Cubic critical, medium range

Average Error: 69.24% → 0.25%

Time: 15.9s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

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 -1.0) (+ b (sqrt (fma c (* a -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 / -1.0) / (b + sqrt(fma(c, (a * -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 / -1.0) / Float64(b + sqrt(fma(c, Float64(a * -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[(N[(c / -1.0), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.24

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

2. Applied egg-rr 68.51

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

3. Simplified 68.33

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

   [Start] 68.51	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}}{3 \cdot a} \]
   associate-/r* [<=] 68.51	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}\right)}}}{3 \cdot a} \]
   fma-def [<=] 68.34	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}\right)}}{3 \cdot a} \]
   +-commutative [=>] 68.34	\[ \frac{\frac{b \cdot b - \color{blue}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right)}}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}\right)}}{3 \cdot a} \]
   fma-def [=>] 68.34	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}\right)}}{3 \cdot a} \]
   distribute-rgt-neg-in [<=] 68.34	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\color{blue}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}}}{3 \cdot a} \]
   rem-square-sqrt [=>] 68.33	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{-\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}}}{3 \cdot a} \]
   fma-def [<=] 68.34	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)}}{3 \cdot a} \]
   +-commutative [=>] 68.34	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right) + b \cdot b}}\right)}}{3 \cdot a} \]
   fma-def [=>] 68.33	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}\right)}}{3 \cdot a} \]

4. Taylor expanded in b around 0 0.85

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

5. Applied egg-rr 0.85

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

6. Simplified 0.84

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

   [Start] 0.85	\[ \frac{\frac{0 + 3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{3 \cdot a} \]
   +-lft-identity [=>] 0.85	\[ \frac{\frac{\color{blue}{3 \cdot \left(c \cdot a\right)}}{-\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{3 \cdot a} \]
   associate-*r* [=>] 0.84	\[ \frac{\frac{\color{blue}{\left(3 \cdot c\right) \cdot a}}{-\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{3 \cdot a} \]
   *-commutative [=>] 0.84	\[ \frac{\frac{\color{blue}{\left(c \cdot 3\right)} \cdot a}{-\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{3 \cdot a} \]

7. Applied egg-rr 63.21

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

8. Simplified 0.25

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

   [Start] 63.21	\[ e^{\mathsf{log1p}\left(\frac{c \cdot \left(a \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \cdot \frac{0.3333333333333333}{a}\right)} - 1 \]
   expm1-def [=>] 16.66	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c \cdot \left(a \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \cdot \frac{0.3333333333333333}{a}\right)\right)} \]
   expm1-log1p [=>] 0.92	\[ \color{blue}{\frac{c \cdot \left(a \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \cdot \frac{0.3333333333333333}{a}} \]
   associate-*l/ [=>] 0.82	\[ \color{blue}{\frac{\left(c \cdot \left(a \cdot -3\right)\right) \cdot \frac{0.3333333333333333}{a}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}} \]
   /-rgt-identity [<=] 0.82	\[ \frac{\color{blue}{\frac{\left(c \cdot \left(a \cdot -3\right)\right) \cdot \frac{0.3333333333333333}{a}}{1}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   associate-/l* [=>] 0.8	\[ \frac{\color{blue}{\frac{c \cdot \left(a \cdot -3\right)}{\frac{1}{\frac{0.3333333333333333}{a}}}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   associate-/r/ [=>] 0.36	\[ \frac{\frac{c \cdot \left(a \cdot -3\right)}{\color{blue}{\frac{1}{0.3333333333333333} \cdot a}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   metadata-eval [=>] 0.36	\[ \frac{\frac{c \cdot \left(a \cdot -3\right)}{\color{blue}{3} \cdot a}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   metadata-eval [<=] 0.36	\[ \frac{\frac{c \cdot \left(a \cdot -3\right)}{\color{blue}{\left(--3\right)} \cdot a}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   distribute-lft-neg-in [<=] 0.36	\[ \frac{\frac{c \cdot \left(a \cdot -3\right)}{\color{blue}{--3 \cdot a}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   *-commutative [<=] 0.36	\[ \frac{\frac{c \cdot \left(a \cdot -3\right)}{-\color{blue}{a \cdot -3}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   mul-1-neg [<=] 0.36	\[ \frac{\frac{c \cdot \left(a \cdot -3\right)}{\color{blue}{-1 \cdot \left(a \cdot -3\right)}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   associate-/l/ [<=] 0.36	\[ \frac{\color{blue}{\frac{\frac{c \cdot \left(a \cdot -3\right)}{a \cdot -3}}{-1}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   associate-/l* [=>] 0.25	\[ \frac{\frac{\color{blue}{\frac{c}{\frac{a \cdot -3}{a \cdot -3}}}}{-1}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   associate-/l/ [=>] 0.25	\[ \frac{\color{blue}{\frac{c}{-1 \cdot \frac{a \cdot -3}{a \cdot -3}}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   *-inverses [=>] 0.25	\[ \frac{\frac{c}{-1 \cdot \color{blue}{1}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
   metadata-eval [=>] 0.25	\[ \frac{\frac{c}{\color{blue}{-1}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]

9. Final simplification 0.25

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

Alternatives

Alternative 1
Error	0.85%
Cost	7808

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

Alternative 2
Error	0.84%
Cost	7808

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

Alternative 3
Error	9%
Cost	7488

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

Alternative 4
Error	9.26%
Cost	7296

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

Alternative 5
Error	18.43%
Cost	320

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

Alternative 6
Error	18.22%
Cost	320

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

Reproduce

herbie shell --seed 2023102 
(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)))