Cubic critical, wide range

Average Accuracy: 18.1% → 99.9%

Time: 27.4s

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 18.1%

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

2. Simplified 18.1%

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

   [Start] 18.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 18.1	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 18.1	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 18.1	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 18.1	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 18.1	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 18.1	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 18.1	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]

3. Applied egg-rr 18.6%

   \[\leadsto \frac{\color{blue}{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}{a} \cdot -0.3333333333333333 \]
   Proof

   [Start] 18.1	\[ \frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot -0.3333333333333333 \]
   flip-- [=>] 18.1	\[ \frac{\color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}{a} \cdot -0.3333333333333333 \]
   clear-num [=>] 18.1	\[ \frac{\color{blue}{\frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}}{a} \cdot -0.3333333333333333 \]
   associate-/r/ [=>] 18.1	\[ \frac{\color{blue}{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}}{a} \cdot -0.3333333333333333 \]
   add-sqr-sqrt [<=] 18.6	\[ \frac{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(b \cdot b - \color{blue}{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{a} \cdot -0.3333333333333333 \]

4. Taylor expanded in b around 0 99.0%

   \[\leadsto \frac{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \color{blue}{\left(3 \cdot \left(c \cdot a\right)\right)}}{a} \cdot -0.3333333333333333 \]

5. Applied egg-rr 19.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{\frac{3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a} \cdot \left(a \cdot c\right)\right) \cdot -0.3333333333333333\right)} - 1} \]
   Proof

   [Start] 99.0	\[ \frac{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(3 \cdot \left(c \cdot a\right)\right)}{a} \cdot -0.3333333333333333 \]
   expm1-log1p-u [=>] 82.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(3 \cdot \left(c \cdot a\right)\right)}{a} \cdot -0.3333333333333333\right)\right)} \]
   expm1-udef [=>] 19.7	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(3 \cdot \left(c \cdot a\right)\right)}{a} \cdot -0.3333333333333333\right)} - 1} \]

6. Simplified 99.4%

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

   [Start] 19.7	\[ e^{\mathsf{log1p}\left(\left(\frac{\frac{3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a} \cdot \left(a \cdot c\right)\right) \cdot -0.3333333333333333\right)} - 1 \]
   expm1-def [=>] 82.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a} \cdot \left(a \cdot c\right)\right) \cdot -0.3333333333333333\right)\right)} \]
   expm1-log1p [=>] 99.1	\[ \color{blue}{\left(\frac{\frac{3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a} \cdot \left(a \cdot c\right)\right) \cdot -0.3333333333333333} \]
   associate-*l/ [=>] 99.1	\[ \color{blue}{\frac{\frac{3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(a \cdot c\right)}{a}} \cdot -0.3333333333333333 \]
   *-commutative [<=] 99.1	\[ \frac{\color{blue}{\left(a \cdot c\right) \cdot \frac{3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}{a} \cdot -0.3333333333333333 \]
   associate-*l/ [=>] 99.1	\[ \color{blue}{\frac{\left(\left(a \cdot c\right) \cdot \frac{3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right) \cdot -0.3333333333333333}{a}} \]

7. Applied egg-rr 99.3%

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

   [Start] 99.4	\[ \frac{\frac{c \cdot \left(-a\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a} \]
   div-inv [=>] 99.3	\[ \color{blue}{\frac{c \cdot \left(-a\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \frac{1}{a}} \]
   clear-num [=>] 99.2	\[ \color{blue}{\frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{c \cdot \left(-a\right)}}} \cdot \frac{1}{a} \]
   associate-*l/ [=>] 99.3	\[ \color{blue}{\frac{1 \cdot \frac{1}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{c \cdot \left(-a\right)}}} \]
   *-un-lft-identity [<=] 99.3	\[ \frac{\color{blue}{\frac{1}{a}}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{c \cdot \left(-a\right)}} \]
   frac-2neg [=>] 99.3	\[ \frac{\frac{1}{a}}{\color{blue}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{-c \cdot \left(-a\right)}}} \]
   distribute-rgt-neg-in [=>] 99.3	\[ \frac{\frac{1}{a}}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\color{blue}{c \cdot \left(-\left(-a\right)\right)}}} \]
   add-sqr-sqrt [=>] 0.0	\[ \frac{\frac{1}{a}}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{c \cdot \left(-\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)}} \]
   sqrt-unprod [=>] 1.7	\[ \frac{\frac{1}{a}}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{c \cdot \left(-\color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)}} \]
   sqr-neg [=>] 1.7	\[ \frac{\frac{1}{a}}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{c \cdot \left(-\sqrt{\color{blue}{a \cdot a}}\right)}} \]
   sqrt-unprod [<=] 1.7	\[ \frac{\frac{1}{a}}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{c \cdot \left(-\color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)}} \]
   add-sqr-sqrt [<=] 1.7	\[ \frac{\frac{1}{a}}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{c \cdot \left(-\color{blue}{a}\right)}} \]
   associate-/r/ [=>] 1.7	\[ \color{blue}{\frac{\frac{1}{a}}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot \left(c \cdot \left(-a\right)\right)} \]
   add-sqr-sqrt [=>] 0.0	\[ \frac{\frac{1}{a}}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot \left(c \cdot \color{blue}{\left(\sqrt{-a} \cdot \sqrt{-a}\right)}\right) \]
   sqrt-unprod [=>] 99.3	\[ \frac{\frac{1}{a}}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot \left(c \cdot \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right) \]

8. Simplified 99.9%

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

   [Start] 99.3	\[ \frac{\frac{1}{a}}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot \left(c \cdot a\right) \]
   associate-*l/ [=>] 99.5	\[ \color{blue}{\frac{\frac{1}{a} \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   associate-*l/ [=>] 99.7	\[ \frac{\color{blue}{\frac{1 \cdot \left(c \cdot a\right)}{a}}}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   associate-*r/ [<=] 99.7	\[ \frac{\color{blue}{1 \cdot \frac{c \cdot a}{a}}}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   associate-/l* [=>] 99.9	\[ \frac{1 \cdot \color{blue}{\frac{c}{\frac{a}{a}}}}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   *-inverses [=>] 99.9	\[ \frac{1 \cdot \frac{c}{\color{blue}{1}}}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   /-rgt-identity [=>] 99.9	\[ \frac{1 \cdot \color{blue}{c}}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   associate-*l/ [<=] 99.5	\[ \color{blue}{\frac{1}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \cdot c} \]
   neg-mul-1 [=>] 99.5	\[ \frac{1}{\color{blue}{-1 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \cdot c \]
   associate-/r* [=>] 99.5	\[ \color{blue}{\frac{\frac{1}{-1}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \cdot c \]
   metadata-eval [=>] 99.5	\[ \frac{\color{blue}{-1}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot c \]
   associate-*l/ [=>] 99.9	\[ \color{blue}{\frac{-1 \cdot c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   neg-mul-1 [<=] 99.9	\[ \frac{\color{blue}{-c}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]

9. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	7552

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

Alternative 2
Accuracy	95.2%
Cost	7488

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

Alternative 3
Accuracy	95.0%
Cost	1152

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

Alternative 4
Accuracy	89.9%
Cost	320

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

Alternative 5
Accuracy	90.2%
Cost	320

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

Reproduce

herbie shell --seed 2023131 
(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)))