)
		tmp = -1.0 + (a * (a * 4.0));
	else
		tmp = a * ((b * b) * -12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.2e+117], N[(-1.0 + N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.19999999999999951e117

   1. Initial program 76.5%

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

      1. associate--l+ 76.5%

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

      2. fma-def 76.5%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in b around 0 60.1%

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

      1. associate--l+ 60.1%

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

      2. associate-*r* 60.1%

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

      3. unpow2 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in a around 0 58.7%

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

      1. fma-neg 58.7%

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

      2. unpow2 58.7%

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

      3. metadata-eval 58.7%

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

   9. Simplified 58.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, a \cdot a, -1\right)} \]
   10. Step-by-step derivation

       1. fma-udef 58.7%

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

       2. *-commutative 58.7%

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

       3. associate-*l* 58.7%

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

   11. Applied egg-rr 58.7%

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

   if 9.19999999999999951e117 < b

   1. Initial program 63.2%

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

      1. associate--l+ 63.2%

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

      2. fma-def 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in a around 0 44.7%

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

      1. fma-def 44.7%

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

      2. +-commutative 44.7%

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

      3. *-commutative 44.7%

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

      4. unpow2 44.7%

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

      5. associate-*l* 63.2%

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

      6. unpow2 63.2%

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

   6. Simplified 63.2%

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

   7. Taylor expanded in a around 0 65.8%

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

      1. +-commutative 65.8%

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

      2. associate-+l+ 65.8%

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

      3. associate--l+ 65.8%

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

      4. associate-*r* 65.8%

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

      5. distribute-rgt-out 84.2%

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

      6. unpow2 84.2%

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

      7. metadata-eval 84.2%

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

      8. associate-*r* 84.2%

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

      9. *-commutative 84.2%

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

      10. *-commutative 84.2%

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

      11. associate-*l* 84.2%

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

      12. metadata-eval 84.2%

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

   9. Simplified 84.2%

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

   10. Taylor expanded in a around inf 55.4%

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

       1. *-commutative 55.4%

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

       2. associate-*l* 55.4%

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

       3. *-commutative 55.4%

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

       4. unpow2 55.4%

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

   12. Simplified 55.4%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+117}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \end{array} \]

Alternative 14: 24.6% 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. fma-def 74.5%

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

3. Simplified 76.4%

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

4. Taylor expanded in b around 0 53.9%

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

   1. associate--l+ 53.9%

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

   2. associate-*r* 53.9%

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

   3. unpow2 53.9%

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

6. Simplified 53.9%

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

7. Taylor expanded in a around 0 28.4%

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

8. Final simplification 28.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023207 
(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))