Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.8% → 98.3%

Time: 11.1s

Alternatives: 10

Speedup: 1.2×

• 60.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-+l+ 99.8%

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

      6. +-commutative 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. cube-mult 99.8%

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

      4. distribute-rgt-in 99.8%

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

      5. associate-*l* 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt-in 99.8%

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

      8. *-un-lft-identity 99.8%

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

      9. fma-def 99.8%

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

   6. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. +-commutative 0.0%

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

      3. +-commutative 0.0%

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

      4. sub-neg 0.0%

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

      5. associate-+l+ 0.0%

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

      6. +-commutative 0.0%

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

      7. fma-def 0.0%

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

   3. Simplified 5.5%

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

      1. fma-udef 5.5%

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

      2. *-un-lft-identity 5.5%

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

      3. cube-mult 5.5%

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

      4. distribute-rgt-in 5.5%

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

      5. associate-*l* 5.5%

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

      6. +-commutative 5.5%

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

      7. distribute-rgt-in 5.5%

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

      8. *-un-lft-identity 5.5%

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

      9. fma-def 5.5%

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

   6. Applied egg-rr 5.5%

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

   7. Taylor expanded in a around inf 94.9%

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

4. Final simplification 98.4%

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

Alternative 2: 98.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

      3. fma-neg 99.8%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-rgt-neg-in 99.8%

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

      10. metadata-eval 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. +-commutative 0.0%

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

      3. +-commutative 0.0%

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

      4. sub-neg 0.0%

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

      5. associate-+l+ 0.0%

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

      6. +-commutative 0.0%

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

      7. fma-def 0.0%

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

   3. Simplified 5.5%

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

      1. fma-udef 5.5%

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

      2. *-un-lft-identity 5.5%

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

      3. cube-mult 5.5%

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

      4. distribute-rgt-in 5.5%

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

      5. associate-*l* 5.5%

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

      6. +-commutative 5.5%

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

      7. distribute-rgt-in 5.5%

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

      8. *-un-lft-identity 5.5%

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

      9. fma-def 5.5%

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

   6. Applied egg-rr 5.5%

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

   7. Taylor expanded in a around inf 94.9%

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

4. Final simplification 98.4%

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

Alternative 3: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. +-commutative 0.0%

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

      3. +-commutative 0.0%

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

      4. sub-neg 0.0%

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

      5. associate-+l+ 0.0%

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

      6. +-commutative 0.0%

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

      7. fma-def 0.0%

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

   3. Simplified 5.5%

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

      1. fma-udef 5.5%

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

      2. *-un-lft-identity 5.5%

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

      3. cube-mult 5.5%

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

      4. distribute-rgt-in 5.5%

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

      5. associate-*l* 5.5%

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

      6. +-commutative 5.5%

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

      7. distribute-rgt-in 5.5%

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

      8. *-un-lft-identity 5.5%

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

      9. fma-def 5.5%

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

   6. Applied egg-rr 5.5%

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

   7. Taylor expanded in a around inf 94.9%

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

4. Final simplification 98.4%

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

Alternative 4: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+73}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -2.1:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+15}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.5e+73)
   (pow a 4.0)
   (if (<= a -2.1)
     (pow b 4.0)
     (if (<= a 3.9e+15) (+ -1.0 (* (* b b) 4.0)) (pow a 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.5e+73) {
		tmp = pow(a, 4.0);
	} else if (a <= -2.1) {
		tmp = pow(b, 4.0);
	} else if (a <= 3.9e+15) {
		tmp = -1.0 + ((b * b) * 4.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.5d+73)) then
        tmp = a ** 4.0d0
    else if (a <= (-2.1d0)) then
        tmp = b ** 4.0d0
    else if (a <= 3.9d+15) then
        tmp = (-1.0d0) + ((b * b) * 4.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.5e+73) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= -2.1) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 3.9e+15) {
		tmp = -1.0 + ((b * b) * 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.5e+73:
		tmp = math.pow(a, 4.0)
	elif a <= -2.1:
		tmp = math.pow(b, 4.0)
	elif a <= 3.9e+15:
		tmp = -1.0 + ((b * b) * 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.5e+73)
		tmp = a ^ 4.0;
	elseif (a <= -2.1)
		tmp = b ^ 4.0;
	elseif (a <= 3.9e+15)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 4.0));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.5e+73)
		tmp = a ^ 4.0;
	elseif (a <= -2.1)
		tmp = b ^ 4.0;
	elseif (a <= 3.9e+15)
		tmp = -1.0 + ((b * b) * 4.0);
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.5e+73], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, -2.1], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 3.9e+15], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.5000000000000003e73 or 3.9e15 < a

   1. Initial program 37.9%

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

      1. associate--l+ 37.9%

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

      2. +-commutative 37.9%

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

      3. +-commutative 37.9%

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

      4. sub-neg 37.9%

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

      5. associate-+l+ 37.9%

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

      6. +-commutative 37.9%

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

      7. fma-def 37.9%

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

   3. Simplified 41.3%

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

      1. fma-udef 41.3%

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

      2. *-un-lft-identity 41.3%

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

      3. cube-mult 41.3%

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

      4. distribute-rgt-in 41.3%

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

      5. associate-*l* 41.3%

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

      6. +-commutative 41.3%

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

      7. distribute-rgt-in 41.3%

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

      8. *-un-lft-identity 41.3%

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

      9. fma-def 41.3%

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

   6. Applied egg-rr 41.3%

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

   7. Taylor expanded in a around inf 95.1%

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

   if -5.5000000000000003e73 < a < -2.10000000000000009

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-+l+ 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-def 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. cube-mult 99.7%

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

      4. distribute-rgt-in 99.7%

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

      5. associate-*l* 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-rgt-in 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. fma-def 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around inf 74.2%

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

   if -2.10000000000000009 < a < 3.9e15

   1. Initial program 99.0%

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

      1. associate--l+ 99.0%

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

      2. fma-def 99.0%

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

      3. fma-neg 99.0%

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

      4. associate-*l* 99.0%

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

      5. fma-def 99.0%

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

      6. +-commutative 99.0%

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

      7. sub-neg 99.0%

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

      8. *-commutative 99.0%

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

      9. distribute-rgt-neg-in 99.0%

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

      10. metadata-eval 99.0%

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

      11. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in a around 0 97.5%

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

   6. Taylor expanded in b around 0 76.0%

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

      1. unpow2 76.0%

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

   8. Applied egg-rr 76.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+73}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -2.1:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+15}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+50)
   (+ -1.0 (+ (pow 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) <= 1e+50) {
		tmp = -1.0 + (pow(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) <= 1d+50) then
        tmp = (-1.0d0) + ((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) <= 1e+50) {
		tmp = -1.0 + (Math.pow(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) <= 1e+50:
		tmp = -1.0 + (math.pow(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) <= 1e+50)
		tmp = Float64(-1.0 + Float64((a ^ 4.0) + Float64(Float64(a * 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) <= 1e+50)
		tmp = -1.0 + ((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], 1e+50], N[(-1.0 + N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 4.0), $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) < 1.0000000000000001e50

   1. Initial program 79.9%

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

      1. associate--l+ 79.9%

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

      2. +-commutative 79.9%

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

      3. +-commutative 79.9%

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

      4. sub-neg 79.9%

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

      5. associate-+l+ 79.9%

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

      6. +-commutative 79.9%

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

      7. fma-def 79.9%

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

   3. Simplified 79.9%

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

      1. fma-udef 79.9%

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

      2. *-un-lft-identity 79.9%

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

      3. cube-mult 79.9%

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

      4. distribute-rgt-in 79.9%

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

      5. associate-*l* 79.9%

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

      6. +-commutative 79.9%

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

      7. distribute-rgt-in 79.9%

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

      8. *-un-lft-identity 79.9%

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

      9. fma-def 79.9%

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

   6. Applied egg-rr 79.9%

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

   7. Taylor expanded in b around 0 77.7%

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

   8. Taylor expanded in a around 0 95.9%

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

   if 1.0000000000000001e50 < (*.f64 b b)

   1. Initial program 61.6%

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

      1. sub-neg 61.6%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in b around inf 94.6%

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

4. Final simplification 95.3%

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

Alternative 6: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+37} \lor \neg \left(a \leq 5.5 \cdot 10^{+15}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -9.5e+37) (not (<= a 5.5e+15)))
   (pow a 4.0)
   (+ -1.0 (* (* b b) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -9.5e+37) || !(a <= 5.5e+15)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * 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 ((a <= (-9.5d+37)) .or. (.not. (a <= 5.5d+15))) then
        tmp = a ** 4.0d0
    else
        tmp = (-1.0d0) + ((b * b) * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -9.5e+37) || !(a <= 5.5e+15)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -9.5e+37) or not (a <= 5.5e+15):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -9.5e+37) || !(a <= 5.5e+15))
		tmp = a ^ 4.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -9.5e+37) || ~((a <= 5.5e+15)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0 + ((b * b) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -9.5e+37], N[Not[LessEqual[a, 5.5e+15]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.4999999999999995e37 or 5.5e15 < a

   1. Initial program 41.8%

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

      1. associate--l+ 41.8%

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

      2. +-commutative 41.8%

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

      3. +-commutative 41.8%

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

      4. sub-neg 41.8%

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

      5. associate-+l+ 41.8%

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

      6. +-commutative 41.8%

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

      7. fma-def 41.8%

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

   3. Simplified 45.1%

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

      1. fma-udef 45.1%

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

      2. *-un-lft-identity 45.1%

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

      3. cube-mult 45.1%

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

      4. distribute-rgt-in 45.1%

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

      5. associate-*l* 45.1%

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

      6. +-commutative 45.1%

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

      7. distribute-rgt-in 45.1%

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

      8. *-un-lft-identity 45.1%

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

      9. fma-def 45.1%

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

   6. Applied egg-rr 45.1%

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

   7. Taylor expanded in a around inf 91.7%

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

   if -9.4999999999999995e37 < a < 5.5e15

   1. Initial program 99.0%

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

      1. associate--l+ 99.0%

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

      2. fma-def 99.0%

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

      3. fma-neg 99.0%

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

      4. associate-*l* 99.0%

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

      5. fma-def 99.0%

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

      6. +-commutative 99.0%

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

      7. sub-neg 99.0%

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

      8. *-commutative 99.0%

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

      9. distribute-rgt-neg-in 99.0%

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

      10. metadata-eval 99.0%

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

      11. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in a around 0 96.9%

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

   6. Taylor expanded in b around 0 73.6%

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

      1. unpow2 73.6%

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

   8. Applied egg-rr 73.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+37} \lor \neg \left(a \leq 5.5 \cdot 10^{+15}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+50) (+ -1.0 (pow a 4.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+50) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+50) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+50) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+50:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+50)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+50)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+50], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.0000000000000001e50

   1. Initial program 79.9%

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

      1. sub-neg 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in a around inf 95.7%

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

   if 1.0000000000000001e50 < (*.f64 b b)

   1. Initial program 61.6%

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

      1. associate--l+ 61.6%

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

      2. +-commutative 61.6%

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

      3. +-commutative 61.6%

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

      4. sub-neg 61.6%

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

      5. associate-+l+ 61.6%

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

      6. +-commutative 61.6%

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

      7. fma-def 61.6%

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

   3. Simplified 64.9%

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

      1. fma-udef 64.9%

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

      2. *-un-lft-identity 64.9%

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

      3. cube-mult 64.9%

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

      4. distribute-rgt-in 64.9%

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

      5. associate-*l* 64.9%

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

      6. +-commutative 64.9%

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

      7. distribute-rgt-in 64.9%

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

      8. *-un-lft-identity 64.9%

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

      9. fma-def 64.9%

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

   6. Applied egg-rr 64.9%

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

   7. Taylor expanded in b around inf 94.6%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+50}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+50) (+ -1.0 (pow a 4.0)) (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+50) {
		tmp = -1.0 + pow(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) <= 1d+50) then
        tmp = (-1.0d0) + (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) <= 1e+50) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+50:
		tmp = -1.0 + math.pow(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) <= 1e+50)
		tmp = Float64(-1.0 + (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) <= 1e+50)
		tmp = -1.0 + (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], 1e+50], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.0000000000000001e50

   1. Initial program 79.9%

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

      1. sub-neg 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in a around inf 95.7%

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

   if 1.0000000000000001e50 < (*.f64 b b)

   1. Initial program 61.6%

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

      1. sub-neg 61.6%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in b around inf 94.6%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+50}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.3% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

   1. associate--l+ 71.3%

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

   2. fma-def 71.3%

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

   3. fma-neg 71.3%

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

   4. associate-*l* 71.3%

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

   5. fma-def 72.9%

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

   6. +-commutative 72.9%

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

   7. sub-neg 72.9%

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

   8. *-commutative 72.9%

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

   9. distribute-rgt-neg-in 72.9%

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

   10. metadata-eval 72.9%

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

   11. metadata-eval 72.9%

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

3. Simplified 72.9%

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

5. Taylor expanded in a around 0 70.7%

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

6. Taylor expanded in b around 0 53.5%

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

   1. unpow2 53.5%

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

8. Applied egg-rr 53.5%

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

9. Final simplification 53.5%

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

Alternative 10: 25.6% accurate, 130.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 71.3%

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

   1. sub-neg 71.3%

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

3. Simplified 72.9%

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

5. Taylor expanded in a around inf 70.2%

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

6. Taylor expanded in a around 0 25.4%

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

7. Final simplification 25.4%

   \[\leadsto -1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))