Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.6% → 81.5%

Time: 12.9s

Alternatives: 6

Speedup: 3.7×

• 60.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b + a \cdot a\\ t_1 := a \cdot a + b \cdot b\\ \mathbf{if}\;a \leq -5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, b \cdot 3\right), b \cdot 4, t\_0 \cdot t\_0 + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - a, \left(a \cdot a\right) \cdot 4, 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right)\right) + t\_1 \cdot t\_1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* b b) (* a a))) (t_1 (+ (* a a) (* b b))))
   (if (<= a -5.0)
     (-
      (fma
       (fma b a (* b 3.0))
       (* b 4.0)
       (+ (* t_0 t_0) (* 4.0 (* (* a a) (- 1.0 a)))))
      1.0)
     (-
      (fma
       (- 1.0 a)
       (* (* a a) 4.0)
       (+ (* 4.0 (* (* b b) (+ a 3.0))) (* t_1 t_1)))
      1.0))))

C

double code(double a, double b) {
	double t_0 = (b * b) + (a * a);
	double t_1 = (a * a) + (b * b);
	double tmp;
	if (a <= -5.0) {
		tmp = fma(fma(b, a, (b * 3.0)), (b * 4.0), ((t_0 * t_0) + (4.0 * ((a * a) * (1.0 - a))))) - 1.0;
	} else {
		tmp = fma((1.0 - a), ((a * a) * 4.0), ((4.0 * ((b * b) * (a + 3.0))) + (t_1 * t_1))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) + Float64(a * a))
	t_1 = Float64(Float64(a * a) + Float64(b * b))
	tmp = 0.0
	if (a <= -5.0)
		tmp = Float64(fma(fma(b, a, Float64(b * 3.0)), Float64(b * 4.0), Float64(Float64(t_0 * t_0) + Float64(4.0 * Float64(Float64(a * a) * Float64(1.0 - a))))) - 1.0);
	else
		tmp = Float64(fma(Float64(1.0 - a), Float64(Float64(a * a) * 4.0), Float64(Float64(4.0 * Float64(Float64(b * b) * Float64(a + 3.0))) + Float64(t_1 * t_1))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.0], N[(N[(N[(b * a + N[(b * 3.0), $MachinePrecision]), $MachinePrecision] * N[(b * 4.0), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(1.0 - a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[(N[(4.0 * N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < -5

   1. Initial program 62.3%

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

   3. Applied egg-rr 0

      \[\leadsto expr\]

   5. Applied egg-rr 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if -5 < a

   1. Initial program 82.7%

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

   3. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot a + b \cdot b\\ \mathbf{if}\;a \leq -4.5 \cdot 10^{+35}:\\ \;\;\;\;\left({\left(\frac{1}{b \cdot b + a \cdot a}\right)}^{-2} + 4 \cdot \mathsf{fma}\left(a + 3, b \cdot b, a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - a, \left(a \cdot a\right) \cdot 4, 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right)\right) + t\_0 \cdot t\_0\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* a a) (* b b))))
   (if (<= a -4.5e+35)
     (-
      (+
       (pow (/ 1.0 (+ (* b b) (* a a))) -2.0)
       (* 4.0 (fma (+ a 3.0) (* b b) (* a (* a (- 1.0 a))))))
      1.0)
     (-
      (fma
       (- 1.0 a)
       (* (* a a) 4.0)
       (+ (* 4.0 (* (* b b) (+ a 3.0))) (* t_0 t_0)))
      1.0))))

C

double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	double tmp;
	if (a <= -4.5e+35) {
		tmp = (pow((1.0 / ((b * b) + (a * a))), -2.0) + (4.0 * fma((a + 3.0), (b * b), (a * (a * (1.0 - a)))))) - 1.0;
	} else {
		tmp = fma((1.0 - a), ((a * a) * 4.0), ((4.0 * ((b * b) * (a + 3.0))) + (t_0 * t_0))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) + Float64(b * b))
	tmp = 0.0
	if (a <= -4.5e+35)
		tmp = Float64(Float64((Float64(1.0 / Float64(Float64(b * b) + Float64(a * a))) ^ -2.0) + Float64(4.0 * fma(Float64(a + 3.0), Float64(b * b), Float64(a * Float64(a * Float64(1.0 - a)))))) - 1.0);
	else
		tmp = Float64(fma(Float64(1.0 - a), Float64(Float64(a * a) * 4.0), Float64(Float64(4.0 * Float64(Float64(b * b) * Float64(a + 3.0))) + Float64(t_0 * t_0))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.5e+35], N[(N[(N[Power[N[(1.0 / N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + N[(4.0 * N[(N[(a + 3.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(1.0 - a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[(N[(4.0 * N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.4999999999999997e35

   1. Initial program 64.0%

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

   3. Applied egg-rr 0

      \[\leadsto expr\]

   5. Applied egg-rr 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if -4.4999999999999997e35 < a

   1. Initial program 81.6%

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

   3. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b + a \cdot a\\ t_1 := a \cdot a + b \cdot b\\ \mathbf{if}\;a \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\left(t\_0 \cdot t\_0 + 4 \cdot \mathsf{fma}\left(a + 3, b \cdot b, a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - a, \left(a \cdot a\right) \cdot 4, 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right)\right) + t\_1 \cdot t\_1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* b b) (* a a))) (t_1 (+ (* a a) (* b b))))
   (if (<= a -1.6e+154)
     (-
      (+ (* t_0 t_0) (* 4.0 (fma (+ a 3.0) (* b b) (* a (* a (- 1.0 a))))))
      1.0)
     (-
      (fma
       (- 1.0 a)
       (* (* a a) 4.0)
       (+ (* 4.0 (* (* b b) (+ a 3.0))) (* t_1 t_1)))
      1.0))))

C

double code(double a, double b) {
	double t_0 = (b * b) + (a * a);
	double t_1 = (a * a) + (b * b);
	double tmp;
	if (a <= -1.6e+154) {
		tmp = ((t_0 * t_0) + (4.0 * fma((a + 3.0), (b * b), (a * (a * (1.0 - a)))))) - 1.0;
	} else {
		tmp = fma((1.0 - a), ((a * a) * 4.0), ((4.0 * ((b * b) * (a + 3.0))) + (t_1 * t_1))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) + Float64(a * a))
	t_1 = Float64(Float64(a * a) + Float64(b * b))
	tmp = 0.0
	if (a <= -1.6e+154)
		tmp = Float64(Float64(Float64(t_0 * t_0) + Float64(4.0 * fma(Float64(a + 3.0), Float64(b * b), Float64(a * Float64(a * Float64(1.0 - a)))))) - 1.0);
	else
		tmp = Float64(fma(Float64(1.0 - a), Float64(Float64(a * a) * 4.0), Float64(Float64(4.0 * Float64(Float64(b * b) * Float64(a + 3.0))) + Float64(t_1 * t_1))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.6e+154], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(4.0 * N[(N[(a + 3.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(1.0 - a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[(N[(4.0 * N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6e154

   1. Initial program 52.6%

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

   3. Applied egg-rr 0

      \[\leadsto expr\]

   5. Applied egg-rr 0

      \[\leadsto expr\]

   if -1.6e154 < a

   1. Initial program 81.5%

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

   3. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b + a \cdot a\\ \left(t\_0 \cdot t\_0 + 4 \cdot \mathsf{fma}\left(a + 3, b \cdot b, a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) - 1 \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* b b) (* a a))))
   (-
    (+ (* t_0 t_0) (* 4.0 (fma (+ a 3.0) (* b b) (* a (* a (- 1.0 a))))))
    1.0)))

C

double code(double a, double b) {
	double t_0 = (b * b) + (a * a);
	return ((t_0 * t_0) + (4.0 * fma((a + 3.0), (b * b), (a * (a * (1.0 - a)))))) - 1.0;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) + Float64(a * a))
	return Float64(Float64(Float64(t_0 * t_0) + Float64(4.0 * fma(Float64(a + 3.0), Float64(b * b), Float64(a * Float64(a * Float64(1.0 - a)))))) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Applied egg-rr 0

   \[\leadsto expr\]

5. Applied egg-rr 0

   \[\leadsto expr\]
6. Add Preprocessing

Alternative 5: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot a + b \cdot b\\ \left(t\_0 \cdot t\_0 + 4 \cdot \mathsf{fma}\left(1 - a, a \cdot a, b \cdot \left(b \cdot \left(a + 3\right)\right)\right)\right) - 1 \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* a a) (* b b))))
   (-
    (+ (* t_0 t_0) (* 4.0 (fma (- 1.0 a) (* a a) (* b (* b (+ a 3.0))))))
    1.0)))

C

double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return ((t_0 * t_0) + (4.0 * fma((1.0 - a), (a * a), (b * (b * (a + 3.0)))))) - 1.0;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) + Float64(b * b))
	return Float64(Float64(Float64(t_0 * t_0) + Float64(4.0 * fma(Float64(1.0 - a), Float64(a * a), Float64(b * Float64(b * Float64(a + 3.0)))))) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Applied egg-rr 0

   \[\leadsto expr\]

5. Applied egg-rr 0

   \[\leadsto expr\]
6. Add Preprocessing

Alternative 6: 74.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot a + b \cdot b\\ \left(t\_0 \cdot t\_0 + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (* a a) (* b b))))
   (-
    (+ (* t_0 t_0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
    1.0)))

C

double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return ((t_0 * t_0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    t_0 = (a * a) + (b * b)
    code = ((t_0 * t_0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) + (b * b);
	return ((t_0 * t_0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	t_0 = (a * a) + (b * b)
	return ((t_0 * t_0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) + Float64(b * b))
	return Float64(Float64(Float64(t_0 * t_0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	t_0 = (a * a) + (b * b);
	tmp = ((t_0 * t_0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Applied egg-rr 0

   \[\leadsto expr\]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))