Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.0

Time: 25.2s

Precision: binary64

Cost: 40512

Math

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

↓

\[\begin{array}{l} t_0 := {a}^{4} + {b}^{4}\\ t_1 := {\left(a \cdot b\right)}^{2}\\ \left(\left(\left(t_1 + \left(b \cdot \left(b \cdot 4\right) + \frac{t_0}{2}\right)\right) - \frac{t_0}{-2}\right) + t_1\right) - 1 \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (pow a 4.0) (pow b 4.0))) (t_1 (pow (* a b) 2.0)))
   (- (+ (- (+ t_1 (+ (* b (* b 4.0)) (/ t_0 2.0))) (/ t_0 -2.0)) t_1) 1.0)))

C

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

↓

double code(double a, double b) {
	double t_0 = pow(a, 4.0) + pow(b, 4.0);
	double t_1 = pow((a * b), 2.0);
	return (((t_1 + ((b * (b * 4.0)) + (t_0 / 2.0))) - (t_0 / -2.0)) + t_1) - 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 * (b * b))) - 1.0d0
end function

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (a ** 4.0d0) + (b ** 4.0d0)
    t_1 = (a * b) ** 2.0d0
    code = (((t_1 + ((b * (b * 4.0d0)) + (t_0 / 2.0d0))) - (t_0 / (-2.0d0))) + t_1) - 1.0d0
end function

Java

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

↓

public static double code(double a, double b) {
	double t_0 = Math.pow(a, 4.0) + Math.pow(b, 4.0);
	double t_1 = Math.pow((a * b), 2.0);
	return (((t_1 + ((b * (b * 4.0)) + (t_0 / 2.0))) - (t_0 / -2.0)) + t_1) - 1.0;
}

Python

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

↓

def code(a, b):
	t_0 = math.pow(a, 4.0) + math.pow(b, 4.0)
	t_1 = math.pow((a * b), 2.0)
	return (((t_1 + ((b * (b * 4.0)) + (t_0 / 2.0))) - (t_0 / -2.0)) + t_1) - 1.0

Julia

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

↓

function code(a, b)
	t_0 = Float64((a ^ 4.0) + (b ^ 4.0))
	t_1 = Float64(a * b) ^ 2.0
	return Float64(Float64(Float64(Float64(t_1 + Float64(Float64(b * Float64(b * 4.0)) + Float64(t_0 / 2.0))) - Float64(t_0 / -2.0)) + t_1) - 1.0)
end

MATLAB

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

↓

function tmp = code(a, b)
	t_0 = (a ^ 4.0) + (b ^ 4.0);
	t_1 = (a * b) ^ 2.0;
	tmp = (((t_1 + ((b * (b * 4.0)) + (t_0 / 2.0))) - (t_0 / -2.0)) + t_1) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[a_, b_] := Block[{t$95$0 = N[(N[Power[a, 4.0], $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[(t$95$1 + N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Initial program 0.2

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

2. Taylor expanded in a around 0 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   rational_best-simplify-47 [=>] 0.0	\[ \left(\color{blue}{\left({b}^{4} + \left({a}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   rational_best-simplify-3 [<=] 0.0	\[ \left(\left({b}^{4} + \color{blue}{\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   exponential-simplify-28 [=>] 0.0	\[ \left(\left({b}^{4} + \left(2 \cdot \color{blue}{{\left(a \cdot b\right)}^{2}} + {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   rational_best-simplify-1 [=>] 0.0	\[ \left(\left({b}^{4} + \left(2 \cdot {\color{blue}{\left(b \cdot a\right)}}^{2} + {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

4. Applied egg-rr 0.0

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

5. Simplified 0.0

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

   [Start] 0.0	\[ \left(\left(b \cdot \left(b \cdot 4\right) + \left({\left(b \cdot a\right)}^{2} + \frac{{b}^{4} + {a}^{4}}{2}\right)\right) - \left(-\left({\left(b \cdot a\right)}^{2} + \frac{{b}^{4} + {a}^{4}}{2}\right)\right)\right) - 1 \]
   rational_best-simplify-14 [=>] 0.0	\[ \left(\left(b \cdot \left(b \cdot 4\right) + \left({\left(b \cdot a\right)}^{2} + \frac{{b}^{4} + {a}^{4}}{2}\right)\right) - \color{blue}{\left(0 - \left({\left(b \cdot a\right)}^{2} + \frac{{b}^{4} + {a}^{4}}{2}\right)\right)}\right) - 1 \]
   rational_best-simplify-3 [=>] 0.0	\[ \left(\left(b \cdot \left(b \cdot 4\right) + \left({\left(b \cdot a\right)}^{2} + \frac{{b}^{4} + {a}^{4}}{2}\right)\right) - \left(0 - \color{blue}{\left(\frac{{b}^{4} + {a}^{4}}{2} + {\left(b \cdot a\right)}^{2}\right)}\right)\right) - 1 \]
   rational_best-simplify-57 [=>] 0.0	\[ \left(\left(b \cdot \left(b \cdot 4\right) + \left({\left(b \cdot a\right)}^{2} + \frac{{b}^{4} + {a}^{4}}{2}\right)\right) - \color{blue}{\left(\left(0 - \frac{{b}^{4} + {a}^{4}}{2}\right) + \left(-{\left(b \cdot a\right)}^{2}\right)\right)}\right) - 1 \]
   rational_best-simplify-14 [<=] 0.0	\[ \left(\left(b \cdot \left(b \cdot 4\right) + \left({\left(b \cdot a\right)}^{2} + \frac{{b}^{4} + {a}^{4}}{2}\right)\right) - \left(\color{blue}{\left(-\frac{{b}^{4} + {a}^{4}}{2}\right)} + \left(-{\left(b \cdot a\right)}^{2}\right)\right)\right) - 1 \]
   rational_best-simplify-57 [=>] 0.0	\[ \color{blue}{\left(\left(\left(b \cdot \left(b \cdot 4\right) + \left({\left(b \cdot a\right)}^{2} + \frac{{b}^{4} + {a}^{4}}{2}\right)\right) - \left(-\frac{{b}^{4} + {a}^{4}}{2}\right)\right) + \left(-\left(-{\left(b \cdot a\right)}^{2}\right)\right)\right)} - 1 \]

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	27072

\[\begin{array}{l} t_0 := {\left(a \cdot b\right)}^{2}\\ \left(\left(t_0 + \left(b \cdot \left(b \cdot 4\right) + \left({a}^{4} + {b}^{4}\right)\right)\right) + t_0\right) - 1 \end{array} \]

Alternative 2
Error	0.0
Cost	20480

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

Alternative 3
Error	0.2
Cost	7808

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

Alternative 4
Error	0.2
Cost	7424

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

Alternative 5
Error	2.4
Cost	7304

\[\begin{array}{l} t_0 := {b}^{4} - 1\\ \mathbf{if}\;b \leq -7.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.2:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.4
Cost	7304

\[\begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ t_1 := \left({a}^{4} + t_0\right) - 1\\ \mathbf{if}\;a \leq -0.056:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 0.0014:\\ \;\;\;\;\left({b}^{4} + t_0\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	2.8
Cost	6920

\[\begin{array}{l} t_0 := {b}^{4} - 1\\ \mathbf{if}\;b \leq -7.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.2:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	12.2
Cost	6656

\[{a}^{4} - 1 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))