Cubic critical, medium range

Average Error: 44.1 → 2.8

Time: 10.7s

Precision: binary64

Cost: 41088

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

↓

\[\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right), \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} \cdot 6.328125, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{c \cdot -0.375}{\frac{b \cdot b}{a} \cdot \frac{b}{c}}\right)\right)\right) \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5625
  (* (* a a) (* (* c c) (* c (pow b -5.0))))
  (fma
   -0.16666666666666666
   (* (/ (pow (* a c) 4.0) (* a (pow b 7.0))) 6.328125)
   (fma -0.5 (/ c b) (/ (* c -0.375) (* (/ (* b b) a) (/ b c)))))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	return fma(-0.5625, ((a * a) * ((c * c) * (c * pow(b, -5.0)))), fma(-0.16666666666666666, ((pow((a * c), 4.0) / (a * pow(b, 7.0))) * 6.328125), fma(-0.5, (c / b), ((c * -0.375) / (((b * b) / a) * (b / c))))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	return fma(-0.5625, Float64(Float64(a * a) * Float64(Float64(c * c) * Float64(c * (b ^ -5.0)))), fma(-0.16666666666666666, Float64(Float64((Float64(a * c) ^ 4.0) / Float64(a * (b ^ 7.0))) * 6.328125), fma(-0.5, Float64(c / b), Float64(Float64(c * -0.375) / Float64(Float64(Float64(b * b) / a) * Float64(b / c))))))
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(c * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(c * -0.375), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.1

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Simplified 44.0

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

3. Taylor expanded in b around inf 2.8

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

4. Simplified 2.8

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 6.328125}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{c \cdot -0.375}{\frac{{b}^{3}}{c \cdot a}}\right)\right)\right)} \]

5. Applied egg-rr 2.8

   \[\leadsto \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}} \cdot 6.328125}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{c \cdot -0.375}{\frac{{b}^{3}}{c \cdot a}}\right)\right)\right) \]

6. Applied egg-rr 2.8

   \[\leadsto \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}} \cdot 6.328125, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{c \cdot -0.375}{\color{blue}{\frac{b \cdot b}{a} \cdot \frac{b}{c}}}\right)\right)\right) \]

7. Applied egg-rr 2.8

   \[\leadsto \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right)}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}} \cdot 6.328125, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{c \cdot -0.375}{\frac{b \cdot b}{a} \cdot \frac{b}{c}}\right)\right)\right) \]

8. Final simplification 2.8

   \[\leadsto \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right), \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} \cdot 6.328125, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{c \cdot -0.375}{\frac{b \cdot b}{a} \cdot \frac{b}{c}}\right)\right)\right) \]

Alternatives

Alternative 1
Error	3.8
Cost	27136

\[\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(-0.375 \cdot \left(c \cdot \frac{c}{{b}^{3}}\right) + \frac{-0.5625 \cdot {c}^{3}}{\frac{{b}^{5}}{a}}\right)\right) \]

Alternative 2
Error	5.7
Cost	7168

\[{\left(\frac{b}{c} \cdot -2 + 1.5 \cdot \frac{a}{b}\right)}^{-1} \]

Alternative 3
Error	11.8
Cost	320

\[-0.5 \cdot \frac{c}{b} \]

Reproduce

herbie shell --seed 2022228 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))