The quadratic formula (r1)

Average Error: 33.6 → 10.2

Time: 9.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -4.11211753122393 \cdot 10^{+143}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \mathbf{elif}\;b \leq 4.4569387361564705 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

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 -4.11211753122393e+143)
   (/ (* b -2.0) (* 2.0 a))
   (if (<= b 4.4569387361564705e-88)
     (/ (- (sqrt (fma b b (* -4.0 (* a c)))) b) (* 2.0 a))
     (- (/ 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 <= -4.11211753122393e+143) {
		tmp = (b * -2.0) / (2.0 * a);
	} else if (b <= 4.4569387361564705e-88) {
		tmp = (sqrt(fma(b, b, (-4.0 * (a * c)))) - b) / (2.0 * a);
	} 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 <= -4.11211753122393e+143)
		tmp = Float64(Float64(b * -2.0) / Float64(2.0 * a));
	elseif (b <= 4.4569387361564705e-88)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(a * c)))) - b) / Float64(2.0 * a));
	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, -4.11211753122393e+143], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4569387361564705e-88], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.6
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 < -4.11211753122393e143

   1. Initial program 60.0

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

   2. Applied egg-rr 60.1

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt[3]{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}}}{2 \cdot a} \]

   3. Taylor expanded in b around -inf 2.2

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a} \]

   4. Simplified 2.2

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{2 \cdot a} \]

   if -4.11211753122393e143 < b < 4.4569387361564705e-88

   1. Initial program 12.4

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

   2. Applied egg-rr 12.4

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

   if 4.4569387361564705e-88 < b

   1. Initial program 52.0

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

   2. Taylor expanded in b around inf 10.0

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

   3. Simplified 10.0

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.11211753122393 \cdot 10^{+143}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \mathbf{elif}\;b \leq 4.4569387361564705 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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)))