Cubic critical, medium range

Average Error: 44.2 → 2.8

Time: 15.3s

Precision: binary64

Cost: 47360

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

Math

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

↓

\[\mathsf{fma}\left(-0.16666666666666666, \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot \frac{{a}^{3}}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(\frac{c \cdot c}{b \cdot b} \cdot \frac{-0.375}{b} + \frac{{c}^{3} \cdot -0.5625}{\frac{{b}^{5}}{a}}\right)\right)\right) \]

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
  -0.16666666666666666
  (* (* (/ (pow c 4.0) (pow b 6.0)) 6.328125) (/ (pow a 3.0) b))
  (fma
   -0.5
   (/ c b)
   (*
    a
    (+
     (* (/ (* c c) (* b b)) (/ -0.375 b))
     (/ (* (pow c 3.0) -0.5625) (/ (pow b 5.0) a)))))))

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(-0.16666666666666666, (((pow(c, 4.0) / pow(b, 6.0)) * 6.328125) * (pow(a, 3.0) / b)), fma(-0.5, (c / b), (a * ((((c * c) / (b * b)) * (-0.375 / b)) + ((pow(c, 3.0) * -0.5625) / (pow(b, 5.0) / a))))));
}

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(-0.16666666666666666, Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 6.328125) * Float64((a ^ 3.0) / b)), fma(-0.5, Float64(c / b), Float64(a * Float64(Float64(Float64(Float64(c * c) / Float64(b * b)) * Float64(-0.375 / b)) + Float64(Float64((c ^ 3.0) * -0.5625) / Float64((b ^ 5.0) / a))))))
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[(-0.16666666666666666 * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] * N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(-0.375 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[c, 3.0], $MachinePrecision] * -0.5625), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.2

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

2. Simplified 44.2

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

3. Applied egg-rr 44.2

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

4. Applied egg-rr 44.2

   \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{{\left(\frac{0.037037037037037035}{{a}^{3}}\right)}^{0.3333333333333333}} \]

5. Taylor expanded in a around 0 2.8

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

6. Simplified 2.8

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot \frac{{a}^{3}}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(\frac{c \cdot \left(c \cdot -0.375\right)}{{b}^{3}} + \frac{{c}^{3} \cdot -0.5625}{\frac{{b}^{5}}{a}}\right)\right)\right)} \]

7. Applied egg-rr 2.8

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot \frac{{a}^{3}}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(\color{blue}{\frac{c \cdot c}{b \cdot b} \cdot \frac{-0.375}{b}} + \frac{{c}^{3} \cdot -0.5625}{\frac{{b}^{5}}{a}}\right)\right)\right) \]

8. Final simplification 2.8

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot \frac{{a}^{3}}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(\frac{c \cdot c}{b \cdot b} \cdot \frac{-0.375}{b} + \frac{{c}^{3} \cdot -0.5625}{\frac{{b}^{5}}{a}}\right)\right)\right) \]

Alternatives

Alternative 1
Error	2.8
Cost	47296

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

Alternative 2
Error	3.7
Cost	26816

\[{\left(\mathsf{fma}\left(-2, \frac{b}{c}, \mathsf{fma}\left(1.5, \frac{a}{b}, \frac{1.125 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{3}}\right)\right)\right)}^{-1} \]

Alternative 3
Error	3.7
Cost	20672

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

Alternative 4
Error	5.7
Cost	7168

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

Alternative 5
Error	11.8
Cost	320

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

Alternative 6
Error	11.7
Cost	320

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

Reproduce

herbie shell --seed 2022284 
(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)))