Cubic critical, wide range

Average Error: 52.2 → 1.6

Time: 7.2s

Precision: binary64

Cost: 53504

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

Math

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

↓

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

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 a) 4.0) 6.328125) (* a (pow b 7.0)))
  (fma
   -0.5625
   (* (/ (* a a) (pow b 5.0)) (pow c 3.0))
   (fma -0.375 (* a (/ (* c c) (pow b 3.0))) (/ (* c -0.5) b)))))

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 * a), 4.0) * 6.328125) / (a * pow(b, 7.0))), fma(-0.5625, (((a * a) / pow(b, 5.0)) * pow(c, 3.0)), fma(-0.375, (a * ((c * c) / pow(b, 3.0))), ((c * -0.5) / b))));
}

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 * a) ^ 4.0) * 6.328125) / Float64(a * (b ^ 7.0))), fma(-0.5625, Float64(Float64(Float64(a * a) / (b ^ 5.0)) * (c ^ 3.0)), fma(-0.375, Float64(a * Float64(Float64(c * c) / (b ^ 3.0))), Float64(Float64(c * -0.5) / b))))
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[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2

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

2. Simplified 52.2

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

3. Taylor expanded in b around inf 1.6

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

4. Simplified 1.6

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

5. Applied egg-rr 1.6

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

6. Final simplification 1.6

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

Alternatives

Alternative 1
Error	1.8
Cost	53504

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

Alternative 2
Error	2.1
Cost	27136

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

Alternative 3
Error	2.6
Cost	21632

\[\frac{-1.125 \cdot \left(\frac{c \cdot c}{b \cdot b} \cdot \frac{a \cdot a}{b}\right) + \left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)}{a \cdot 3} \]

Alternative 4
Error	3.2
Cost	13696

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

Alternative 5
Error	3.4
Cost	7616

\[\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\left(b \cdot b\right) \cdot \left(b \cdot \frac{1}{c}\right)}, \frac{-0.5}{\frac{b}{c}}\right) \]

Alternative 6
Error	6.7
Cost	320

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

Alternative 7
Error	6.5
Cost	320

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

Reproduce

herbie shell --seed 2022212 
(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)))