The quadratic formula (r1)

Average Error: 34.2 → 10.2

Time: 8.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -4.637978985458623 \cdot 10^{+120}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.181936558569041 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(1, b \cdot b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.637978985458623e+120)
   (/ (- b) a)
   (if (<= b 3.181936558569041e-61)
     (/ (- (sqrt (fma 1.0 (* b b) (* c (* a -4.0)))) b) (* a 2.0))
     (/ (- c) b))))

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 tmp;
	if (b <= -4.637978985458623e+120) {
		tmp = -b / a;
	} else if (b <= 3.181936558569041e-61) {
		tmp = (sqrt(fma(1.0, (b * b), (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

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

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.637978985458623e+120)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.181936558569041e-61)
		tmp = Float64(Float64(sqrt(fma(1.0, Float64(b * b), Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

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

↓

code[a_, b_, c_] := If[LessEqual[b, -4.637978985458623e+120], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.181936558569041e-61], N[(N[(N[Sqrt[N[(1.0 * N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	21.1
Herbie	10.2

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

Derivation

1. Split input into 3 regimes

2. if b < -4.63797898545862286e120

   1. Initial program 52.9

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

   2. Taylor expanded in b around -inf 3.3

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

   3. Simplified 3.3

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

   if -4.63797898545862286e120 < b < 3.18193655856904112e-61

   1. Initial program 13.4

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

   2. Applied egg-rr 13.4

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

   if 3.18193655856904112e-61 < b

   1. Initial program 53.9

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

   2. Taylor expanded in b around inf 8.5

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

   3. Simplified 8.5

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.637978985458623 \cdot 10^{+120}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.181936558569041 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(1, b \cdot b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022146 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))