Cubic critical, wide range

Average Error: 52.5 → 0.4

Time: 15.9s

Precision: binary64

Cost: 14016

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

Math

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

↓

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

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

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

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

Derivation

1. Initial program 52.5

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

2. Applied egg-rr 52.3

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

3. Simplified 52.3

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

   [Start] 52.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a} \]
   fma-udef [=>] 52.2	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + 3 \cdot \left(a \cdot c\right)\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a} \]
   fma-udef [<=] 52.3	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a} \]
   associate-*r* [=>] 52.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(3 \cdot a\right) \cdot c}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a} \]
   *-commutative [=>] 52.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot 3\right)} \cdot c\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a} \]
   fma-def [<=] 52.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -3\right)}}}}{3 \cdot a} \]
   +-commutative [=>] 52.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right)}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right) + b \cdot b}}}}{3 \cdot a} \]
   fma-def [=>] 52.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right)}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}}{3 \cdot a} \]

4. Taylor expanded in b around 0 0.5

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

5. Simplified 0.4

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

   [Start] 0.5	\[ \frac{\frac{-3 \cdot \left(c \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{3 \cdot a} \]
   *-commutative [=>] 0.5	\[ \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot -3}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{3 \cdot a} \]
   associate-*l* [=>] 0.4	\[ \frac{\frac{\color{blue}{c \cdot \left(a \cdot -3\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}}{3 \cdot a} \]

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	3.1
Cost	7488

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

Alternative 2
Error	3.1
Cost	7424

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

Alternative 3
Error	3.2
Cost	1344

\[\frac{\frac{c \cdot \left(a \cdot -3\right)}{-1.5 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 3} \]

Alternative 4
Error	3.3
Cost	1088

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

Alternative 5
Error	6.4
Cost	320

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

Alternative 6
Error	6.4
Cost	320

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

Alternative 7
Error	6.2
Cost	320

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

Alternative 8
Error	52.0
Cost	256

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

Reproduce

herbie shell --seed 2023053 
(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)))