Cubic critical, wide range

Average Accuracy: 17.5% → 99.9%

Time: 16.3s

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

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

↓

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

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(c / Float64(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 17.5%

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

2. Simplified 17.5%

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

   [Start] 17.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 17.5	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 17.5	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 17.5	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 17.5	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 17.5	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 17.5	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 17.5	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]

3. Applied egg-rr 18.1%

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

4. Simplified 18.1%

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

   [Start] 18.1	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right) \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a} \cdot -0.3333333333333333 \]
   associate-*r/ [=>] 18.1	\[ \frac{\color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right) \cdot 1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}{a} \cdot -0.3333333333333333 \]
   *-rgt-identity [=>] 18.1	\[ \frac{\frac{\color{blue}{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a} \cdot -0.3333333333333333 \]

5. Taylor expanded in b around 0 99.1%

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

6. Applied egg-rr 20.1%

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

7. Simplified 99.7%

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

   [Start] 20.1	\[ e^{\mathsf{log1p}\left(\frac{-1 \cdot \frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a}\right)} - 1 \]
   expm1-def [=>] 83.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 \cdot \frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a}\right)\right)} \]
   expm1-log1p [=>] 99.4	\[ \color{blue}{\frac{-1 \cdot \frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a}} \]
   /-rgt-identity [<=] 99.4	\[ \frac{\color{blue}{\frac{-1 \cdot \frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{1}}}{a} \]
   *-commutative [=>] 99.4	\[ \frac{\frac{\color{blue}{\frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot -1}}{1}}{a} \]
   associate-/l* [=>] 99.4	\[ \frac{\color{blue}{\frac{\frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{\frac{1}{-1}}}}{a} \]
   metadata-eval [=>] 99.4	\[ \frac{\frac{\frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{\color{blue}{-1}}}{a} \]
   associate-/r* [<=] 99.4	\[ \color{blue}{\frac{\frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{-1 \cdot a}} \]
   associate-/l* [=>] 99.4	\[ \frac{\color{blue}{\frac{c}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}}{-1 \cdot a} \]
   neg-mul-1 [<=] 99.4	\[ \frac{\frac{c}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}{\color{blue}{-a}} \]
   associate-/l/ [=>] 99.7	\[ \color{blue}{\frac{c}{\left(-a\right) \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]

8. Applied egg-rr 90.0%

   \[\leadsto \frac{c}{\color{blue}{\left(0 - e^{\mathsf{log1p}\left(\frac{a}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}\right) + 1}} \]

9. Simplified 99.9%

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

   [Start] 90.0	\[ \frac{c}{\left(0 - e^{\mathsf{log1p}\left(\frac{a}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}\right) + 1} \]
   associate-+l- [=>] 90.0	\[ \frac{c}{\color{blue}{0 - \left(e^{\mathsf{log1p}\left(\frac{a}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} - 1\right)}} \]
   expm1-def [=>] 96.0	\[ \frac{c}{0 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)\right)}} \]
   expm1-log1p [=>] 99.9	\[ \frac{c}{0 - \color{blue}{\frac{a}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   sub0-neg [=>] 99.9	\[ \frac{c}{\color{blue}{-\frac{a}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   distribute-lft-neg-in [=>] 99.9	\[ \frac{c}{\color{blue}{\left(-\frac{a}{a}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   *-inverses [=>] 99.9	\[ \frac{c}{\left(-\color{blue}{1}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   metadata-eval [=>] 99.9	\[ \frac{c}{\color{blue}{-1} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   neg-mul-1 [<=] 99.9	\[ \frac{c}{\color{blue}{-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]

10. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7552

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

Alternative 2
Accuracy	95.6%
Cost	832

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

Alternative 3
Accuracy	90.3%
Cost	320

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

Alternative 4
Accuracy	90.6%
Cost	320

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

Alternative 5
Accuracy	3.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023125 
(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)))