Cubic critical, wide range

Average Error: 53.0 → 0.4

Time: 14.2s

Precision: binary64

Cost: 13824

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

↓

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

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(Float64(-b) - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))) / 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 53.0

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

2. Simplified 53.0

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

   [Start] 53.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 53.0	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 53.0	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 53.0	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 53.0	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 53.0	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 53.0	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 53.0	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]

3. Applied egg-rr 53.2

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

4. Simplified 52.5

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

   [Start] 53.2	\[ \frac{\left(\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l* [=>] 53.2	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-0.3333333333333333}}} \]
   associate-/r/ [=>] 53.2	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333} \]

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 51.4

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

7. Simplified 0.4

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

   [Start] 51.4	\[ e^{\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} - 1 \]
   expm1-def [=>] 10.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)\right)} \]
   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	\[ \frac{\frac{c}{a}}{\color{blue}{-\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   *-lft-identity [<=] 0.4	\[ \frac{\frac{c}{a}}{-\color{blue}{1 \cdot \left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   distribute-rgt-in [=>] 0.4	\[ \frac{\frac{c}{a}}{-\color{blue}{\left(\frac{b}{a} \cdot 1 + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot 1\right)}} \]
   *-commutative [<=] 0.4	\[ \frac{\frac{c}{a}}{-\left(\color{blue}{1 \cdot \frac{b}{a}} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot 1\right)} \]
   associate-*r/ [=>] 0.4	\[ \frac{\frac{c}{a}}{-\left(\color{blue}{\frac{1 \cdot b}{a}} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot 1\right)} \]
   associate-*l/ [<=] 0.4	\[ \frac{\frac{c}{a}}{-\left(\color{blue}{\frac{1}{a} \cdot b} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot 1\right)} \]
   associate-*l/ [=>] 0.4	\[ \frac{\frac{c}{a}}{-\left(\frac{1}{a} \cdot b + \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot 1}{a}}\right)} \]
   associate-*r/ [<=] 0.4	\[ \frac{\frac{c}{a}}{-\left(\frac{1}{a} \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \frac{1}{a}}\right)} \]
   *-commutative [<=] 0.4	\[ \frac{\frac{c}{a}}{-\left(\frac{1}{a} \cdot b + \color{blue}{\frac{1}{a} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} \]
   distribute-lft-in [<=] 0.5	\[ \frac{\frac{c}{a}}{-\color{blue}{\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]

8. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	13824

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

Alternative 2
Error	2.8
Cost	7424

\[\left(a \cdot -0.375\right) \cdot \left(c \cdot \left(c \cdot {b}^{-3}\right)\right) + \frac{\frac{c}{b}}{-2} \]

Alternative 3
Error	3.1
Cost	7296

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

Alternative 4
Error	6.0
Cost	320

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

Alternative 5
Error	6.0
Cost	320

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

Alternative 6
Error	5.8
Cost	320

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

Alternative 7
Error	61.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023045 
(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)))