Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.5% → 99.9%

Time: 7.6s

Alternatives: 10

Speedup: 8.6×

• 59.8% 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: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* b b) (* a a)) 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(((b * b) + (a * a)), 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(b * b) + Float64(a * a)) ^ 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[(b * b), $MachinePrecision] + N[(a * a), $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 4.3%

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

   4. Taylor expanded in a around 0 4.3%

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

      1. +-commutative 4.3%

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

      2. associate-+r+ 4.3%

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

      3. fma-def 4.3%

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

      4. *-commutative 4.3%

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

      5. unpow2 4.3%

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

      6. unpow2 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in a around inf 43.5%

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

      1. +-commutative 43.5%

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

      2. *-commutative 43.5%

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

      3. cube-mult 43.5%

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

      4. unpow2 43.5%

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

      5. associate-*r* 43.5%

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

      6. unpow2 43.5%

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

      7. associate-+l+ 43.5%

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

      8. *-commutative 43.5%

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

      9. metadata-eval 43.5%

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

      10. pow-sqr 43.5%

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

      11. *-commutative 43.5%

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

      12. unpow2 43.5%

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

      13. associate-*l* 43.5%

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

      14. distribute-lft-out 100.0%

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

      15. distribute-lft-out 100.0%

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

      16. unpow2 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. associate--l+ 72.9%

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

   2. fma-def 72.9%

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

3. Simplified 74.0%

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

4. Taylor expanded in a around 0 63.3%

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

   1. +-commutative 63.3%

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

   2. associate-+r+ 63.3%

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

   3. fma-def 63.3%

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

   4. *-commutative 63.3%

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

   5. unpow2 63.3%

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

   6. unpow2 63.3%

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

6. Simplified 63.3%

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

7. Taylor expanded in a around 0 85.8%

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

   1. unpow2 85.8%

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

9. Simplified 85.8%

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

    1. fma-udef 85.8%

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

    2. unswap-sqr 99.4%

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

    3. pow2 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+22} \lor \neg \left(a \leq 4.5 \cdot 10^{+17}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\mathsf{fma}\left(2, b \cdot b, 4\right) + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.4e+22) (not (<= a 4.5e+17)))
   (* (* a a) (+ (fma 2.0 (* b b) 4.0) (* a (+ a 4.0))))
   (+ (+ (pow b 4.0) (* 4.0 (* b b))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.4e+22) || !(a <= 4.5e+17)) {
		tmp = (a * a) * (fma(2.0, (b * b), 4.0) + (a * (a + 4.0)));
	} else {
		tmp = (pow(b, 4.0) + (4.0 * (b * b))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.4e+22) || !(a <= 4.5e+17))
		tmp = Float64(Float64(a * a) * Float64(fma(2.0, Float64(b * b), 4.0) + Float64(a * Float64(a + 4.0))));
	else
		tmp = Float64(Float64((b ^ 4.0) + Float64(4.0 * Float64(b * b))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.4e+22], N[Not[LessEqual[a, 4.5e+17]], $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[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4e22 or 4.5e17 < a

   1. Initial program 41.4%

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

      1. associate--l+ 41.4%

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

      2. fma-def 41.4%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in a around 0 34.7%

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

      1. +-commutative 34.7%

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

      2. associate-+r+ 34.7%

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

      3. fma-def 34.7%

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

      4. *-commutative 34.7%

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

      5. unpow2 34.7%

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

      6. unpow2 34.7%

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

   6. Simplified 34.7%

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

   7. Taylor expanded in a around inf 65.0%

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

      1. +-commutative 65.0%

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

      2. *-commutative 65.0%

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

      3. cube-mult 65.0%

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

      4. unpow2 65.0%

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

      5. associate-*r* 65.0%

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

      6. unpow2 65.0%

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

      7. associate-+l+ 65.0%

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

      8. *-commutative 65.0%

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

      9. metadata-eval 65.0%

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

      10. pow-sqr 64.9%

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

      11. *-commutative 64.9%

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

      12. unpow2 64.9%

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

      13. associate-*l* 64.9%

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

      14. distribute-lft-out 97.9%

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

      15. distribute-lft-out 97.9%

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

      16. unpow2 97.9%

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

   9. Simplified 97.9%

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

   if -1.4e22 < a < 4.5e17

   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 87.7%

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

      1. +-commutative 87.7%

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

      2. associate-+r+ 87.7%

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

      3. fma-def 87.7%

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

      4. *-commutative 87.7%

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

      5. unpow2 87.7%

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

      6. unpow2 87.7%

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

   6. Simplified 87.7%

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

   7. Taylor expanded in a around 0 87.7%

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

      1. sub-neg 87.7%

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

      2. associate-+r+ 87.7%

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

      3. +-commutative 87.7%

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

      4. associate-*r* 87.7%

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

      5. distribute-rgt-out 98.5%

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

      6. metadata-eval 98.5%

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

      7. associate-*r* 98.5%

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

      8. metadata-eval 98.5%

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

      9. distribute-lft-in 98.5%

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

      10. +-commutative 98.5%

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

      11. unpow2 98.5%

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

      12. distribute-lft-in 98.5%

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

      13. metadata-eval 98.5%

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

      14. associate-*r* 98.5%

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

      15. metadata-eval 98.5%

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

      16. *-commutative 98.5%

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

      17. metadata-eval 98.5%

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

   9. Simplified 98.5%

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

   10. Taylor expanded in a around 0 98.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+22} \lor \neg \left(a \leq 4.5 \cdot 10^{+17}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\mathsf{fma}\left(2, b \cdot b, 4\right) + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 4: 93.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+30}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+69}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.2e+30)
   (pow a 4.0)
   (if (<= a 6.4e+69) (+ (+ (pow b 4.0) (* 4.0 (* b b))) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.2e+30) {
		tmp = pow(a, 4.0);
	} else if (a <= 6.4e+69) {
		tmp = (pow(b, 4.0) + (4.0 * (b * b))) + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-4.2d+30)) then
        tmp = a ** 4.0d0
    else if (a <= 6.4d+69) then
        tmp = ((b ** 4.0d0) + (4.0d0 * (b * b))) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.2e+30) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 6.4e+69) {
		tmp = (Math.pow(b, 4.0) + (4.0 * (b * b))) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.2e+30:
		tmp = math.pow(a, 4.0)
	elif a <= 6.4e+69:
		tmp = (math.pow(b, 4.0) + (4.0 * (b * b))) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.2e+30)
		tmp = a ^ 4.0;
	elseif (a <= 6.4e+69)
		tmp = Float64(Float64((b ^ 4.0) + Float64(4.0 * Float64(b * b))) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.2e+30)
		tmp = a ^ 4.0;
	elseif (a <= 6.4e+69)
		tmp = ((b ^ 4.0) + (4.0 * (b * b))) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.2e+30], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 6.4e+69], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.2e30 or 6.3999999999999997e69 < a

   1. Initial program 39.9%

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

      1. associate--l+ 39.9%

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

      2. fma-def 39.9%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in a around inf 95.8%

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

   if -4.2e30 < a < 6.3999999999999997e69

   1. Initial program 95.8%

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

      1. associate--l+ 95.8%

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

      2. fma-def 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in a around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. associate-+r+ 84.7%

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

      3. fma-def 84.7%

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

      4. *-commutative 84.7%

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

      5. unpow2 84.7%

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

      6. unpow2 84.7%

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

   6. Simplified 84.7%

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

   7. Taylor expanded in a around 0 81.5%

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

      1. sub-neg 81.5%

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

      2. associate-+r+ 81.5%

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

      3. +-commutative 81.5%

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

      4. associate-*r* 81.5%

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

      5. distribute-rgt-out 91.5%

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

      6. metadata-eval 91.5%

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

      7. associate-*r* 91.5%

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

      8. metadata-eval 91.5%

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

      9. distribute-lft-in 91.5%

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

      10. +-commutative 91.5%

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

      11. unpow2 91.5%

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

      12. distribute-lft-in 91.5%

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

      13. metadata-eval 91.5%

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

      14. associate-*r* 91.5%

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

      15. metadata-eval 91.5%

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

      16. *-commutative 91.5%

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

      17. metadata-eval 91.5%

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

   9. Simplified 91.5%

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

   10. Taylor expanded in a around 0 95.4%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+30}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+69}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5: 93.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+30} \lor \neg \left(a \leq 2.45 \cdot 10^{+70}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2e+30) (not (<= a 2.45e+70)))
   (pow a 4.0)
   (+ -1.0 (* b (* b (fma b b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2e+30) || !(a <= 2.45e+70)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0 + (b * (b * fma(b, b, 4.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2e+30) || !(a <= 2.45e+70))
		tmp = a ^ 4.0;
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * fma(b, b, 4.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2e+30], N[Not[LessEqual[a, 2.45e+70]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e30 or 2.45000000000000014e70 < a

   1. Initial program 39.9%

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

      1. associate--l+ 39.9%

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

      2. fma-def 39.9%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in a around inf 95.8%

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

   if -2e30 < a < 2.45000000000000014e70

   1. Initial program 95.8%

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

      1. associate--l+ 95.8%

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

      2. fma-def 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in a around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. associate-+r+ 84.7%

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

      3. fma-def 84.7%

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

      4. *-commutative 84.7%

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

      5. unpow2 84.7%

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

      6. unpow2 84.7%

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

   6. Simplified 84.7%

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

   7. Taylor expanded in a around 0 87.8%

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

      1. unpow2 87.8%

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

   9. Simplified 87.8%

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

       1. fma-udef 87.8%

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

       2. unswap-sqr 99.1%

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

       3. pow2 99.1%

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

   11. Applied egg-rr 99.1%

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

   12. Taylor expanded in a around 0 95.4%

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

       1. associate--l+ 95.4%

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

       2. unpow2 95.4%

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

       3. *-commutative 95.4%

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

       4. associate-*r* 95.4%

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

       5. fma-neg 95.4%

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

       6. metadata-eval 95.4%

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

       7. fma-udef 95.4%

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

       8. associate-+l+ 95.4%

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

       9. metadata-eval 95.4%

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

       10. pow-sqr 95.3%

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

       11. unpow2 95.3%

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

       12. unpow2 95.3%

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

       13. associate-*r* 95.3%

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

       14. distribute-lft-in 95.3%

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

       15. associate-*l* 95.3%

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

       16. fma-def 95.3%

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

   14. Simplified 95.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+30} \lor \neg \left(a \leq 2.45 \cdot 10^{+70}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \]

Alternative 6: 93.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+31} \lor \neg \left(a \leq 4.2 \cdot 10^{+69}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -8e+31) (not (<= a 4.2e+69)))
   (pow a 4.0)
   (+ -1.0 (* (* b b) (+ 4.0 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -8e+31) || !(a <= 4.2e+69)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-8d+31)) .or. (.not. (a <= 4.2d+69))) then
        tmp = a ** 4.0d0
    else
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -8e+31) || !(a <= 4.2e+69)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -8e+31) or not (a <= 4.2e+69):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -8e+31) || !(a <= 4.2e+69))
		tmp = a ^ 4.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -8e+31) || ~((a <= 4.2e+69)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -8e+31], N[Not[LessEqual[a, 4.2e+69]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.9999999999999997e31 or 4.2000000000000003e69 < a

   1. Initial program 39.9%

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

      1. associate--l+ 39.9%

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

      2. fma-def 39.9%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in a around inf 95.8%

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

   if -7.9999999999999997e31 < a < 4.2000000000000003e69

   1. Initial program 95.8%

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

      1. associate--l+ 95.8%

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

      2. fma-def 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in a around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. associate-+r+ 84.7%

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

      3. fma-def 84.7%

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

      4. *-commutative 84.7%

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

      5. unpow2 84.7%

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

      6. unpow2 84.7%

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

   6. Simplified 84.7%

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

   7. Taylor expanded in a around 0 81.5%

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

      1. sub-neg 81.5%

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

      2. associate-+r+ 81.5%

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

      3. +-commutative 81.5%

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

      4. associate-*r* 81.5%

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

      5. distribute-rgt-out 91.5%

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

      6. metadata-eval 91.5%

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

      7. associate-*r* 91.5%

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

      8. metadata-eval 91.5%

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

      9. distribute-lft-in 91.5%

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

      10. +-commutative 91.5%

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

      11. unpow2 91.5%

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

      12. distribute-lft-in 91.5%

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

      13. metadata-eval 91.5%

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

      14. associate-*r* 91.5%

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

      15. metadata-eval 91.5%

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

      16. *-commutative 91.5%

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

      17. metadata-eval 91.5%

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

   9. Simplified 91.5%

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

   10. Taylor expanded in a around 0 95.4%

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

       1. metadata-eval 95.4%

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

       2. pow-sqr 95.3%

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

       3. pow2 95.3%

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

       4. pow2 95.3%

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

       5. distribute-lft-out 95.3%

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

   12. Applied egg-rr 95.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+31} \lor \neg \left(a \leq 4.2 \cdot 10^{+69}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 7: 85.2% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -6.5e+153) (not (<= a 6.5e+153)))
   (+ -1.0 (* 4.0 (* a a)))
   (+ -1.0 (* (* b b) (+ 4.0 (* b b))))))

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-6.5d+153)) .or. (.not. (a <= 6.5d+153))) then
        tmp = (-1.0d0) + (4.0d0 * (a * a))
    else
        tmp = (-1.0d0) + ((b * b) * (4.0d0 + (b * b)))
    end if
    code = tmp
end function

Java

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

Python

def code(a, b):
	tmp = 0
	if (a <= -6.5e+153) or not (a <= 6.5e+153):
		tmp = -1.0 + (4.0 * (a * a))
	else:
		tmp = -1.0 + ((b * b) * (4.0 + (b * b)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -6.5e+153], N[Not[LessEqual[a, 6.5e+153]], $MachinePrecision]], N[(-1.0 + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 30.5%

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

      1. associate--l+ 30.5%

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

      2. fma-def 30.5%

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

   3. Simplified 30.5%

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

   4. Taylor expanded in b around 0 54.2%

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

      1. associate--l+ 54.2%

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

      2. associate-*r* 54.2%

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

      3. unpow2 54.2%

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

   6. Simplified 54.2%

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

   7. Taylor expanded in a around 0 54.2%

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

      1. distribute-lft-out 54.2%

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

      2. unpow2 54.2%

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

      3. cube-mult 54.2%

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

      4. unpow2 54.2%

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

      5. distribute-lft-in 54.2%

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

      6. fma-neg 54.2%

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

      7. metadata-eval 54.2%

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

      8. distribute-lft-in 54.2%

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

      9. unpow2 54.2%

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

      10. distribute-rgt1-in 54.2%

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

      11. *-commutative 54.2%

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

      12. unpow2 54.2%

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

      13. fma-udef 54.2%

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

      14. associate-*l* 54.2%

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

      15. associate-*r* 54.2%

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

      16. *-commutative 54.2%

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

      17. *-commutative 54.2%

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

      18. distribute-lft1-in 54.2%

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

      19. fma-udef 54.2%

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

   9. Simplified 54.2%

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

   10. Taylor expanded in a around 0 100.0%

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

       1. unpow2 100.0%

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

   12. Simplified 100.0%

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

   if -6.49999999999999972e153 < a < 6.49999999999999972e153

   1. Initial program 85.6%

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

      1. associate--l+ 85.6%

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

      2. fma-def 85.6%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in a around 0 78.7%

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

      1. +-commutative 78.7%

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

      2. associate-+r+ 78.7%

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

      3. fma-def 78.7%

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

      4. *-commutative 78.7%

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

      5. unpow2 78.7%

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

      6. unpow2 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in a around 0 67.3%

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

      1. sub-neg 67.3%

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

      2. associate-+r+ 67.3%

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

      3. +-commutative 67.3%

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

      4. associate-*r* 67.3%

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

      5. distribute-rgt-out 74.9%

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

      6. metadata-eval 74.9%

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

      7. associate-*r* 74.9%

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

      8. metadata-eval 74.9%

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

      9. distribute-lft-in 74.9%

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

      10. +-commutative 74.9%

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

      11. unpow2 74.9%

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

      12. distribute-lft-in 74.9%

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

      13. metadata-eval 74.9%

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

      14. associate-*r* 74.9%

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

      15. metadata-eval 74.9%

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

      16. *-commutative 74.9%

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

      17. metadata-eval 74.9%

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

   9. Simplified 74.9%

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

   10. Taylor expanded in a around 0 83.0%

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

       1. metadata-eval 83.0%

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

       2. pow-sqr 82.9%

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

       3. pow2 82.9%

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

       4. pow2 82.9%

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

       5. distribute-lft-out 82.9%

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

   12. Applied egg-rr 82.9%

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

4. Final simplification 86.8%

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

Alternative 8: 60.7% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+153} \lor \neg \left(a \leq -1.76 \cdot 10^{-289}\right):\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -7e+153) (not (<= a -1.76e-289)))
   (+ -1.0 (* 4.0 (* a a)))
   (+ -1.0 (* a (* (* b b) -12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -7e+153) || !(a <= -1.76e-289)) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = -1.0 + (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 ((a <= (-7d+153)) .or. (.not. (a <= (-1.76d-289)))) then
        tmp = (-1.0d0) + (4.0d0 * (a * a))
    else
        tmp = (-1.0d0) + (a * ((b * b) * (-12.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -7e+153) || !(a <= -1.76e-289)) {
		tmp = -1.0 + (4.0 * (a * a));
	} else {
		tmp = -1.0 + (a * ((b * b) * -12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -7e+153) or not (a <= -1.76e-289):
		tmp = -1.0 + (4.0 * (a * a))
	else:
		tmp = -1.0 + (a * ((b * b) * -12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -7e+153) || !(a <= -1.76e-289))
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * a)));
	else
		tmp = Float64(-1.0 + Float64(a * Float64(Float64(b * b) * -12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -7e+153) || ~((a <= -1.76e-289)))
		tmp = -1.0 + (4.0 * (a * a));
	else
		tmp = -1.0 + (a * ((b * b) * -12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -7e+153], N[Not[LessEqual[a, -1.76e-289]], $MachinePrecision]], N[(-1.0 + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(a * N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.9999999999999998e153 or -1.7600000000000001e-289 < a

   1. Initial program 63.6%

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

      1. associate--l+ 63.6%

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

      2. fma-def 63.6%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in b around 0 56.9%

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

      1. associate--l+ 56.9%

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

      2. associate-*r* 56.9%

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

      3. unpow2 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in a around 0 50.9%

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

      1. distribute-lft-out 50.9%

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

      2. unpow2 50.9%

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

      3. cube-mult 50.9%

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

      4. unpow2 50.9%

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

      5. distribute-lft-in 50.9%

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

      6. fma-neg 50.9%

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

      7. metadata-eval 50.9%

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

      8. distribute-lft-in 50.9%

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

      9. unpow2 50.9%

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

      10. distribute-rgt1-in 50.9%

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

      11. *-commutative 50.9%

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

      12. unpow2 50.9%

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

      13. fma-udef 50.9%

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

      14. associate-*l* 50.9%

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

      15. associate-*r* 50.9%

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

      16. *-commutative 50.9%

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

      17. *-commutative 50.9%

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

      18. distribute-lft1-in 50.9%

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

      19. fma-udef 50.9%

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

   9. Simplified 50.9%

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

   10. Taylor expanded in a around 0 61.0%

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

       1. unpow2 61.0%

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

   12. Simplified 61.0%

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

   if -6.9999999999999998e153 < a < -1.7600000000000001e-289

   1. Initial program 87.6%

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

      1. associate--l+ 87.6%

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

      2. fma-def 87.6%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in a around 0 82.8%

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

      1. +-commutative 82.8%

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

      2. associate-+r+ 82.8%

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

      3. fma-def 82.8%

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

      4. *-commutative 82.8%

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

      5. unpow2 82.8%

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

      6. unpow2 82.8%

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

   6. Simplified 82.8%

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

   7. Taylor expanded in a around 0 78.2%

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

      1. sub-neg 78.2%

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

      2. associate-+r+ 78.2%

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

      3. +-commutative 78.2%

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

      4. associate-*r* 78.2%

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

      5. distribute-rgt-out 78.2%

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

      6. metadata-eval 78.2%

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

      7. associate-*r* 78.2%

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

      8. metadata-eval 78.2%

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

      9. distribute-lft-in 78.2%

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

      10. +-commutative 78.2%

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

      11. unpow2 78.2%

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

      12. distribute-lft-in 78.2%

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

      13. metadata-eval 78.2%

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

      14. associate-*r* 78.2%

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

      15. metadata-eval 78.2%

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

      16. *-commutative 78.2%

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

      17. metadata-eval 78.2%

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

   9. Simplified 78.2%

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

   10. Taylor expanded in a around inf 59.6%

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

       1. *-commutative 59.6%

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

       2. unpow2 59.6%

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

       3. associate-*l* 59.6%

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

       4. *-commutative 59.6%

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

   12. Simplified 59.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+153} \lor \neg \left(a \leq -1.76 \cdot 10^{-289}\right):\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \end{array} \]

Alternative 9: 51.6% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. associate--l+ 72.9%

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

   2. fma-def 72.9%

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

3. Simplified 74.0%

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

4. Taylor expanded in b around 0 53.6%

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

   1. associate--l+ 53.6%

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

   2. associate-*r* 53.6%

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

   3. unpow2 53.6%

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

6. Simplified 53.6%

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

7. Taylor expanded in a around 0 44.5%

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

   1. distribute-lft-out 44.5%

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

   2. unpow2 44.5%

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

   3. cube-mult 44.5%

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

   4. unpow2 44.5%

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

   5. distribute-lft-in 44.5%

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

   6. fma-neg 44.5%

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

   7. metadata-eval 44.5%

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

   8. distribute-lft-in 44.5%

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

   9. unpow2 44.5%

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

   10. distribute-rgt1-in 44.5%

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

   11. *-commutative 44.5%

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

   12. unpow2 44.5%

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

   13. fma-udef 44.5%

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

   14. associate-*l* 44.5%

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

   15. associate-*r* 44.5%

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

   16. *-commutative 44.5%

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

   17. *-commutative 44.5%

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

   18. distribute-lft1-in 44.5%

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

   19. fma-udef 44.5%

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

9. Simplified 44.5%

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

10. Taylor expanded in a around 0 51.4%

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

    1. unpow2 51.4%

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

12. Simplified 51.4%

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

13. Final simplification 51.4%

    \[\leadsto -1 + 4 \cdot \left(a \cdot a\right) \]

Alternative 10: 25.7% accurate, 130.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 72.9%

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

   1. associate--l+ 72.9%

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

   2. fma-def 72.9%

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

3. Simplified 74.0%

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

4. Taylor expanded in b around 0 53.6%

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

   1. associate--l+ 53.6%

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

   2. associate-*r* 53.6%

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

   3. unpow2 53.6%

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

6. Simplified 53.6%

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

7. Taylor expanded in a around 0 26.9%

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

8. Final simplification 26.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023200 
(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))