Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.4% → 100.0%

Time: 9.2s

Alternatives: 11

Speedup: 9.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(N[(b * N[(b * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + 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]), $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 3 a))))) < +inf.0

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. fma-def 99.8%

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

      3. fma-def 99.8%

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

      4. +-commutative 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-*l* 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-def 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-pow2 100.0%

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

      4. hypot-udef 100.0%

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

      5. expm1-log1p-u 98.2%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + 4 \cdot \left(b \cdot \left(b \cdot \left(a + 3\right)\right) + a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      6. expm1-udef 98.2%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + 4 \cdot \left(b \cdot \left(b \cdot \left(a + 3\right)\right) + a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

   7. Applied egg-rr 98.2%

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} - 1\right)} + 4 \cdot \left(b \cdot \left(b \cdot \left(a + 3\right)\right) + a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]
   8. Step-by-step derivation

      1. expm1-def 98.2%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\right)} + 4 \cdot \left(b \cdot \left(b \cdot \left(a + 3\right)\right) + a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      2. expm1-log1p 100.0%

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

   9. Simplified 100.0%

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

   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 3 a)))))

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 3.1%

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

      4. +-commutative 3.1%

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

      5. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

      1. fma-udef 0.0%

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

      2. +-commutative 0.0%

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

      3. +-commutative 0.0%

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

      4. associate-*l* 0.0%

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

      5. +-commutative 0.0%

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

      6. associate-*l* 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in a around inf 39.1%

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

      1. +-commutative 39.1%

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

      2. associate-+l+ 39.1%

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

      3. *-commutative 39.1%

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

      4. metadata-eval 39.1%

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

      5. pow-plus 39.1%

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

      6. *-commutative 39.1%

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

      7. distribute-lft-in 100.0%

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

      8. unpow3 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

      11. distribute-lft-out 100.0%

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

      12. unpow2 100.0%

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

      13. fma-def 100.0%

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

      14. unpow2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 99.8%

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

   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 3 a)))))

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 3.1%

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

      4. +-commutative 3.1%

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

      5. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

      1. fma-udef 0.0%

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

      2. +-commutative 0.0%

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

      3. +-commutative 0.0%

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

      4. associate-*l* 0.0%

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

      5. +-commutative 0.0%

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

      6. associate-*l* 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in a around inf 39.1%

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

      1. +-commutative 39.1%

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

      2. associate-+l+ 39.1%

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

      3. *-commutative 39.1%

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

      4. metadata-eval 39.1%

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

      5. pow-plus 39.1%

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

      6. *-commutative 39.1%

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

      7. distribute-lft-in 100.0%

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

      8. unpow3 100.0%

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

      9. unpow2 100.0%

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

      10. associate-*l* 100.0%

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

      11. distribute-lft-out 100.0%

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

      12. unpow2 100.0%

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

      13. fma-def 100.0%

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

      14. unpow2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 99.9%

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

Alternative 3: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 50000:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \mathsf{fma}\left(a, a + -4, 4\right)\right)\\ \mathbf{elif}\;b \cdot b \leq 10^{+112}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(12 + a \cdot 4\right)\right)\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+141}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 50000.0)
   (+ -1.0 (* a (* a (fma a (+ a -4.0) 4.0))))
   (if (<= (* b b) 1e+112)
     (+ -1.0 (+ (pow b 4.0) (* (* b b) (+ 12.0 (* a 4.0)))))
     (if (<= (* b b) 2e+141)
       (+ -1.0 (* (* a a) (* a a)))
       (+ -1.0 (pow b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 50000.0) {
		tmp = -1.0 + (a * (a * fma(a, (a + -4.0), 4.0)));
	} else if ((b * b) <= 1e+112) {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * (12.0 + (a * 4.0))));
	} else if ((b * b) <= 2e+141) {
		tmp = -1.0 + ((a * a) * (a * a));
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 50000.0)
		tmp = Float64(-1.0 + Float64(a * Float64(a * fma(a, Float64(a + -4.0), 4.0))));
	elseif (Float64(b * b) <= 1e+112)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * Float64(12.0 + Float64(a * 4.0)))));
	elseif (Float64(b * b) <= 2e+141)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(a * a)));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 50000.0], N[(-1.0 + N[(a * N[(a * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 1e+112], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(12.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 2e+141], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 b b) < 5e4

   1. Initial program 84.9%

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

      1. sub-neg 84.9%

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

      2. fma-def 84.9%

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

      3. fma-def 84.9%

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

      4. +-commutative 84.9%

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

      5. metadata-eval 84.9%

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

   3. Simplified 84.9%

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

      1. fma-udef 84.9%

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

      2. +-commutative 84.9%

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

      3. +-commutative 84.9%

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

      4. associate-*l* 84.9%

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

      5. +-commutative 84.9%

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

      6. associate-*l* 84.9%

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

   5. Applied egg-rr 84.9%

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

   6. Taylor expanded in a around inf 84.6%

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

      1. +-commutative 84.6%

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

      2. associate-+l+ 84.6%

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

      3. *-commutative 84.6%

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

      4. metadata-eval 84.6%

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

      5. pow-plus 84.6%

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

      6. *-commutative 84.6%

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

      7. distribute-lft-in 99.5%

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

      8. unpow3 99.5%

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

      9. unpow2 99.5%

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

      10. associate-*l* 99.4%

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

      11. distribute-lft-out 99.4%

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

      12. unpow2 99.4%

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

      13. fma-def 99.4%

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

      14. unpow2 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in b around 0 99.4%

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

       1. unpow2 99.4%

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

       2. *-commutative 99.4%

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

       3. associate-*l* 99.4%

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

       4. +-commutative 99.4%

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

       5. sub-neg 99.4%

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

       6. metadata-eval 99.4%

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

       7. fma-def 99.4%

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

   11. Simplified 99.4%

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

   if 5e4 < (*.f64 b b) < 9.9999999999999993e111

   1. Initial program 71.8%

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

      1. sub-neg 71.8%

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

      2. fma-def 71.8%

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

      3. fma-def 71.8%

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

      4. +-commutative 71.8%

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

      5. metadata-eval 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in a around 0 65.3%

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

      1. associate-+r+ 65.3%

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

      2. associate-*r* 65.3%

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

      3. distribute-rgt-out 65.3%

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

      4. metadata-eval 65.3%

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

      5. distribute-lft-in 65.3%

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

      6. unpow2 65.3%

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

      7. distribute-rgt-in 65.3%

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

      8. metadata-eval 65.3%

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

   6. Simplified 65.3%

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

   if 9.9999999999999993e111 < (*.f64 b b) < 2.00000000000000003e141

   1. Initial program 33.3%

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

      1. sub-neg 33.3%

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

      2. fma-def 33.3%

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

      3. fma-def 33.3%

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

      4. +-commutative 33.3%

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

      5. metadata-eval 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in a around inf 100.0%

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

      1. sqr-pow 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow2 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 2.00000000000000003e141 < (*.f64 b b)

   1. Initial program 64.2%

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

      1. sub-neg 64.2%

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

      2. fma-def 64.2%

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

      3. fma-def 66.3%

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

      4. +-commutative 66.3%

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

      5. metadata-eval 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in b around inf 100.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 50000:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \mathsf{fma}\left(a, a + -4, 4\right)\right)\\ \mathbf{elif}\;b \cdot b \leq 10^{+112}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(12 + a \cdot 4\right)\right)\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+141}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 4: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+25} \lor \neg \left(a \leq 8.7 \cdot 10^{-16}\right):\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(\mathsf{fma}\left(2, b \cdot b, 4\right) + a \cdot \left(a + -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 12 + {b}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -8e+25) (not (<= a 8.7e-16)))
   (+ -1.0 (* (* a a) (+ (fma 2.0 (* b b) 4.0) (* a (+ a -4.0)))))
   (+ -1.0 (+ (* (* b b) 12.0) (pow b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -8e+25) || !(a <= 8.7e-16)) {
		tmp = -1.0 + ((a * a) * (fma(2.0, (b * b), 4.0) + (a * (a + -4.0))));
	} else {
		tmp = -1.0 + (((b * b) * 12.0) + pow(b, 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -8e+25) || !(a <= 8.7e-16))
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(fma(2.0, Float64(b * b), 4.0) + Float64(a * Float64(a + -4.0)))));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(b * b) * 12.0) + (b ^ 4.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -8e+25], N[Not[LessEqual[a, 8.7e-16]], $MachinePrecision]], N[(-1.0 + 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]), $MachinePrecision], N[(-1.0 + N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.00000000000000072e25 or 8.70000000000000036e-16 < a

   1. Initial program 52.8%

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

      1. sub-neg 52.8%

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

      2. fma-def 52.8%

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

      3. fma-def 54.3%

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

      4. +-commutative 54.3%

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

      5. metadata-eval 54.3%

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

   3. Simplified 54.3%

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

      1. fma-udef 52.8%

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

      2. +-commutative 52.8%

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

      3. +-commutative 52.8%

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

      4. associate-*l* 52.8%

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

      5. +-commutative 52.8%

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

      6. associate-*l* 52.8%

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

   5. Applied egg-rr 52.8%

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

   6. Taylor expanded in a around inf 70.1%

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

      1. +-commutative 70.1%

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

      2. associate-+l+ 70.2%

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

      3. *-commutative 70.2%

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

      4. metadata-eval 70.2%

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

      5. pow-plus 70.1%

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

      6. *-commutative 70.1%

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

      7. distribute-lft-in 99.2%

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

      8. unpow3 99.2%

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

      9. unpow2 99.2%

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

      10. associate-*l* 99.2%

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

      11. distribute-lft-out 99.2%

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

      12. unpow2 99.2%

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

      13. fma-def 99.2%

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

      14. unpow2 99.2%

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

   8. Simplified 99.2%

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

   if -8.00000000000000072e25 < a < 8.70000000000000036e-16

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

      2. fma-def 99.1%

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

      3. fma-def 99.1%

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

      4. +-commutative 99.1%

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

      5. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 86.3%

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

      1. associate-+r+ 86.3%

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

      2. associate-*r* 86.3%

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

      3. distribute-rgt-out 96.9%

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

      4. metadata-eval 96.9%

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

      5. distribute-lft-in 96.9%

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

      6. unpow2 96.9%

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

      7. distribute-rgt-in 96.9%

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

      8. metadata-eval 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in a around 0 97.7%

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

      1. unpow2 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+25} \lor \neg \left(a \leq 8.7 \cdot 10^{-16}\right):\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(\mathsf{fma}\left(2, b \cdot b, 4\right) + a \cdot \left(a + -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 12 + {b}^{4}\right)\\ \end{array} \]

Alternative 5: 92.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+141}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \mathsf{fma}\left(a, a + -4, 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+141)
   (+ -1.0 (* a (* a (fma a (+ a -4.0) 4.0))))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+141) {
		tmp = -1.0 + (a * (a * fma(a, (a + -4.0), 4.0)));
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+141)
		tmp = Float64(-1.0 + Float64(a * Float64(a * fma(a, Float64(a + -4.0), 4.0))));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+141], N[(-1.0 + N[(a * N[(a * N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000003e141

   1. Initial program 81.4%

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

      1. sub-neg 81.4%

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

      2. fma-def 81.5%

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

      3. fma-def 81.5%

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

      4. +-commutative 81.5%

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

      5. metadata-eval 81.5%

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

   3. Simplified 81.5%

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

      1. fma-udef 81.5%

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

      2. +-commutative 81.5%

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

      3. +-commutative 81.5%

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

      4. associate-*l* 81.5%

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

      5. +-commutative 81.5%

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

      6. associate-*l* 81.5%

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

   5. Applied egg-rr 81.5%

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

   6. Taylor expanded in a around inf 75.7%

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

      1. +-commutative 75.7%

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

      2. associate-+l+ 75.8%

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

      3. *-commutative 75.8%

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

      4. metadata-eval 75.8%

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

      5. pow-plus 75.7%

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

      6. *-commutative 75.7%

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

      7. distribute-lft-in 92.8%

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

      8. unpow3 92.8%

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

      9. unpow2 92.8%

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

      10. associate-*l* 92.7%

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

      11. distribute-lft-out 92.7%

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

      12. unpow2 92.7%

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

      13. fma-def 92.7%

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

      14. unpow2 92.7%

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

   8. Simplified 92.7%

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

   9. Taylor expanded in b around 0 92.7%

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

       1. unpow2 92.7%

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

       2. *-commutative 92.7%

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

       3. associate-*l* 92.7%

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

       4. +-commutative 92.7%

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

       5. sub-neg 92.7%

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

       6. metadata-eval 92.7%

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

       7. fma-def 92.7%

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

   11. Simplified 92.7%

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

   if 2.00000000000000003e141 < (*.f64 b b)

   1. Initial program 64.2%

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

      1. sub-neg 64.2%

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

      2. fma-def 64.2%

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

      3. fma-def 66.3%

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

      4. +-commutative 66.3%

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

      5. metadata-eval 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in b around inf 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+141}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \mathsf{fma}\left(a, a + -4, 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 6: 92.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+141}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+141)
   (+ -1.0 (* (* a a) (+ 4.0 (* a (+ a -4.0)))))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+141) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + -4.0))));
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+141) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 + (a * (a + (-4.0d0)))))
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+141) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + -4.0))));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+141:
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + -4.0))))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+141)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + -4.0)))));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+141)
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + -4.0))));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+141], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000003e141

   1. Initial program 81.4%

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

      1. sub-neg 81.4%

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

      2. fma-def 81.5%

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

      3. fma-def 81.5%

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

      4. +-commutative 81.5%

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

      5. metadata-eval 81.5%

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

   3. Simplified 81.5%

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

      1. fma-udef 81.5%

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

      2. +-commutative 81.5%

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

      3. +-commutative 81.5%

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

      4. associate-*l* 81.5%

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

      5. +-commutative 81.5%

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

      6. associate-*l* 81.5%

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

   5. Applied egg-rr 81.5%

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

   6. Taylor expanded in a around inf 75.7%

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

      1. +-commutative 75.7%

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

      2. associate-+l+ 75.8%

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

      3. *-commutative 75.8%

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

      4. metadata-eval 75.8%

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

      5. pow-plus 75.7%

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

      6. *-commutative 75.7%

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

      7. distribute-lft-in 92.8%

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

      8. unpow3 92.8%

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

      9. unpow2 92.8%

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

      10. associate-*l* 92.7%

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

      11. distribute-lft-out 92.7%

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

      12. unpow2 92.7%

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

      13. fma-def 92.7%

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

      14. unpow2 92.7%

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

   8. Simplified 92.7%

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

   9. Taylor expanded in b around 0 92.7%

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

       1. sub-neg 92.7%

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

       2. metadata-eval 92.7%

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

       3. unpow2 92.7%

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

   11. Simplified 92.7%

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

   if 2.00000000000000003e141 < (*.f64 b b)

   1. Initial program 64.2%

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

      1. sub-neg 64.2%

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

      2. fma-def 64.2%

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

      3. fma-def 66.3%

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

      4. +-commutative 66.3%

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

      5. metadata-eval 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in b around inf 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+141}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 7: 84.6% accurate, 7.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+303)
   (+ -1.0 (* (* a a) (+ 4.0 (* a (+ a -4.0)))))
   (+ -1.0 (* b (* b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+303) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + -4.0))));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+303) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 + (a * (a + (-4.0d0)))))
    else
        tmp = (-1.0d0) + (b * (b * 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+303) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + -4.0))));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+303:
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + -4.0))))
	else:
		tmp = -1.0 + (b * (b * 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+303)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + -4.0)))));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+303)
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + -4.0))));
	else
		tmp = -1.0 + (b * (b * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+303], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e303

   1. Initial program 77.9%

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

      1. sub-neg 77.9%

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

      2. fma-def 77.9%

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

      3. fma-def 78.4%

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

      4. +-commutative 78.4%

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

      5. metadata-eval 78.4%

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

   3. Simplified 78.4%

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

      1. fma-udef 77.9%

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

      2. +-commutative 77.9%

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

      3. +-commutative 77.9%

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

      4. associate-*l* 77.9%

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

      5. +-commutative 77.9%

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

      6. associate-*l* 77.9%

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

   5. Applied egg-rr 77.9%

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

   6. Taylor expanded in a around inf 65.4%

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

      1. +-commutative 65.4%

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

      2. associate-+l+ 65.4%

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

      3. *-commutative 65.4%

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

      4. metadata-eval 65.4%

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

      5. pow-plus 65.4%

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

      6. *-commutative 65.4%

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

      7. distribute-lft-in 81.3%

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

      8. unpow3 81.3%

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

      9. unpow2 81.3%

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

      10. associate-*l* 81.2%

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

      11. distribute-lft-out 81.2%

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

      12. unpow2 81.2%

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

      13. fma-def 81.2%

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

      14. unpow2 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in b around 0 80.4%

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

       1. sub-neg 80.4%

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

       2. metadata-eval 80.4%

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

       3. unpow2 80.4%

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

   11. Simplified 80.4%

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

   if 1e303 < (*.f64 b b)

   1. Initial program 63.6%

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

      1. sub-neg 63.6%

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

      2. fma-def 63.6%

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

      3. fma-def 65.5%

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

      4. +-commutative 65.5%

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

      5. metadata-eval 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in a around 0 52.7%

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

      1. associate-+r+ 52.7%

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

      2. associate-*r* 52.7%

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

      3. distribute-rgt-out 76.4%

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

      4. metadata-eval 76.4%

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

      5. distribute-lft-in 76.4%

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

      6. unpow2 76.4%

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

      7. distribute-rgt-in 76.4%

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

      8. metadata-eval 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in b around 0 76.4%

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

      1. *-commutative 76.4%

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

      2. *-commutative 76.4%

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

      3. unpow2 76.4%

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

      4. associate-*l* 76.4%

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

      5. *-commutative 76.4%

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

      6. +-commutative 76.4%

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

      7. fma-def 76.4%

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

   9. Simplified 76.4%

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

   10. Taylor expanded in a around 0 100.0%

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

       1. *-commutative 100.0%

          \[\leadsto \color{blue}{{b}^{2} \cdot 12} + -1 \]

       2. unpow2 100.0%

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

       3. associate-*l* 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 84.6%

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

Alternative 8: 83.8% accurate, 9.8× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+303)
   (+ -1.0 (* (* a a) (* a a)))
   (+ -1.0 (* b (* b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+303) {
		tmp = -1.0 + ((a * a) * (a * a));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+303) then
        tmp = (-1.0d0) + ((a * a) * (a * a))
    else
        tmp = (-1.0d0) + (b * (b * 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+303) {
		tmp = -1.0 + ((a * a) * (a * a));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+303:
		tmp = -1.0 + ((a * a) * (a * a))
	else:
		tmp = -1.0 + (b * (b * 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+303)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(a * a)));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+303)
		tmp = -1.0 + ((a * a) * (a * a));
	else
		tmp = -1.0 + (b * (b * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+303], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e303

   1. Initial program 77.9%

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

      1. sub-neg 77.9%

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

      2. fma-def 77.9%

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

      3. fma-def 78.4%

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

      4. +-commutative 78.4%

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

      5. metadata-eval 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in a around inf 78.2%

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

      1. sqr-pow 78.1%

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

      2. metadata-eval 78.1%

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

      3. pow2 78.1%

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

      4. metadata-eval 78.1%

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

      5. pow2 78.1%

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

   6. Applied egg-rr 78.1%

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

   if 1e303 < (*.f64 b b)

   1. Initial program 63.6%

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

      1. sub-neg 63.6%

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

      2. fma-def 63.6%

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

      3. fma-def 65.5%

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

      4. +-commutative 65.5%

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

      5. metadata-eval 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in a around 0 52.7%

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

      1. associate-+r+ 52.7%

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

      2. associate-*r* 52.7%

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

      3. distribute-rgt-out 76.4%

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

      4. metadata-eval 76.4%

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

      5. distribute-lft-in 76.4%

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

      6. unpow2 76.4%

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

      7. distribute-rgt-in 76.4%

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

      8. metadata-eval 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in b around 0 76.4%

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

      1. *-commutative 76.4%

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

      2. *-commutative 76.4%

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

      3. unpow2 76.4%

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

      4. associate-*l* 76.4%

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

      5. *-commutative 76.4%

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

      6. +-commutative 76.4%

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

      7. fma-def 76.4%

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

   9. Simplified 76.4%

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

   10. Taylor expanded in a around 0 100.0%

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

       1. *-commutative 100.0%

          \[\leadsto \color{blue}{{b}^{2} \cdot 12} + -1 \]

       2. unpow2 100.0%

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

       3. associate-*l* 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 82.8%

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

Alternative 9: 69.1% accurate, 11.6× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+303) (+ -1.0 (* (* a a) 4.0)) (+ -1.0 (* b (* b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+303) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+303) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = (-1.0d0) + (b * (b * 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+303) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+303:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = -1.0 + (b * (b * 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+303)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+303)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = -1.0 + (b * (b * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+303], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e303

   1. Initial program 77.9%

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

      1. sub-neg 77.9%

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

      2. fma-def 77.9%

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

      3. fma-def 78.4%

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

      4. +-commutative 78.4%

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

      5. metadata-eval 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in b around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. associate-*r* 64.6%

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

      3. fma-def 68.6%

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

      4. unpow2 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in a around 0 57.4%

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

      1. unpow2 57.4%

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

   9. Simplified 57.4%

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

   if 1e303 < (*.f64 b b)

   1. Initial program 63.6%

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

      1. sub-neg 63.6%

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

      2. fma-def 63.6%

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

      3. fma-def 65.5%

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

      4. +-commutative 65.5%

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

      5. metadata-eval 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in a around 0 52.7%

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

      1. associate-+r+ 52.7%

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

      2. associate-*r* 52.7%

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

      3. distribute-rgt-out 76.4%

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

      4. metadata-eval 76.4%

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

      5. distribute-lft-in 76.4%

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

      6. unpow2 76.4%

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

      7. distribute-rgt-in 76.4%

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

      8. metadata-eval 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in b around 0 76.4%

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

      1. *-commutative 76.4%

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

      2. *-commutative 76.4%

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

      3. unpow2 76.4%

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

      4. associate-*l* 76.4%

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

      5. *-commutative 76.4%

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

      6. +-commutative 76.4%

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

      7. fma-def 76.4%

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

   9. Simplified 76.4%

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

   10. Taylor expanded in a around 0 100.0%

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

       1. *-commutative 100.0%

          \[\leadsto \color{blue}{{b}^{2} \cdot 12} + -1 \]

       2. unpow2 100.0%

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

       3. associate-*l* 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 66.6%

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

Alternative 10: 51.1% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. sub-neg 74.9%

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

   2. fma-def 74.9%

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

   3. fma-def 75.6%

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

   4. +-commutative 75.6%

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

   5. metadata-eval 75.6%

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

3. Simplified 75.6%

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

4. Taylor expanded in b around 0 55.3%

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

   1. +-commutative 55.3%

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

   2. associate-*r* 55.3%

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

   3. fma-def 58.8%

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

   4. unpow2 58.8%

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

6. Simplified 58.8%

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

7. Taylor expanded in a around 0 50.1%

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

   1. unpow2 50.1%

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

9. Simplified 50.1%

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

10. Final simplification 50.1%

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

Alternative 11: 24.7% accurate, 128.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.9%

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

   1. sub-neg 74.9%

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

   2. fma-def 74.9%

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

   3. fma-def 75.6%

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

   4. +-commutative 75.6%

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

   5. metadata-eval 75.6%

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

3. Simplified 75.6%

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

4. Taylor expanded in a around inf 68.7%

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

5. Taylor expanded in a around 0 22.4%

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

6. Final simplification 22.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023201 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))