Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.0

Time: 7.9s

Precision: binary64

Math

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

↓

\[\mathsf{fma}\left(4 \cdot a, a - a \cdot a, {a}^{4}\right) + \left(\mathsf{fma}\left(12 + a \cdot \left(4 + a \cdot 2\right), b \cdot b, {b}^{4}\right) + -1\right) \]

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))

↓

(FPCore (a b)
 :precision binary64
 (+
  (fma (* 4.0 a) (- a (* a a)) (pow a 4.0))
  (+ (fma (+ 12.0 (* a (+ 4.0 (* a 2.0)))) (* b b) (pow b 4.0)) -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) * (3.0 + a))))) - 1.0;
}

↓

double code(double a, double b) {
	return fma((4.0 * a), (a - (a * a)), pow(a, 4.0)) + (fma((12.0 + (a * (4.0 + (a * 2.0)))), (b * b), pow(b, 4.0)) + -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(3.0 + a))))) - 1.0)
end

↓

function code(a, b)
	return Float64(fma(Float64(4.0 * a), Float64(a - Float64(a * a)), (a ^ 4.0)) + Float64(fma(Float64(12.0 + Float64(a * Float64(4.0 + Float64(a * 2.0)))), Float64(b * b), (b ^ 4.0)) + -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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[a_, b_] := N[(N[(N[(4.0 * a), $MachinePrecision] * N[(a - N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.0 + N[(a * N[(4.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.0

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

3. Applied egg-rr 0.2

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

4. Taylor expanded in b around 0 0.0

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

5. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot a, a - a \cdot a, {a}^{4}\right) + \left(\mathsf{fma}\left(12 + a \cdot \left(a \cdot 2 + 4\right), b \cdot b, {b}^{4}\right) + -1\right)} \]

6. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(4 \cdot a, a - a \cdot a, {a}^{4}\right) + \left(\mathsf{fma}\left(12 + a \cdot \left(4 + a \cdot 2\right), b \cdot b, {b}^{4}\right) + -1\right) \]

Reproduce

herbie shell --seed 2022210 
(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))