Cubic critical, wide range

Average Error: 52.4 → 1.5

Time: 5.8s

Precision: binary64

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

Math

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

↓

\[-0.5 \cdot \frac{c}{b} - \frac{\mathsf{fma}\left(1.125, \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (-
  (* -0.5 (/ c b))
  (/
   (fma
    1.125
    (/ (pow (* c a) 2.0) (pow b 3.0))
    (fma
     1.6875
     (/ (pow (* c a) 3.0) (pow b 5.0))
     (* 3.1640625 (/ (pow (* c a) 4.0) (pow b 7.0)))))
   (* 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) {
	return (-0.5 * (c / b)) - (fma(1.125, (pow((c * a), 2.0) / pow(b, 3.0)), fma(1.6875, (pow((c * a), 3.0) / pow(b, 5.0)), (3.1640625 * (pow((c * a), 4.0) / pow(b, 7.0))))) / (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)
	return Float64(Float64(-0.5 * Float64(c / b)) - Float64(fma(1.125, Float64((Float64(c * a) ^ 2.0) / (b ^ 3.0)), fma(1.6875, Float64((Float64(c * a) ^ 3.0) / (b ^ 5.0)), Float64(3.1640625 * Float64((Float64(c * a) ^ 4.0) / (b ^ 7.0))))) / 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_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(1.125 * N[(N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.6875 * N[(N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(3.1640625 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.4

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

2. Taylor expanded in b around inf 1.9

   \[\leadsto \frac{\color{blue}{-\left(1.5 \cdot \frac{c \cdot a}{b} + \left(1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(3.1640625 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}} + 1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)\right)}}{3 \cdot a} \]

3. Simplified 1.9

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{c \cdot a}{b} - \mathsf{fma}\left(1.125, \frac{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}}\right)\right)}}{3 \cdot a} \]

4. Applied egg-rr 1.9

   \[\leadsto \color{blue}{\frac{-1.5 \cdot \frac{c \cdot a}{b}}{a \cdot 3} - \frac{\mathsf{fma}\left(1.125, \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3}} \]

5. Taylor expanded in c around 0 1.5

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} - \frac{\mathsf{fma}\left(1.125, \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3} \]

6. Final simplification 1.5

   \[\leadsto -0.5 \cdot \frac{c}{b} - \frac{\mathsf{fma}\left(1.125, \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}, \mathsf{fma}\left(1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, 3.1640625 \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 3} \]

Reproduce

herbie shell --seed 2022146 
(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)))