Cubic critical, medium range

Average Accuracy: 30.8% → 95.8%

Time: 24.6s

Precision: binary64

Cost: 53504

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

Math

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

↓

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

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
  -0.5
  (/ c b)
  (fma
   (/ -0.16666666666666666 a)
   (/ (* (pow (* c a) 4.0) 6.328125) (pow b 7.0))
   (fma
    -0.375
    (* (/ a (pow b 3.0)) (* c c))
    (* -0.5625 (* (/ a (pow b 5.0)) (* a (pow c 3.0))))))))

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(-0.5, (c / b), fma((-0.16666666666666666 / a), ((pow((c * a), 4.0) * 6.328125) / pow(b, 7.0)), fma(-0.375, ((a / pow(b, 3.0)) * (c * c)), (-0.5625 * ((a / pow(b, 5.0)) * (a * pow(c, 3.0)))))));
}

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(-0.5, Float64(c / b), fma(Float64(-0.16666666666666666 / a), Float64(Float64((Float64(c * a) ^ 4.0) * 6.328125) / (b ^ 7.0)), fma(-0.375, Float64(Float64(a / (b ^ 3.0)) * Float64(c * c)), Float64(-0.5625 * Float64(Float64(a / (b ^ 5.0)) * Float64(a * (c ^ 3.0)))))))
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[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.16666666666666666 / a), $MachinePrecision] * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

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

2. Simplified 30.9%

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

   [Start] 30.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   /-rgt-identity [<=] 30.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
   metadata-eval [<=] 30.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
   associate-/l* [<=] 30.8	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{3 \cdot a}} \]
   associate-*r/ [<=] 30.8	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{3 \cdot a}} \]
   *-commutative [=>] 30.8	\[ \color{blue}{\frac{--1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]
   associate-*l/ [=>] 30.8	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{3 \cdot a}} \]
   associate-*r/ [<=] 30.8	\[ \color{blue}{\left(--1\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [=>] 30.8	\[ \color{blue}{1} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   metadata-eval [<=] 30.8	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 30.8	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 30.8	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 30.8	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 30.8	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   metadata-eval [=>] 30.8	\[ \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]
   neg-mul-1 [=>] 30.8	\[ -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

3. Taylor expanded in b around inf 95.3%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.6875 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)\right)} \]

4. Simplified 95.5%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{0.5 \cdot \left({\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)\right)}{a \cdot {b}^{7}}\right)\right)\right)} \]
   Proof

   [Start] 95.3	\[ -0.3333333333333333 \cdot \left(1.6875 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)\right) \]
   fma-def [=>] 95.3	\[ -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)} \]
   unpow2 [=>] 95.3	\[ -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot \color{blue}{\left(a \cdot a\right)}}{{b}^{5}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]
   fma-def [=>] 95.3	\[ -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)}\right) \]
   *-commutative [=>] 95.3	\[ -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(1.125, \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]
   unpow2 [=>] 95.3	\[ -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(1.125, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]
   fma-def [=>] 95.5	\[ -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)}\right)\right) \]

5. Taylor expanded in c around 0 95.7%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

6. Simplified 95.8%

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

   [Start] 95.7	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
   +-commutative [=>] 95.7	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)} \]
   associate-+r+ [=>] 95.7	\[ \color{blue}{\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}} \]
   associate-+r+ [=>] 95.7	\[ \color{blue}{\left(\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.5 \cdot \frac{c}{b}\right) + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} \]
   associate-/r* [=>] 95.7	\[ \left(\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.5 \cdot \frac{c}{b}\right) + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + -0.16666666666666666 \cdot \color{blue}{\frac{\frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a}}{{b}^{7}}} \]
   associate-*r/ [=>] 95.7	\[ \left(\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.5 \cdot \frac{c}{b}\right) + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + \color{blue}{\frac{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a}}{{b}^{7}}} \]

7. Final simplification 95.8%

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

Alternatives

Alternative 1
Accuracy	95.7%
Cost	47296

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

Alternative 2
Accuracy	94.0%
Cost	33536

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

Alternative 3
Accuracy	94.3%
Cost	33536

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

Alternative 4
Accuracy	91.1%
Cost	13696

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

Alternative 5
Accuracy	90.8%
Cost	1600

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

Alternative 6
Accuracy	81.5%
Cost	320

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

Alternative 7
Accuracy	81.5%
Cost	320

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

Alternative 8
Accuracy	81.7%
Cost	320

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

Reproduce

herbie shell --seed 2023146 
(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)))