Cubic critical, medium range

Average Error: 43.5 → 0.4

Time: 16.0s

Precision: binary64

Cost: 13824

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

Math

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

↓

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

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 a) (/ (+ b (sqrt (fma a (* c -3.0) (* b b)))) (- a))))

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 / a) / ((b + sqrt(fma(a, (c * -3.0), (b * b)))) / -a);
}

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

Derivation

1. Initial program 43.5

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

2. Simplified 43.5

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

   [Start] 43.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 43.5	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 43.5	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 43.5	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 43.5	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 43.5	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 43.5	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 43.5	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]

3. Applied egg-rr 43.8

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

4. Simplified 43.1

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

   [Start] 43.8	\[ \frac{\left(\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l* [=>] 43.8	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-0.3333333333333333}}} \]
   associate-/r/ [=>] 43.8	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333} \]

5. Taylor expanded in b around 0 0.6

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

6. Applied egg-rr 39.9

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

7. Simplified 0.4

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

   [Start] 39.9	\[ e^{\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} - 1 \]
   expm1-def [=>] 10.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)\right)} \]
   expm1-log1p [=>] 0.4	\[ \color{blue}{\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]
   /-rgt-identity [<=] 0.4	\[ \frac{\color{blue}{\frac{\frac{c}{a} \cdot -1}{1}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l* [=>] 0.4	\[ \frac{\color{blue}{\frac{\frac{c}{a}}{\frac{1}{-1}}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   metadata-eval [=>] 0.4	\[ \frac{\frac{\frac{c}{a}}{\color{blue}{-1}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l/ [=>] 0.4	\[ \color{blue}{\frac{\frac{c}{a}}{\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -1}} \]
   *-commutative [<=] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{-1 \cdot \left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   neg-mul-1 [<=] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{-\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   neg-sub0 [=>] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{0 - \left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   +-commutative [=>] 0.4	\[ \frac{\frac{c}{a}}{0 - \color{blue}{\left(\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} + \frac{b}{a}\right)}} \]
   associate--r+ [=>] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{\left(0 - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) - \frac{b}{a}}} \]
   neg-sub0 [<=] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{\left(-\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)} - \frac{b}{a}} \]
   distribute-neg-frac [=>] 0.4	\[ \frac{\frac{c}{a}}{\color{blue}{\frac{-\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} - \frac{b}{a}} \]

8. Applied egg-rr 0.4

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

9. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	7680

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

Alternative 2
Error	6.1
Cost	832

\[\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \]

Alternative 3
Error	12.4
Cost	320

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

Reproduce

herbie shell --seed 2023059 
(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)))