Cubic critical, medium range

Average Error: 44.0 → 2.9

Time: 42.4s

Precision: binary64

Cost: 49920

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

\[\begin{array}{l} t_0 := {\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2}\\ t_1 := \left(\left(2.25 \cdot \left(c \cdot c\right)\right) \cdot a\right) \cdot a\\ -0.16666666666666666 \cdot \frac{{\left(\frac{1}{b}\right)}^{7} \cdot \left(2.25 \cdot \left(t_0 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + {\left(-0.5 \cdot t_0\right)}^{2}\right)}{a} + \left(-0.25 \cdot \left(c \cdot \left(t_1 \cdot {\left(\frac{1}{b}\right)}^{5}\right)\right) + \left(-0.16666666666666666 \cdot \frac{t_1 \cdot \frac{\frac{\frac{1}{b}}{b}}{b}}{a} + -0.5 \cdot \frac{c}{b}\right)\right) \end{array} \]

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 (pow (* -1.5 (* c a)) 2.0)) (t_1 (* (* (* 2.25 (* c c)) a) a)))
   (+
    (*
     -0.16666666666666666
     (/
      (*
       (pow (/ 1.0 b) 7.0)
       (+ (* 2.25 (* t_0 (* (pow a 2.0) (pow c 2.0)))) (pow (* -0.5 t_0) 2.0)))
      a))
    (+
     (* -0.25 (* c (* t_1 (pow (/ 1.0 b) 5.0))))
     (+
      (* -0.16666666666666666 (/ (* t_1 (/ (/ (/ 1.0 b) b) b)) a))
      (* -0.5 (/ c 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 = pow((-1.5 * (c * a)), 2.0);
	double t_1 = ((2.25 * (c * c)) * a) * a;
	return (-0.16666666666666666 * ((pow((1.0 / b), 7.0) * ((2.25 * (t_0 * (pow(a, 2.0) * pow(c, 2.0)))) + pow((-0.5 * t_0), 2.0))) / a)) + ((-0.25 * (c * (t_1 * pow((1.0 / b), 5.0)))) + ((-0.16666666666666666 * ((t_1 * (((1.0 / b) / b) / b)) / a)) + (-0.5 * (c / b))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    t_0 = ((-1.5d0) * (c * a)) ** 2.0d0
    t_1 = ((2.25d0 * (c * c)) * a) * a
    code = ((-0.16666666666666666d0) * ((((1.0d0 / b) ** 7.0d0) * ((2.25d0 * (t_0 * ((a ** 2.0d0) * (c ** 2.0d0)))) + (((-0.5d0) * t_0) ** 2.0d0))) / a)) + (((-0.25d0) * (c * (t_1 * ((1.0d0 / b) ** 5.0d0)))) + (((-0.16666666666666666d0) * ((t_1 * (((1.0d0 / b) / b) / b)) / a)) + ((-0.5d0) * (c / b))))
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double t_0 = Math.pow((-1.5 * (c * a)), 2.0);
	double t_1 = ((2.25 * (c * c)) * a) * a;
	return (-0.16666666666666666 * ((Math.pow((1.0 / b), 7.0) * ((2.25 * (t_0 * (Math.pow(a, 2.0) * Math.pow(c, 2.0)))) + Math.pow((-0.5 * t_0), 2.0))) / a)) + ((-0.25 * (c * (t_1 * Math.pow((1.0 / b), 5.0)))) + ((-0.16666666666666666 * ((t_1 * (((1.0 / b) / b) / b)) / a)) + (-0.5 * (c / b))));
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

↓

def code(a, b, c):
	t_0 = math.pow((-1.5 * (c * a)), 2.0)
	t_1 = ((2.25 * (c * c)) * a) * a
	return (-0.16666666666666666 * ((math.pow((1.0 / b), 7.0) * ((2.25 * (t_0 * (math.pow(a, 2.0) * math.pow(c, 2.0)))) + math.pow((-0.5 * t_0), 2.0))) / a)) + ((-0.25 * (c * (t_1 * math.pow((1.0 / b), 5.0)))) + ((-0.16666666666666666 * ((t_1 * (((1.0 / b) / b) / b)) / a)) + (-0.5 * (c / b))))

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(-1.5 * Float64(c * a)) ^ 2.0
	t_1 = Float64(Float64(Float64(2.25 * Float64(c * c)) * a) * a)
	return Float64(Float64(-0.16666666666666666 * Float64(Float64((Float64(1.0 / b) ^ 7.0) * Float64(Float64(2.25 * Float64(t_0 * Float64((a ^ 2.0) * (c ^ 2.0)))) + (Float64(-0.5 * t_0) ^ 2.0))) / a)) + Float64(Float64(-0.25 * Float64(c * Float64(t_1 * (Float64(1.0 / b) ^ 5.0)))) + Float64(Float64(-0.16666666666666666 * Float64(Float64(t_1 * Float64(Float64(Float64(1.0 / b) / b) / b)) / a)) + Float64(-0.5 * Float64(c / b)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

↓

function tmp = code(a, b, c)
	t_0 = (-1.5 * (c * a)) ^ 2.0;
	t_1 = ((2.25 * (c * c)) * a) * a;
	tmp = (-0.16666666666666666 * ((((1.0 / b) ^ 7.0) * ((2.25 * (t_0 * ((a ^ 2.0) * (c ^ 2.0)))) + ((-0.5 * t_0) ^ 2.0))) / a)) + ((-0.25 * (c * (t_1 * ((1.0 / b) ^ 5.0)))) + ((-0.16666666666666666 * ((t_1 * (((1.0 / b) / b) / b)) / a)) + (-0.5 * (c / b))));
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[Power[N[(-1.5 * N[(c * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(2.25 * N[(c * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]}, N[(N[(-0.16666666666666666 * N[(N[(N[Power[N[(1.0 / b), $MachinePrecision], 7.0], $MachinePrecision] * N[(N[(2.25 * N[(t$95$0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(-0.5 * t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(c * N[(t$95$1 * N[Power[N[(1.0 / b), $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[(t$95$1 * N[(N[(N[(1.0 / b), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 44.0

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

2. Simplified 44.0

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

3. Taylor expanded in b around inf 2.9

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{\left(\frac{1}{b}\right)}^{7} \cdot \left(2.25 \cdot \left({\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + {\left(-0.5 \cdot {\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2}\right)}^{2}\right)}{a} + \left(-0.25 \cdot \left(c \cdot \left({\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2} \cdot {\left(\frac{1}{b}\right)}^{5}\right)\right) + \left(-0.16666666666666666 \cdot \frac{{\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2} \cdot {\left(\frac{1}{b}\right)}^{3}}{a} + -0.5 \cdot \frac{c}{b}\right)\right)} \]

4. Applied egg-rr 2.9

   \[\leadsto -0.16666666666666666 \cdot \frac{{\left(\frac{1}{b}\right)}^{7} \cdot \left(2.25 \cdot \left({\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + {\left(-0.5 \cdot {\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2}\right)}^{2}\right)}{a} + \left(-0.25 \cdot \left(c \cdot \left({\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2} \cdot {\left(\frac{1}{b}\right)}^{5}\right)\right) + \left(-0.16666666666666666 \cdot \frac{{\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2} \cdot \color{blue}{\frac{\frac{\frac{1}{b}}{b}}{b}}}{a} + -0.5 \cdot \frac{c}{b}\right)\right) \]

5. Applied egg-rr 2.9

   \[\leadsto -0.16666666666666666 \cdot \frac{{\left(\frac{1}{b}\right)}^{7} \cdot \left(2.25 \cdot \left({\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + {\left(-0.5 \cdot {\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2}\right)}^{2}\right)}{a} + \left(-0.25 \cdot \left(c \cdot \left({\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2} \cdot {\left(\frac{1}{b}\right)}^{5}\right)\right) + \left(-0.16666666666666666 \cdot \frac{\color{blue}{\left(\left(\left(2.25 \cdot \left(c \cdot c\right)\right) \cdot a\right) \cdot a\right)} \cdot \frac{\frac{\frac{1}{b}}{b}}{b}}{a} + -0.5 \cdot \frac{c}{b}\right)\right) \]

6. Applied egg-rr 2.9

   \[\leadsto -0.16666666666666666 \cdot \frac{{\left(\frac{1}{b}\right)}^{7} \cdot \left(2.25 \cdot \left({\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + {\left(-0.5 \cdot {\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2}\right)}^{2}\right)}{a} + \left(-0.25 \cdot \left(c \cdot \left(\color{blue}{\left(\left(\left(2.25 \cdot \left(c \cdot c\right)\right) \cdot a\right) \cdot a\right)} \cdot {\left(\frac{1}{b}\right)}^{5}\right)\right) + \left(-0.16666666666666666 \cdot \frac{\left(\left(\left(2.25 \cdot \left(c \cdot c\right)\right) \cdot a\right) \cdot a\right) \cdot \frac{\frac{\frac{1}{b}}{b}}{b}}{a} + -0.5 \cdot \frac{c}{b}\right)\right) \]

Alternatives

Alternative 1
Error	3.8
Cost	27968

\[\begin{array}{l} t_0 := {\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2}\\ -0.25 \cdot \left(c \cdot \left(t_0 \cdot {\left(\frac{1}{b}\right)}^{5}\right)\right) + \left(-0.16666666666666666 \cdot \frac{t_0 \cdot {\left(\frac{1}{b}\right)}^{3}}{a} + -0.5 \cdot \frac{c}{b}\right) \end{array} \]

Alternative 2
Error	10.0
Cost	21252

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 3
Error	10.0
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 4
Error	10.0
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 5
Error	10.0
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 6
Error	10.0
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 7
Error	10.0
Cost	14916

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 8
Error	5.8
Cost	14144

\[-0.16666666666666666 \cdot \frac{{\left(-1.5 \cdot \left(c \cdot a\right)\right)}^{2} \cdot {\left(\frac{1}{b}\right)}^{3}}{a} + -0.5 \cdot \frac{c}{b} \]

Alternative 9
Error	11.9
Cost	320

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

Reproduce

herbie shell --seed 2023010 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))