Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.2% → 97.6%

Time: 9.2s

Alternatives: 8

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: 74.2% 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: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot \left(2 \cdot {a}^{2} + \left({b}^{2} + 4 \cdot \left(1 + a \cdot -3\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+18)
   (+ (* (* a a) (+ 4.0 (* a (+ a 4.0)))) -1.0)
   (*
    (pow b 2.0)
    (+ (* 2.0 (pow a 2.0)) (+ (pow b 2.0) (* 4.0 (+ 1.0 (* a -3.0))))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+18) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = pow(b, 2.0) * ((2.0 * pow(a, 2.0)) + (pow(b, 2.0) + (4.0 * (1.0 + (a * -3.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+18) then
        tmp = ((a * a) * (4.0d0 + (a * (a + 4.0d0)))) + (-1.0d0)
    else
        tmp = (b ** 2.0d0) * ((2.0d0 * (a ** 2.0d0)) + ((b ** 2.0d0) + (4.0d0 * (1.0d0 + (a * (-3.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+18) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = Math.pow(b, 2.0) * ((2.0 * Math.pow(a, 2.0)) + (Math.pow(b, 2.0) + (4.0 * (1.0 + (a * -3.0)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+18:
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0
	else:
		tmp = math.pow(b, 2.0) * ((2.0 * math.pow(a, 2.0)) + (math.pow(b, 2.0) + (4.0 * (1.0 + (a * -3.0)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+18)
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + 4.0)))) + -1.0);
	else
		tmp = Float64((b ^ 2.0) * Float64(Float64(2.0 * (a ^ 2.0)) + Float64((b ^ 2.0) + Float64(4.0 * Float64(1.0 + Float64(a * -3.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+18)
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	else
		tmp = (b ^ 2.0) * ((2.0 * (a ^ 2.0)) + ((b ^ 2.0) + (4.0 * (1.0 + (a * -3.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+18], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] * N[(N[(2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] + N[(4.0 * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e18

   1. Initial program 76.0%

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

      1. sub-neg 76.0%

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

      2. +-commutative 76.0%

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

      3. fma-define 76.0%

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

      4. +-commutative 76.0%

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

      5. associate-*l* 76.0%

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

      6. cancel-sign-sub-inv 76.0%

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

      7. metadata-eval 76.0%

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

      8. fma-define 76.0%

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

      9. metadata-eval 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in b around 0 74.5%

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

   6. Taylor expanded in a around 0 98.4%

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

      1. pow2 98.4%

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

   8. Applied egg-rr 98.4%

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

   if 5e18 < (*.f64 b b)

   1. Initial program 65.7%

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

      1. associate--l+ 65.7%

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

      2. +-commutative 65.7%

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

      3. +-commutative 65.7%

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

      4. sub-neg 65.7%

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

      5. associate-+l+ 65.7%

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

      6. +-commutative 65.7%

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

      7. associate-+l+ 65.7%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in b around inf 84.3%

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

   6. Taylor expanded in b around 0 98.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot \left(2 \cdot {a}^{2} + \left({b}^{2} + 4 \cdot \left(1 + a \cdot -3\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b):
	t_0 = math.pow(((b * b) + (a * a)), 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 = ((a * a) * ((a * a) + 4.0)) + -1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((b * b) + (a * a)) ^ 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 * a) * ((a * a) + 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < +inf.0

   1. Initial program 99.9%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. fma-define 8.2%

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

      4. +-commutative 8.2%

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

      5. associate-*l* 8.2%

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

      6. cancel-sign-sub-inv 8.2%

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

      7. metadata-eval 8.2%

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

      8. fma-define 8.2%

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

      9. metadata-eval 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in b around 0 16.9%

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

   6. Taylor expanded in a around 0 93.7%

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

      1. pow2 93.7%

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

   8. Applied egg-rr 93.7%

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

   9. Taylor expanded in a around inf 93.7%

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

4. Final simplification 98.1%

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

Alternative 3: 81.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3300000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3300000000.0)
   (+ (* (* a a) (+ 4.0 (* a (+ a 4.0)))) -1.0)
   (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3300000000.0) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3300000000.0d0) then
        tmp = ((a * a) * (4.0d0 + (a * (a + 4.0d0)))) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3300000000.0) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3300000000.0:
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3300000000.0)
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + 4.0)))) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3300000000.0)
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3300000000.0], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.3e9

   1. Initial program 73.7%

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

      1. sub-neg 73.7%

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

      2. +-commutative 73.7%

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

      3. fma-define 75.8%

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

      4. +-commutative 75.8%

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

      5. associate-*l* 75.8%

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

      6. cancel-sign-sub-inv 75.8%

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

      7. metadata-eval 75.8%

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

      8. fma-define 75.8%

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

      9. metadata-eval 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in b around 0 60.4%

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

   6. Taylor expanded in a around 0 83.9%

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

      1. pow2 83.9%

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

   8. Applied egg-rr 83.9%

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

   if 3.3e9 < b

   1. Initial program 64.5%

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

      1. associate--l+ 64.5%

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

      2. +-commutative 64.5%

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

      3. +-commutative 64.5%

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

      4. sub-neg 64.5%

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

      5. associate-+l+ 64.5%

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

      6. +-commutative 64.5%

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

      7. associate-+l+ 64.5%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in a around 0 91.2%

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

      1. unpow2 91.2%

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

   7. Applied egg-rr 91.2%

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

   8. Taylor expanded in b around inf 91.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3300000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.7% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.45:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.45)
   (+ (* (* a a) 4.0) -1.0)
   (+ (* (* a a) (+ 4.0 (* a 4.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.45) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = ((a * a) * (4.0 + (a * 4.0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.45d0)) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else
        tmp = ((a * a) * (4.0d0 + (a * 4.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.45) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = ((a * a) * (4.0 + (a * 4.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.45:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = ((a * a) * (4.0 + (a * 4.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.45)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * 4.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.45)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = ((a * a) * (4.0 + (a * 4.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.45], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4500000000000002

   1. Initial program 27.2%

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

      1. sub-neg 27.2%

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

      2. +-commutative 27.2%

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

      3. fma-define 35.0%

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

      4. +-commutative 35.0%

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

      5. associate-*l* 35.0%

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

      6. cancel-sign-sub-inv 35.0%

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

      7. metadata-eval 35.0%

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

      8. fma-define 35.0%

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

      9. metadata-eval 35.0%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in b around 0 17.5%

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

   6. Taylor expanded in a around 0 58.7%

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

      1. pow2 90.1%

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

   8. Applied egg-rr 58.7%

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

   if -2.4500000000000002 < a

   1. Initial program 90.4%

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

      1. sub-neg 90.4%

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

      2. +-commutative 90.4%

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

      3. fma-define 90.4%

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

      4. +-commutative 90.4%

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

      5. associate-*l* 90.4%

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

      6. cancel-sign-sub-inv 90.4%

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

      7. metadata-eval 90.4%

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

      8. fma-define 90.4%

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

      9. metadata-eval 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in b around 0 63.6%

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

   6. Taylor expanded in a around 0 56.5%

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

      1. pow2 63.6%

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

   8. Applied egg-rr 56.5%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.45:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot 4\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.3% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

   1. sub-neg 71.4%

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

   2. +-commutative 71.4%

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

   3. fma-define 73.7%

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

   4. +-commutative 73.7%

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

   5. associate-*l* 73.7%

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

   6. cancel-sign-sub-inv 73.7%

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

   7. metadata-eval 73.7%

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

   8. fma-define 73.7%

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

   9. metadata-eval 73.7%

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

3. Simplified 73.7%

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

5. Taylor expanded in b around 0 49.7%

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

6. Taylor expanded in a around 0 71.6%

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

   1. pow2 71.6%

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

8. Applied egg-rr 71.6%

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

9. Final simplification 71.6%

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

Alternative 6: 69.7% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

   1. sub-neg 71.4%

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

   2. +-commutative 71.4%

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

   3. fma-define 73.7%

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

   4. +-commutative 73.7%

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

   5. associate-*l* 73.7%

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

   6. cancel-sign-sub-inv 73.7%

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

   7. metadata-eval 73.7%

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

   8. fma-define 73.7%

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

   9. metadata-eval 73.7%

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

3. Simplified 73.7%

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

5. Taylor expanded in b around 0 49.7%

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

6. Taylor expanded in a around 0 71.6%

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

   1. pow2 71.6%

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

8. Applied egg-rr 71.6%

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

9. Taylor expanded in a around inf 70.7%

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

10. Final simplification 70.7%

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

Alternative 7: 51.6% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

   1. sub-neg 71.4%

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

   2. +-commutative 71.4%

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

   3. fma-define 73.7%

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

   4. +-commutative 73.7%

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

   5. associate-*l* 73.7%

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

   6. cancel-sign-sub-inv 73.7%

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

   7. metadata-eval 73.7%

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

   8. fma-define 73.7%

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

   9. metadata-eval 73.7%

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

3. Simplified 73.7%

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

5. Taylor expanded in b around 0 49.7%

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

6. Taylor expanded in a around 0 52.0%

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

   1. pow2 71.6%

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

8. Applied egg-rr 52.0%

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

9. Final simplification 52.0%

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

Alternative 8: 25.2% 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 71.4%

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

   1. associate--l+ 71.4%

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

   2. +-commutative 71.4%

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

   3. +-commutative 71.4%

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

   4. sub-neg 71.4%

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

   5. associate-+l+ 71.4%

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

   6. +-commutative 71.4%

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

   7. associate-+l+ 71.4%

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

3. Simplified 72.2%

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

5. Taylor expanded in a around 0 63.1%

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

6. Taylor expanded in b around 0 21.7%

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

Reproduce

herbie shell --seed 2024137 
(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))