Cubic critical, narrow range

Average Error: 28.6 → 5.1

Time: 5.2s

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

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -88.30613369776961:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \left(\sqrt[3]{\frac{0.3333333333333333}{a}} \cdot \sqrt[3]{\frac{0.1111111111111111}{a \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, -0.5625, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(\frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.375, \left(\frac{{a}^{3}}{{b}^{7}} \cdot {c}^{4}\right) \cdot -1.0546875\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<=
      (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
      -88.30613369776961)
   (*
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (* (cbrt (/ 0.3333333333333333 a)) (cbrt (/ 0.1111111111111111 (* a a)))))
   (fma
    (* (/ (* a a) (pow b 5.0)) (pow c 3.0))
    -0.5625
    (fma
     -0.5
     (/ c b)
     (fma
      (/ a (/ (pow b 3.0) (* c c)))
      -0.375
      (* (* (/ (pow a 3.0) (pow b 7.0)) (pow c 4.0)) -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 tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -88.30613369776961) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * (cbrt((0.3333333333333333 / a)) * cbrt((0.1111111111111111 / (a * a))));
	} else {
		tmp = fma((((a * a) / pow(b, 5.0)) * pow(c, 3.0)), -0.5625, fma(-0.5, (c / b), fma((a / (pow(b, 3.0) / (c * c))), -0.375, (((pow(a, 3.0) / pow(b, 7.0)) * pow(c, 4.0)) * -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)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -88.30613369776961)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * Float64(cbrt(Float64(0.3333333333333333 / a)) * cbrt(Float64(0.1111111111111111 / Float64(a * a)))));
	else
		tmp = fma(Float64(Float64(Float64(a * a) / (b ^ 5.0)) * (c ^ 3.0)), -0.5625, fma(-0.5, Float64(c / b), fma(Float64(a / Float64((b ^ 3.0) / Float64(c * c))), -0.375, Float64(Float64(Float64((a ^ 3.0) / (b ^ 7.0)) * (c ^ 4.0)) * -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_] := 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], -88.30613369776961], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(N[Power[N[(0.3333333333333333 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(0.1111111111111111 / N[(a * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * -1.0546875), $MachinePrecision]), $MachinePrecision]), $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)) < -88.30613369776961

   1. Initial program 7.6

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

   2. Simplified 7.5

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

   3. Applied egg-rr 7.5

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{\left(\sqrt[3]{\frac{0.3333333333333333}{a}} \cdot \sqrt[3]{\frac{0.1111111111111111}{a \cdot a}}\right)} \]

   if -88.30613369776961 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.9

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

   2. Simplified 29.9

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

   3. Taylor expanded in b around inf 5.3

      \[\leadsto \color{blue}{\left(-\left(1.5 \cdot \frac{c \cdot a}{b} + \left(1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(3.1640625 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}} + 1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

   4. Taylor expanded in c around 0 5.0

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

   5. Simplified 5.0

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -88.30613369776961:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \left(\sqrt[3]{\frac{0.3333333333333333}{a}} \cdot \sqrt[3]{\frac{0.1111111111111111}{a \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, -0.5625, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(\frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.375, \left(\frac{{a}^{3}}{{b}^{7}} \cdot {c}^{4}\right) \cdot -1.0546875\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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)))