Cubic critical, narrow range

Average Accuracy: 55.6% → 99.6%

Time: 23.5s

Precision: binary64

Cost: 13824

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 55.6%

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

2. Simplified 55.6%

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

   [Start] 55.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   remove-double-neg [<=] 55.6	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
   sub-neg [<=] 55.6	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   div-sub [=>] 54.8	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   neg-mul-1 [=>] 54.8	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   associate-*l/ [<=] 54.8	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   distribute-frac-neg [=>] 54.8	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   fma-neg [=>] 55.6	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
   /-rgt-identity [<=] 55.6	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   metadata-eval [<=] 55.6	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   associate-/l* [<=] 55.6	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   *-commutative [<=] 55.6	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   neg-mul-1 [<=] 55.6	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   fma-neg [<=] 54.8	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   neg-mul-1 [=>] 54.8	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

3. Applied egg-rr 57.1%

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

   [Start] 55.6	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a} \]
   *-commutative [=>] 55.6	\[ \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   clear-num [=>] 55.6	\[ \color{blue}{\frac{1}{\frac{a}{-0.3333333333333333}}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \]
   flip-- [=>] 55.6	\[ \frac{1}{\frac{a}{-0.3333333333333333}} \cdot \color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   frac-times [=>] 55.6	\[ \color{blue}{\frac{1 \cdot \left(b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\frac{a}{-0.3333333333333333} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   associate-/l* [=>] 55.6	\[ \color{blue}{\frac{1}{\frac{\frac{a}{-0.3333333333333333} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}} \]
   div-inv [=>] 55.6	\[ \frac{1}{\frac{\color{blue}{\left(a \cdot \frac{1}{-0.3333333333333333}\right)} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   metadata-eval [=>] 55.6	\[ \frac{1}{\frac{\left(a \cdot \color{blue}{-3}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   add-sqr-sqrt [<=] 57.1	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{b \cdot b - \color{blue}{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]

4. Taylor expanded in b around 0 99.1%

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

5. Applied egg-rr 62.0%

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

   [Start] 99.1	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{3 \cdot \left(c \cdot a\right)}} \]
   expm1-log1p-u [=>] 85.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{3 \cdot \left(c \cdot a\right)}}\right)\right)} \]
   expm1-udef [=>] 62.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{3 \cdot \left(c \cdot a\right)}}\right)} - 1} \]

6. Simplified 99.6%

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

   [Start] 62.0	\[ e^{\mathsf{log1p}\left(\frac{a \cdot c}{\left(-1 \cdot a\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)}\right)} - 1 \]
   expm1-def [=>] 85.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot c}{\left(-1 \cdot a\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)}\right)\right)} \]
   expm1-log1p [=>] 99.3	\[ \color{blue}{\frac{a \cdot c}{\left(-1 \cdot a\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)}} \]
   times-frac [=>] 99.6	\[ \color{blue}{\frac{a}{-1 \cdot a} \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}} \]
   mul-1-neg [=>] 99.6	\[ \frac{a}{\color{blue}{-a}} \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}} \]
   *-commutative [=>] 99.6	\[ \frac{a}{-a} \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(a, \color{blue}{c \cdot -3}, b \cdot b\right)}} \]

7. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	85.5%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.078:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 2
Accuracy	85.5%
Cost	15556

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 3} \leq -0.078:\\ \;\;\;\;\frac{\sqrt{\left(b \cdot b + \left(3 \cdot \left(a \cdot c\right) + t_0\right)\right) + t_0} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 3
Accuracy	85.5%
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.078:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 4
Accuracy	85.4%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 0.78:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{1.5}{\frac{b}{a}}\right)}\\ \end{array} \]

Alternative 5
Accuracy	85.4%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 0.75:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)}}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{1.5}{\frac{b}{a}}\right)}\\ \end{array} \]

Alternative 6
Accuracy	82.0%
Cost	7104

\[\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{1.5}{\frac{b}{a}}\right)} \]

Alternative 7
Accuracy	82.0%
Cost	832

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

Alternative 8
Accuracy	64.3%
Cost	320

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

Reproduce

herbie shell --seed 2023125 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))