Bouland and Aaronson, Equation (26)

Average Accuracy: 39.9% → 40.0%

Time: 9.0s

Precision: binary64

Cost: 13504

Math

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

↓

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

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (+ (+ (pow (hypot a b) 4.0) (* 4.0 (* b b))) -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) {
	return (pow(hypot(a, b), 4.0) + (4.0 * (b * b))) + -1.0;
}

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) {
	return (Math.pow(Math.hypot(a, b), 4.0) + (4.0 * (b * b))) + -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):
	return (math.pow(math.hypot(a, b), 4.0) + (4.0 * (b * b))) + -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)
	return Float64(Float64((hypot(a, b) ^ 4.0) + Float64(4.0 * Float64(b * b))) + -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)
	tmp = ((hypot(a, b) ^ 4.0) + (4.0 * (b * b))) + -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_] := N[(N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 39.9%

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

2. Applied egg-rr 40.0%

   \[\leadsto \left(\color{blue}{\left(0 + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   Proof

   [Start] 39.9	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   add-log-exp [=>] 25.8	\[ \left(\color{blue}{\log \left(e^{{\left(a \cdot a + b \cdot b\right)}^{2}}\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   *-un-lft-identity [=>] 25.8	\[ \left(\log \color{blue}{\left(1 \cdot e^{{\left(a \cdot a + b \cdot b\right)}^{2}}\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   log-prod [=>] 25.8	\[ \left(\color{blue}{\left(\log 1 + \log \left(e^{{\left(a \cdot a + b \cdot b\right)}^{2}}\right)\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   metadata-eval [=>] 25.8	\[ \left(\left(\color{blue}{0} + \log \left(e^{{\left(a \cdot a + b \cdot b\right)}^{2}}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   add-log-exp [<=] 39.9	\[ \left(\left(0 + \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   add-sqr-sqrt [=>] 39.9	\[ \left(\left(0 + {\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}}^{2}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   unpow-prod-down [=>] 39.9	\[ \left(\left(0 + \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   pow-prod-up [=>] 40.0	\[ \left(\left(0 + \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{\left(2 + 2\right)}}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   hypot-def [=>] 40.0	\[ \left(\left(0 + {\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{\left(2 + 2\right)}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   metadata-eval [=>] 40.0	\[ \left(\left(0 + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{\color{blue}{4}}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

3. Simplified 40.0%

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

   [Start] 40.0	\[ \left(\left(0 + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   +-lft-identity [=>] 40.0	\[ \left(\color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

4. Final simplification 40.0%

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

Alternatives

Alternative 1
Accuracy	39.9%
Cost	7424

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

Alternative 2
Accuracy	38.0%
Cost	7305

\[\begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq -125000000 \lor \neg \left(a \leq 1.6 \cdot 10^{-101}\right):\\ \;\;\;\;\left(t_0 + {a}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 3
Accuracy	38.0%
Cost	7305

\[\begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq -125000000 \lor \neg \left(a \leq 4 \cdot 10^{-101}\right):\\ \;\;\;\;\left(t_0 + {a}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + {b}^{4}\right) + -1\\ \end{array} \]

Alternative 4
Accuracy	32.9%
Cost	960

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

Reproduce

herbie shell --seed 2023153 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))