Cubic critical, narrow range

Average Error: 28.7 → 5.0

Time: 4.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -4:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), -0.5625, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{{b}^{3}}{a \cdot \left(c \cdot c\right)}}\right)\right) + \left(\frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}\right) \cdot -1.0546875\\ \end{array} \]

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 (fma b b (* c (* a -3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -4.0)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* 3.0 a))
     (+
      (fma
       (* (/ (pow c 3.0) (pow b 5.0)) (* a a))
       -0.5625
       (fma -0.5 (/ c b) (/ -0.375 (/ (pow b 3.0) (* a (* c c))))))
      (* (* (/ (pow c 4.0) (pow b 7.0)) (pow a 3.0)) -1.0546875)))))

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 = fma(b, b, (c * (a * -3.0)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -4.0) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(((pow(c, 3.0) / pow(b, 5.0)) * (a * a)), -0.5625, fma(-0.5, (c / b), (-0.375 / (pow(b, 3.0) / (a * (c * c)))))) + (((pow(c, 4.0) / pow(b, 7.0)) * pow(a, 3.0)) * -1.0546875);
	}
	return tmp;
}

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 = fma(b, b, Float64(c * Float64(a * -3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -4.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(a * a)), -0.5625, fma(-0.5, Float64(c / b), Float64(-0.375 / Float64((b ^ 3.0) / Float64(a * Float64(c * c)))))) + Float64(Float64(Float64((c ^ 4.0) / (b ^ 7.0)) * (a ^ 3.0)) * -1.0546875));
	end
	return tmp
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[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -4.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 / N[(N[Power[b, 3.0], $MachinePrecision] / N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] * -1.0546875), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4

   1. Initial program 10.2

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

   2. Applied egg-rr 9.5

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

   if -4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 30.9

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

   2. Taylor expanded in b around inf 4.4

      \[\leadsto \color{blue}{-\left(0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right)\right)} \]

   3. Simplified 4.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), -0.5625, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{{b}^{3}}{a \cdot \left(c \cdot c\right)}}\right)\right) + \left(\frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}\right) \cdot -1.0546875} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), -0.5625, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{{b}^{3}}{a \cdot \left(c \cdot c\right)}}\right)\right) + \left(\frac{{c}^{4}}{{b}^{7}} \cdot {a}^{3}\right) \cdot -1.0546875\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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)))