Cubic critical, medium range

Average Accuracy: 31.9% → 99.2%

Time: 19.5s

Precision: binary64

Cost: 14528

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (/
  (+ (* a (* c 3.0)) (- (* b b) (* b b)))
  (* -3.0 (* a (+ 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 ((a * (c * 3.0)) + ((b * b) - (b * b))) / (-3.0 * (a * (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(Float64(a * Float64(c * 3.0)) + Float64(Float64(b * b) - Float64(b * b))) / Float64(-3.0 * Float64(a * 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[(N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-3.0 * N[(a * 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 31.9%

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

2. Simplified 31.9%

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

   [Start] 31.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   remove-double-neg [<=] 31.9	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
   sub-neg [<=] 31.9	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   div-sub [=>] 31.4	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   neg-mul-1 [=>] 31.4	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   associate-*l/ [<=] 31.7	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   distribute-frac-neg [=>] 31.7	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   fma-neg [=>] 33.2	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
   /-rgt-identity [<=] 33.2	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   metadata-eval [<=] 33.2	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   associate-/l* [<=] 33.2	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   *-commutative [<=] 33.2	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   neg-mul-1 [<=] 33.2	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   fma-neg [<=] 31.7	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   neg-mul-1 [=>] 31.7	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

3. Applied egg-rr 32.8%

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

   [Start] 31.9	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a} \]
   flip-- [=>] 31.9	\[ \color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \cdot \frac{-0.3333333333333333}{a} \]
   frac-times [=>] 31.9	\[ \color{blue}{\frac{\left(b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot -0.3333333333333333}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}} \]
   associate-/l* [=>] 31.9	\[ \color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{\frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}{-0.3333333333333333}}} \]
   add-sqr-sqrt [<=] 32.8	\[ \frac{b \cdot b - \color{blue}{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{\frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}{-0.3333333333333333}} \]

4. Applied egg-rr 99.1%

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

   [Start] 32.8	\[ \frac{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}{\frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}{-0.3333333333333333}} \]
   sub-neg [=>] 32.8	\[ \frac{\color{blue}{b \cdot b + \left(-\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}{\frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}{-0.3333333333333333}} \]
   +-commutative [=>] 32.8	\[ \frac{\color{blue}{\left(-\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right) + b \cdot b}}{\frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}{-0.3333333333333333}} \]
   fma-udef [=>] 32.8	\[ \frac{\left(-\color{blue}{\left(a \cdot \left(c \cdot -3\right) + b \cdot b\right)}\right) + b \cdot b}{\frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}{-0.3333333333333333}} \]
   distribute-neg-in [=>] 32.8	\[ \frac{\color{blue}{\left(\left(-a \cdot \left(c \cdot -3\right)\right) + \left(-b \cdot b\right)\right)} + b \cdot b}{\frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}{-0.3333333333333333}} \]
   associate-+l+ [=>] 99.1	\[ \frac{\color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}}{\frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}{-0.3333333333333333}} \]

5. Applied egg-rr 99.2%

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

   [Start] 99.1	\[ \frac{\left(-a \cdot \left(c \cdot -3\right)\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}{\frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}{-0.3333333333333333}} \]
   div-inv [=>] 99.2	\[ \frac{\left(-a \cdot \left(c \cdot -3\right)\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}{\color{blue}{\left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a\right) \cdot \frac{1}{-0.3333333333333333}}} \]
   *-commutative [=>] 99.2	\[ \frac{\left(-a \cdot \left(c \cdot -3\right)\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}{\color{blue}{\left(a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} \cdot \frac{1}{-0.3333333333333333}} \]
   metadata-eval [=>] 99.2	\[ \frac{\left(-a \cdot \left(c \cdot -3\right)\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}{\left(a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \color{blue}{-3}} \]

6. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	99.1%
Cost	14464

\[\frac{c \cdot \left(a \cdot 3\right) + \left(b \cdot b\right) \cdot 0}{\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) - b} \cdot \frac{0.3333333333333333}{a} \]

Alternative 2
Accuracy	99.1%
Cost	14016

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

Alternative 3
Accuracy	90.6%
Cost	832

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

Alternative 4
Accuracy	80.9%
Cost	320

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

Reproduce

herbie shell --seed 2023126 
(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)))