Cubic critical, wide range

Average Error: 81.12% → 0.13%

Time: 19.8s

Precision: binary64

Cost: 13568

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (/ (- c) (+ b (sqrt (fma a (* c -3.0) (* b 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 -c / (b + sqrt(fma(a, (c * -3.0), (b * 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 Float64(Float64(-c) / Float64(b + sqrt(fma(a, Float64(c * -3.0), Float64(b * 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[((-c) / N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.12

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

2. Simplified 81.12

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

   [Start] 81.12	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 81.12	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 81.12	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 81.12	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 81.12	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 81.12	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 81.12	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 81.12	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]

3. Applied egg-rr 81.39

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

4. Simplified 80.34

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

   [Start] 81.39	\[ \frac{\left(\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l* [=>] 81.39	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-0.3333333333333333}}} \]
   associate-/r/ [=>] 81.39	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333} \]

5. Taylor expanded in b around 0 0.88

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

6. Applied egg-rr 79.55

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} - 1} \]

7. Simplified 0.58

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

   [Start] 79.55	\[ e^{\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} - 1 \]
   expm1-def [=>] 16.91	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)\right)} \]
   expm1-log1p [=>] 0.58	\[ \color{blue}{\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]
   /-rgt-identity [<=] 0.58	\[ \frac{\color{blue}{\frac{\frac{c}{a} \cdot -1}{1}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l* [=>] 0.58	\[ \frac{\color{blue}{\frac{\frac{c}{a}}{\frac{1}{-1}}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   metadata-eval [=>] 0.58	\[ \frac{\frac{\frac{c}{a}}{\color{blue}{-1}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l/ [=>] 0.58	\[ \color{blue}{\frac{\frac{c}{a}}{\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -1}} \]
   *-commutative [<=] 0.58	\[ \frac{\frac{c}{a}}{\color{blue}{-1 \cdot \left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   neg-mul-1 [<=] 0.58	\[ \frac{\frac{c}{a}}{\color{blue}{-\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   *-lft-identity [<=] 0.58	\[ \frac{\frac{c}{a}}{-\color{blue}{1 \cdot \left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   distribute-lft-in [=>] 0.58	\[ \frac{\frac{c}{a}}{-\color{blue}{\left(1 \cdot \frac{b}{a} + 1 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   associate-*r/ [=>] 0.58	\[ \frac{\frac{c}{a}}{-\left(\color{blue}{\frac{1 \cdot b}{a}} + 1 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)} \]
   associate-*l/ [<=] 0.63	\[ \frac{\frac{c}{a}}{-\left(\color{blue}{\frac{1}{a} \cdot b} + 1 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)} \]
   associate-*r/ [=>] 0.63	\[ \frac{\frac{c}{a}}{-\left(\frac{1}{a} \cdot b + \color{blue}{\frac{1 \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} \]
   associate-*l/ [<=] 0.69	\[ \frac{\frac{c}{a}}{-\left(\frac{1}{a} \cdot b + \color{blue}{\frac{1}{a} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} \]
   distribute-lft-in [<=] 0.71	\[ \frac{\frac{c}{a}}{-\color{blue}{\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   associate-*l/ [=>] 0.58	\[ \frac{\frac{c}{a}}{-\color{blue}{\frac{1 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{a}}} \]

8. Applied egg-rr 0.6

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{c}{a} \cdot \frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} \]

9. Simplified 0.13

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

   [Start] 0.6	\[ -1 \cdot \left(\frac{c}{a} \cdot \frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right) \]
   mul-1-neg [=>] 0.6	\[ \color{blue}{-\frac{c}{a} \cdot \frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   associate-*l/ [=>] 0.6	\[ -\color{blue}{\frac{c \cdot \frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a}} \]
   *-rgt-identity [<=] 0.6	\[ -\frac{c \cdot \color{blue}{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot 1\right)}}{a} \]
   associate-*r/ [<=] 0.61	\[ -\color{blue}{c \cdot \frac{\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot 1}{a}} \]
   associate-*r/ [<=] 0.71	\[ -c \cdot \color{blue}{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \frac{1}{a}\right)} \]
   *-commutative [<=] 0.71	\[ -c \cdot \color{blue}{\left(\frac{1}{a} \cdot \frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} \]
   associate-*r/ [=>] 0.56	\[ -c \cdot \color{blue}{\frac{\frac{1}{a} \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   associate-/l* [=>] 0.71	\[ -c \cdot \color{blue}{\frac{\frac{1}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]
   /-rgt-identity [<=] 0.71	\[ -\color{blue}{\frac{c \cdot \frac{\frac{1}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}{1}} \]
   associate-/l* [=>] 0.64	\[ -\color{blue}{\frac{c}{\frac{1}{\frac{\frac{1}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}}} \]
   associate-/l* [<=] 0.44	\[ -\frac{c}{\frac{1}{\color{blue}{\frac{\frac{1}{a} \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}} \]

10. Final simplification 0.13

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

Alternatives

Alternative 1
Error	0.58%
Cost	7552

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

Alternative 2
Error	5.27%
Cost	960

\[\frac{\frac{c}{a}}{\frac{c}{b} \cdot 1.5 + \frac{b}{a} \cdot -2} \]

Alternative 3
Error	10.29%
Cost	320

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

Reproduce

herbie shell --seed 2023135 
(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)))