Cubic critical, narrow range

Average Error: 28.6 → 5.1

Time: 16.2s

Precision: binary64

Cost: 54724

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

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ t_1 := \mathsf{fma}\left(b, b, t_0\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a} \leq -2.5:\\ \;\;\;\;\frac{\frac{t_1 - b \cdot b}{b + \sqrt{t_1}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.16666666666666666}{a}, {\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{{b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \frac{\frac{\left(a \cdot c\right) \cdot \left(c \cdot -0.375\right)}{b}}{b \cdot b}\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -3.0))) (t_1 (fma b b t_0)))
   (if (<= (/ (- (sqrt (+ (* b b) t_0)) b) (* 3.0 a)) -2.5)
     (/ (/ (- t_1 (* b b)) (+ b (sqrt t_1))) (* 3.0 a))
     (fma
      (/ -0.16666666666666666 a)
      (* (pow (* a c) 4.0) (/ 6.328125 (pow b 7.0)))
      (fma
       -0.5
       (/ c b)
       (fma
        -0.5625
        (* (* a a) (/ (pow c 3.0) (pow b 5.0)))
        (/ (/ (* (* a c) (* c -0.375)) b) (* 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) {
	double t_0 = c * (a * -3.0);
	double t_1 = fma(b, b, t_0);
	double tmp;
	if (((sqrt(((b * b) + t_0)) - b) / (3.0 * a)) <= -2.5) {
		tmp = ((t_1 - (b * b)) / (b + sqrt(t_1))) / (3.0 * a);
	} else {
		tmp = fma((-0.16666666666666666 / a), (pow((a * c), 4.0) * (6.328125 / pow(b, 7.0))), fma(-0.5, (c / b), fma(-0.5625, ((a * a) * (pow(c, 3.0) / pow(b, 5.0))), ((((a * c) * (c * -0.375)) / b) / (b * b)))));
	}
	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 = Float64(c * Float64(a * -3.0))
	t_1 = fma(b, b, t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) + t_0)) - b) / Float64(3.0 * a)) <= -2.5)
		tmp = Float64(Float64(Float64(t_1 - Float64(b * b)) / Float64(b + sqrt(t_1))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(-0.16666666666666666 / a), Float64((Float64(a * c) ^ 4.0) * Float64(6.328125 / (b ^ 7.0))), fma(-0.5, Float64(c / b), fma(-0.5625, Float64(Float64(a * a) * Float64((c ^ 3.0) / (b ^ 5.0))), Float64(Float64(Float64(Float64(a * c) * Float64(c * -0.375)) / b) / Float64(b * b)))));
	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[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * b + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.5], N[(N[(N[(t$95$1 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.16666666666666666 / a), $MachinePrecision] * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * c), $MachinePrecision] * N[(c * -0.375), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

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)) < -2.5

   1. Initial program 9.9

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

   2. Applied egg-rr 10.1

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{b}^{4} - \left(c \cdot \left(a \cdot -3\right)\right) \cdot \left(c \cdot \left(a \cdot -3\right)\right)}{b \cdot b - c \cdot \left(a \cdot -3\right)}}}}{3 \cdot a} \]

   3. Applied egg-rr 9.2

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

   if -2.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 31.2

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

   2. Applied egg-rr 31.3

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{b}^{4} - \left(c \cdot \left(a \cdot -3\right)\right) \cdot \left(c \cdot \left(a \cdot -3\right)\right)}{b \cdot b - c \cdot \left(a \cdot -3\right)}}}}{3 \cdot a} \]

   3. Taylor expanded in c around 0 4.6

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(5.0625 \cdot \frac{{a}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{a}^{2}}{{b}^{3}}\right)}^{2}\right)}{a \cdot b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   4. Simplified 4.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.16666666666666666}{a}, {c}^{4} \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{b}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \frac{-0.375}{{b}^{3}} \cdot \left(c \cdot \left(c \cdot a\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in c around 0 4.6

      \[\leadsto \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \color{blue}{6.328125 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \frac{-0.375}{{b}^{3}} \cdot \left(c \cdot \left(c \cdot a\right)\right)\right)\right)\right) \]

   6. Simplified 4.6

      \[\leadsto \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \color{blue}{{\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{7}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \frac{-0.375}{{b}^{3}} \cdot \left(c \cdot \left(c \cdot a\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 4.6

      \[\leadsto \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, {\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{{b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \color{blue}{\frac{\frac{\left(-0.375 \cdot c\right) \cdot \left(c \cdot a\right)}{b}}{b \cdot b}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -2.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.16666666666666666}{a}, {\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{{b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}, \frac{\frac{\left(a \cdot c\right) \cdot \left(c \cdot -0.375\right)}{b}}{b \cdot b}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	9.6
Cost	35980

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ t_1 := \frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a}\\ \mathbf{if}\;t_1 \leq -0.425:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{elif}\;t_1 \leq -0.0179:\\ \;\;\;\;\frac{\frac{c}{b}}{a} \cdot \frac{a \cdot \mathsf{fma}\left(\frac{c}{b}, -1.125 \cdot \frac{a}{b}, -1.5\right)}{3}\\ \mathbf{elif}\;t_1 \leq -0.004:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, c \cdot \frac{-0.5}{b}\right)\\ \end{array} \]

Alternative 2
Error	9.5
Cost	35980

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ t_1 := \frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a}\\ \mathbf{if}\;t_1 \leq -0.425:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{elif}\;t_1 \leq -0.0179:\\ \;\;\;\;\frac{\frac{c}{b}}{a} \cdot \frac{a \cdot \mathsf{fma}\left(\frac{c}{b}, -1.125 \cdot \frac{a}{b}, -1.5\right)}{3}\\ \mathbf{elif}\;t_1 \leq -0.004:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(c \cdot \left(a \cdot c\right)\right) \cdot \frac{-0.375}{{b}^{3}}\right)\\ \end{array} \]

Alternative 3
Error	9.6
Cost	35916

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ t_1 := \frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right)\\ t_2 := \frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a}\\ \mathbf{if}\;t_2 \leq -0.425:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -0.0179:\\ \;\;\;\;\frac{\frac{c}{b}}{a} \cdot \frac{a \cdot \mathsf{fma}\left(\frac{c}{b}, -1.125 \cdot \frac{a}{b}, -1.5\right)}{3}\\ \mathbf{elif}\;t_2 \leq -0.004:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-0.5}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c}}\right)\\ \end{array} \]

Alternative 4
Error	9.6
Cost	35916

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ t_1 := \frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a}\\ \mathbf{if}\;t_1 \leq -0.425:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{elif}\;t_1 \leq -0.0179:\\ \;\;\;\;\frac{\frac{c}{b}}{a} \cdot \frac{a \cdot \mathsf{fma}\left(\frac{c}{b}, -1.125 \cdot \frac{a}{b}, -1.5\right)}{3}\\ \mathbf{elif}\;t_1 \leq -0.004:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-0.5}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c}}\right)\\ \end{array} \]

Alternative 5
Error	6.6
Cost	34564

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ t_1 := \mathsf{fma}\left(b, b, t_0\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a} \leq -2.5:\\ \;\;\;\;\frac{\frac{t_1 - b \cdot b}{b + \sqrt{t_1}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left(\frac{-0.5625 \cdot {c}^{3}}{\frac{{b}^{5}}{a}} + \frac{c \cdot \left(c \cdot -0.375\right)}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 6
Error	9.6
Cost	29644

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ t_1 := c \cdot \left(\frac{-0.5}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c}}\right)\\ \mathbf{if}\;t_0 \leq -0.425:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq -0.0179:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -0.004:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	9.6
Cost	29644

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.425:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq -0.0179:\\ \;\;\;\;\frac{\frac{c}{\frac{b}{a}} \cdot \mathsf{fma}\left(a \cdot \frac{-1.125}{b}, \frac{c}{b}, -1.5\right)}{3 \cdot a}\\ \mathbf{elif}\;t_0 \leq -0.004:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-0.5}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c}}\right)\\ \end{array} \]

Alternative 8
Error	9.6
Cost	29644

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.425:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq -0.0179:\\ \;\;\;\;\frac{\frac{c}{b}}{a} \cdot \frac{a \cdot \mathsf{fma}\left(\frac{c}{b}, -1.125 \cdot \frac{a}{b}, -1.5\right)}{3}\\ \mathbf{elif}\;t_0 \leq -0.004:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-0.5}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c}}\right)\\ \end{array} \]

Alternative 9
Error	6.7
Cost	28228

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ t_1 := \mathsf{fma}\left(b, b, t_0\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{3 \cdot a} \leq -2.5:\\ \;\;\;\;\frac{\frac{t_1 - b \cdot b}{b + \sqrt{t_1}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot \left(a \cdot c\right), \frac{-0.375}{{b}^{3}} + \frac{c \cdot -0.5625}{\frac{{b}^{5}}{a}}, c \cdot \frac{-0.5}{b}\right)\\ \end{array} \]

Alternative 10
Error	6.7
Cost	28100

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a} \leq -2.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot \left(a \cdot c\right), \frac{-0.375}{{b}^{3}} + \frac{c \cdot -0.5625}{\frac{{b}^{5}}{a}}, c \cdot \frac{-0.5}{b}\right)\\ \end{array} \]

Alternative 11
Error	12.1
Cost	7296

\[c \cdot \left(\frac{-0.5}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c}}\right) \]

Alternative 12
Error	12.0
Cost	7232

\[\frac{c}{b} \cdot \mathsf{fma}\left(c \cdot \frac{a}{b \cdot b}, -0.375, -0.5\right) \]

Alternative 13
Error	12.1
Cost	1728

\[\begin{array}{l} t_0 := a \cdot \frac{c}{b}\\ \frac{\left(\frac{c}{b} \cdot \left(-1.125 \cdot \frac{a}{b}\right)\right) \cdot t_0 + -1.5 \cdot t_0}{3 \cdot a} \end{array} \]

Alternative 14
Error	22.8
Cost	320

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

Alternative 15
Error	22.8
Cost	320

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

Alternative 16
Error	22.8
Cost	320

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

Reproduce

herbie shell --seed 2022221 
(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)))