Cubic critical, wide range

Average Error: 52.9 → 0.5

Time: 15.2s

Precision: binary64

Cost: 13888

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

Math

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

↓

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (* (* 3.0 c) (/ 1.0 (* -3.0 (+ 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 (3.0 * c) * (1.0 / (-3.0 * (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(3.0 * c) * Float64(1.0 / Float64(-3.0 * 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[(N[(3.0 * c), $MachinePrecision] * N[(1.0 / N[(-3.0 * N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9

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

2. Simplified 52.9

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

   [Start] 52.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 52.9	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 52.9	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 52.9	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 52.9	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 52.9	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 52.9	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 52.9	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]

3. Applied egg-rr 52.6

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

4. Simplified 52.6

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

   [Start] 52.6	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right) \cdot -0.3333333333333333}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   times-frac [=>] 52.6	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}{a} \cdot \frac{-0.3333333333333333}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]

5. Taylor expanded in b around 0 0.5

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

6. Applied egg-rr 0.5

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

7. Simplified 0.5

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

   [Start] 0.5	\[ \left(3 \cdot c\right) \cdot {\left(-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}^{-1} \]
   unpow-1 [=>] 0.5	\[ \left(3 \cdot c\right) \cdot \color{blue}{\frac{1}{-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]

8. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	13888

\[\left(3 \cdot c\right) \cdot \left(\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot -0.3333333333333333\right) \]

Alternative 2
Error	0.5
Cost	7488

\[\left(3 \cdot c\right) \cdot \frac{-0.3333333333333333}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}} \]

Alternative 3
Error	2.8
Cost	7424

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

Alternative 4
Error	3.0
Cost	1216

\[\left(3 \cdot c\right) \cdot \frac{1}{-3 \cdot \left(-1.5 \cdot \frac{c \cdot a}{b} + b \cdot 2\right)} \]

Alternative 5
Error	3.0
Cost	1088

\[\left(3 \cdot c\right) \cdot \frac{-0.3333333333333333}{-1.5 \cdot \frac{c \cdot a}{b} + b \cdot 2} \]

Alternative 6
Error	3.0
Cost	1088

\[\left(3 \cdot c\right) \cdot \frac{1}{b \cdot -6 + \frac{c \cdot a}{b} \cdot 4.5} \]

Alternative 7
Error	6.0
Cost	320

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

Alternative 8
Error	6.0
Cost	320

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

Alternative 9
Error	5.8
Cost	320

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

Reproduce

herbie shell --seed 2023066 
(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)))