Cubic critical, medium range

Average Error: 43.5 → 0.2

Time: 19.5s

Precision: binary64

Cost: 13568

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

Math

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

↓

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

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

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

   [Start] 43.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   remove-double-neg [<=] 43.5	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
   sub-neg [<=] 43.5	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   div-sub [=>] 43.8	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   neg-mul-1 [=>] 43.8	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   associate-*l/ [<=] 43.6	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   distribute-frac-neg [=>] 43.6	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   fma-neg [=>] 42.7	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
   /-rgt-identity [<=] 42.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   metadata-eval [<=] 42.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   associate-/l* [<=] 42.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   *-commutative [<=] 42.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   neg-mul-1 [<=] 42.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   fma-neg [<=] 43.6	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   neg-mul-1 [=>] 43.6	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

3. Applied egg-rr 42.9

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

4. Simplified 42.9

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

   [Start] 42.9	\[ \frac{\frac{\frac{-0.3333333333333333}{a} \cdot \left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}{\sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   associate-/l/ [=>] 42.9	\[ \color{blue}{\frac{\frac{-0.3333333333333333}{a} \cdot \left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}} \]
   *-commutative [=>] 42.9	\[ \frac{\color{blue}{\left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right) \cdot \frac{-0.3333333333333333}{a}}}{\sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]

5. Taylor expanded in b around 0 0.5

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

6. Simplified 0.5

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

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

7. Applied egg-rr 0.2

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

8. Final simplification 0.2

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

Alternatives

Alternative 1
Error	5.9
Cost	896

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

Alternative 2
Error	12.3
Cost	320

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

Reproduce

herbie shell --seed 2023073 
(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)))