quadp (p42, positive)

Average Error: 34.2 → 11.1

Time: 8.8s

Precision: binary64

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

↓

\[\begin{array}{l} t_0 := -\frac{c}{b}\\ t_1 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ t_2 := 4 \cdot \left(c \cdot a\right)\\ \mathbf{if}\;b \leq -4.825734430233833 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.130866333133747 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(1, b \cdot b, \mathsf{fma}\left(-4, c \cdot a, t_2\right) - t_2\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3.1907078857048223 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 7879913229977634:\\ \;\;\;\;\frac{t_1 - b \cdot b}{b + \sqrt{t_1}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b)))
        (t_1 (fma a (* c -4.0) (* b b)))
        (t_2 (* 4.0 (* c a))))
   (if (<= b -4.825734430233833e+108)
     (- (/ c b) (/ b a))
     (if (<= b 1.130866333133747e-127)
       (/
        (- (sqrt (fma 1.0 (* b b) (- (fma -4.0 (* c a) t_2) t_2))) b)
        (* a 2.0))
       (if (<= b 3.1907078857048223e-29)
         t_0
         (if (<= b 7879913229977634.0)
           (* (/ (- t_1 (* b b)) (+ b (sqrt t_1))) (/ 0.5 a))
           t_0))))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	double t_0 = -(c / b);
	double t_1 = fma(a, (c * -4.0), (b * b));
	double t_2 = 4.0 * (c * a);
	double tmp;
	if (b <= -4.825734430233833e+108) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.130866333133747e-127) {
		tmp = (sqrt(fma(1.0, (b * b), (fma(-4.0, (c * a), t_2) - t_2))) - b) / (a * 2.0);
	} else if (b <= 3.1907078857048223e-29) {
		tmp = t_0;
	} else if (b <= 7879913229977634.0) {
		tmp = ((t_1 - (b * b)) / (b + sqrt(t_1))) * (0.5 / a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	t_0 = Float64(-Float64(c / b))
	t_1 = fma(a, Float64(c * -4.0), Float64(b * b))
	t_2 = Float64(4.0 * Float64(c * a))
	tmp = 0.0
	if (b <= -4.825734430233833e+108)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.130866333133747e-127)
		tmp = Float64(Float64(sqrt(fma(1.0, Float64(b * b), Float64(fma(-4.0, Float64(c * a), t_2) - t_2))) - b) / Float64(a * 2.0));
	elseif (b <= 3.1907078857048223e-29)
		tmp = t_0;
	elseif (b <= 7879913229977634.0)
		tmp = Float64(Float64(Float64(t_1 - Float64(b * b)) / Float64(b + sqrt(t_1))) * Float64(0.5 / a));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := Block[{t$95$0 = (-N[(c / b), $MachinePrecision])}, Block[{t$95$1 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.825734430233833e+108], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.130866333133747e-127], N[(N[(N[Sqrt[N[(1.0 * N[(b * b), $MachinePrecision] + N[(N[(-4.0 * N[(c * a), $MachinePrecision] + t$95$2), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.1907078857048223e-29], t$95$0, If[LessEqual[b, 7879913229977634.0], N[(N[(N[(t$95$1 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	21.4
Herbie	11.1

\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if b < -4.8257344302338334e108

   1. Initial program 50.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 3.7

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -4.8257344302338334e108 < b < 1.13086633313374705e-127

   1. Initial program 11.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Applied egg-rr 11.4

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(1, b \cdot b, -\left(4 \cdot \left(a \cdot c\right) - \mathsf{fma}\left(-4, a \cdot c, 4 \cdot \left(a \cdot c\right)\right)\right)\right)}}}{2 \cdot a} \]

   if 1.13086633313374705e-127 < b < 3.19070788570482231e-29 or 7879913229977634 < b

   1. Initial program 52.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 10.4

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   3. Simplified 10.4

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   if 3.19070788570482231e-29 < b < 7879913229977634

   1. Initial program 42.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Simplified 43.0

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]

   3. Applied egg-rr 43.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}} \cdot \frac{0.5}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.825734430233833 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.130866333133747 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(1, b \cdot b, \mathsf{fma}\left(-4, c \cdot a, 4 \cdot \left(c \cdot a\right)\right) - 4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3.1907078857048223 \cdot 10^{-29}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 7879913229977634:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))