Cubic critical, medium range

Average Error: 32.3% → 99.7%

Time: 15.7s

Precision: binary64

Cost: 13568.00

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

Math

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

↓

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

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 32.3

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

2. Simplified 32.3

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

   [Start] 32.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   remove-double-neg [<=] 32.3	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
   sub-neg [<=] 32.3	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   div-sub [=>] 31.8	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   neg-mul-1 [=>] 31.8	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   associate-*l/ [<=] 32.1	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   distribute-frac-neg [=>] 32.1	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   fma-neg [=>] 33.5	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
   /-rgt-identity [<=] 33.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   metadata-eval [<=] 33.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   associate-/l* [<=] 33.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   *-commutative [<=] 33.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   neg-mul-1 [<=] 33.5	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   fma-neg [<=] 32.1	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   neg-mul-1 [=>] 32.1	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

3. Applied egg-rr 33.2

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

4. Taylor expanded in b around 0 99.1

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

5. Simplified 99.3

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

   [Start] 99.1	\[ \frac{3 \cdot \left(c \cdot a\right)}{\frac{a \cdot -3}{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}} \]
   *-commutative [=>] 99.1	\[ \frac{\color{blue}{\left(c \cdot a\right) \cdot 3}}{\frac{a \cdot -3}{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}} \]
   associate-*l* [=>] 99.3	\[ \frac{\color{blue}{c \cdot \left(a \cdot 3\right)}}{\frac{a \cdot -3}{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}} \]
   *-commutative [=>] 99.3	\[ \frac{c \cdot \color{blue}{\left(3 \cdot a\right)}}{\frac{a \cdot -3}{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}} \]

6. Applied egg-rr 99.4

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

7. Applied egg-rr 52.2

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

8. Simplified 99.7

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

   [Start] 52.2	\[ -\left(e^{\mathsf{log1p}\left(\left(\frac{c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(3 \cdot a\right)\right) \cdot \frac{0.3333333333333333}{a}\right)} - 1\right) \]
   expm1-def [=>] 98.9	\[ -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(3 \cdot a\right)\right) \cdot \frac{0.3333333333333333}{a}\right)\right)} \]
   expm1-log1p [=>] 99.2	\[ -\color{blue}{\left(\frac{c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(3 \cdot a\right)\right) \cdot \frac{0.3333333333333333}{a}} \]
   associate-*l* [=>] 99.1	\[ -\color{blue}{\frac{c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \left(\left(3 \cdot a\right) \cdot \frac{0.3333333333333333}{a}\right)} \]
   *-commutative [<=] 99.1	\[ -\frac{c}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \color{blue}{\left(\frac{0.3333333333333333}{a} \cdot \left(3 \cdot a\right)\right)} \]
   associate-*l/ [=>] 99.1	\[ -\color{blue}{\frac{c \cdot \left(\frac{0.3333333333333333}{a} \cdot \left(3 \cdot a\right)\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   associate-*l/ [=>] 99.3	\[ -\frac{c \cdot \color{blue}{\frac{0.3333333333333333 \cdot \left(3 \cdot a\right)}{a}}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   associate-*r/ [=>] 99.3	\[ -\frac{\color{blue}{\frac{c \cdot \left(0.3333333333333333 \cdot \left(3 \cdot a\right)\right)}{a}}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   associate-*l* [<=] 99.2	\[ -\frac{\frac{\color{blue}{\left(c \cdot 0.3333333333333333\right) \cdot \left(3 \cdot a\right)}}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   associate-/l/ [=>] 99.1	\[ -\color{blue}{\frac{\left(c \cdot 0.3333333333333333\right) \cdot \left(3 \cdot a\right)}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}} \]
   times-frac [=>] 99.1	\[ -\color{blue}{\frac{c \cdot 0.3333333333333333}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \frac{3 \cdot a}{a}} \]
   associate-*l/ [=>] 99.2	\[ -\color{blue}{\frac{\left(c \cdot 0.3333333333333333\right) \cdot \frac{3 \cdot a}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   associate-/l* [=>] 99.4	\[ -\frac{\left(c \cdot 0.3333333333333333\right) \cdot \color{blue}{\frac{3}{\frac{a}{a}}}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   *-inverses [=>] 99.4	\[ -\frac{\left(c \cdot 0.3333333333333333\right) \cdot \frac{3}{\color{blue}{1}}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   metadata-eval [=>] 99.4	\[ -\frac{\left(c \cdot 0.3333333333333333\right) \cdot \color{blue}{3}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   associate-*l* [=>] 99.7	\[ -\frac{\color{blue}{c \cdot \left(0.3333333333333333 \cdot 3\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   metadata-eval [=>] 99.7	\[ -\frac{c \cdot \color{blue}{1}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   *-rgt-identity [=>] 99.7	\[ -\frac{\color{blue}{c}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   remove-double-neg [<=] 99.7	\[ -\frac{\color{blue}{-\left(-c\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]

9. Final simplification 99.7

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

Alternatives

Alternative 1
Error	99.4%
Cost	7744.00

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

Alternative 2
Error	90.3%
Cost	1088.00

\[\frac{\frac{-0.3333333333333333}{a}}{\frac{\frac{0.6666666666666666}{\frac{c}{b}}}{a} + 0.5 \cdot \frac{-1}{b}} \]

Alternative 3
Error	90.2%
Cost	960.00

\[\frac{0.3333333333333333}{a \cdot \left(\frac{0.5}{b} + \frac{b}{c \cdot a} \cdot -0.6666666666666666\right)} \]

Alternative 4
Error	90.2%
Cost	960.00

\[\frac{\frac{-0.3333333333333333}{a}}{\frac{b \cdot 0.6666666666666666}{c \cdot a} + \frac{-0.5}{b}} \]

Alternative 5
Error	80.6%
Cost	320.00

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

Reproduce

herbie shell --seed 2023098 
(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)))