Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 8.3s

Precision: binary64

Cost: 14400

Math

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

↓

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

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))

↓

(FPCore (a b)
 :precision binary64
 (+
  (+
   (pow (hypot b a) 4.0)
   (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (+ 1.0 (* a -3.0))))))
  -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;
}

↓

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

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;
}

↓

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

↓

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

↓

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

↓

function tmp = code(a, b)
	tmp = ((hypot(b, a) ^ 4.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 + (a * -3.0)))))) + -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]

↓

code[a_, b_] := N[(N[(N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 4.0], $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] + -1.0), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Applied egg-rr 2.1

   \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

3. Simplified 0.0

   \[\leadsto \left(\color{blue}{{\left(\mathsf{hypot}\left(b, a\right)\right)}^{4}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   Proof
   (pow.f64 (hypot.f64 b a) 4): 0 points increase in error, 0 points decrease in error
   (pow.f64 (hypot.f64 b a) (Rewrite<= metadata-eval (*.f64 2 2))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (hypot.f64 b a) 2) (pow.f64 (hypot.f64 b a) 2))): 50 points increase in error, 18 points decrease in error
   (Rewrite<= unpow2_binary64 (pow.f64 (pow.f64 (hypot.f64 b a) 2) 2)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (pow.f64 (pow.f64 (hypot.f64 b a) 2) 2)))): 91 points increase in error, 5 points decrease in error
   (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (pow.f64 (pow.f64 (hypot.f64 b a) 2) 2))) 1)): 4 points increase in error, 80 points decrease in error

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.4
Cost	7944

\[\begin{array}{l} t_0 := a \cdot a + b \cdot b\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;b \leq -0.00082:\\ \;\;\;\;\left(t_1 + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-55}:\\ \;\;\;\;\left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + 4 \cdot t_0\right) + -1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7936

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

Alternative 3
Error	1.5
Cost	7688

\[\begin{array}{l} t_0 := \left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) + -1\\ \mathbf{if}\;a \leq -4.7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7688

\[\begin{array}{l} t_0 := \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{if}\;b \leq -0.00082:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;\left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.6
Cost	7560

\[\begin{array}{l} t_0 := \left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) + -1\\ \mathbf{if}\;a \leq -0.0004:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	2.8
Cost	7432

\[\begin{array}{l} \mathbf{if}\;a \leq -260000000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + {a}^{3} \cdot \left(a + 4\right)\right) + -1\right) + -1\\ \end{array} \]

Alternative 7
Error	2.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -260000000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \end{array} \]

Alternative 8
Error	2.8
Cost	7048

\[\begin{array}{l} \mathbf{if}\;a \leq -280000000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 125:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \end{array} \]

Alternative 9
Error	3.0
Cost	6660

\[\begin{array}{l} \mathbf{if}\;a \leq -260000000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot a\right) + -1\\ \end{array} \]

Alternative 10
Error	3.0
Cost	968

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(4 + a \cdot a\right) + -1\\ \mathbf{if}\;a \leq -260000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	13.1
Cost	704

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

Alternative 12
Error	22.8
Cost	448

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

Alternative 13
Error	23.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022338 
(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))