Cubic critical, wide range

Average Error: 52.7 → 1.3

Time: 12.3s

Precision: binary64

Cost: 47296

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

↓

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

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
   -0.5
   (/ c b)
   (fma
    -0.16666666666666666
    (/ (* (pow (* a c) 4.0) 6.328125) (* a (pow b 7.0)))
    (* -0.375 (* a (/ c (/ (pow b 3.0) 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(-0.5, (c / b), fma(-0.16666666666666666, ((pow((a * c), 4.0) * 6.328125) / (a * pow(b, 7.0))), (-0.375 * (a * (c / (pow(b, 3.0) / 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(-0.5, Float64(c / b), fma(-0.16666666666666666, Float64(Float64((Float64(a * c) ^ 4.0) * 6.328125) / Float64(a * (b ^ 7.0))), Float64(-0.375 * Float64(a * Float64(c / Float64((b ^ 3.0) / 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[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7

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

2. Taylor expanded in b around inf 1.3

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

3. Simplified 1.3

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

4. Applied egg-rr 1.3

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

5. Applied egg-rr 1.3

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

6. Final simplification 1.3

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

Alternatives

Alternative 1
Error	1.8
Cost	33536

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

Alternative 2
Error	2.9
Cost	7424

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

Alternative 3
Error	2.9
Cost	7232

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

Alternative 4
Error	3.3
Cost	1344

\[\frac{\frac{a}{\frac{b}{c}} \cdot \left(\frac{c \cdot -1.125}{\frac{b \cdot b}{a}} + -1.5\right)}{a \cdot 3} \]

Alternative 5
Error	6.3
Cost	320

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

Alternative 6
Error	6.0
Cost	320

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

Reproduce

herbie shell --seed 2022284 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))