Cubic critical, narrow range

Average Error: 28.3 → 0.5

Time: 8.3s

Precision: binary64

Cost: 14016

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

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(-3 \cdot a\right)\\ \frac{\frac{t_0}{b + \sqrt{\mathsf{fma}\left(b, b, t_0\right)}}}{a \cdot 3} \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* -3.0 a))))
   (/ (/ t_0 (+ b (sqrt (fma b b t_0)))) (* 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) {
	double t_0 = c * (-3.0 * a);
	return (t_0 / (b + sqrt(fma(b, b, t_0)))) / (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)
	t_0 = Float64(c * Float64(-3.0 * a))
	return Float64(Float64(t_0 / Float64(b + sqrt(fma(b, b, t_0)))) / 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_] := Block[{t$95$0 = N[(c * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[(b + N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 28.3

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

2. Applied egg-rr 27.6

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

3. Taylor expanded in b around 0 0.6

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

4. Simplified 0.5

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

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	9.7
Cost	21060

\[\begin{array}{l} t_0 := a \cdot \left(c \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{c \cdot \left(-3 \cdot a\right) + b \cdot b} - b}{a \cdot 3} \leq -0.3:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{-1.5 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 3}\\ \end{array} \]

Alternative 2
Error	9.7
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{c \cdot \left(-3 \cdot a\right) + b \cdot b} - b}{a \cdot 3} \leq -0.3:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot -3\right)}{-1.5 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 3}\\ \end{array} \]

Alternative 3
Error	0.6
Cost	14016

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

Alternative 4
Error	11.8
Cost	1344

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

Alternative 5
Error	23.0
Cost	320

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

Alternative 6
Error	23.0
Cost	320

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

Reproduce

herbie shell --seed 2022212 
(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)))