The quadratic formula (r1)

Average Error: 34.2 → 9.7

Time: 6.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{+148}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \mathbf{elif}\;b \leq 3.05 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -4.3e+148)
   (/ (* b -2.0) (* 2.0 a))
   (if (<= b 3.05e-55)
     (/ (- (sqrt (fma a (* -4.0 c) (* b b))) 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 <= -4.3e+148) {
		tmp = (b * -2.0) / (2.0 * a);
	} else if (b <= 3.05e-55) {
		tmp = (sqrt(fma(a, (-4.0 * c), (b * b))) - 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 <= -4.3e+148)
		tmp = Float64(Float64(b * -2.0) / Float64(2.0 * a));
	elseif (b <= 3.05e-55)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-4.0 * c), Float64(b * b))) - 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, -4.3e+148], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.05e-55], N[(N[(N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Target

Original	34.2
Target	20.6
Herbie	9.7

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

Derivation

1. Split input into 3 regimes

2. if b < -4.3000000000000002e148

   1. Initial program 62.1

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

   2. Applied egg-rr 62.1

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}}}{2 \cdot a} \]

   3. Taylor expanded in b around -inf 2.5

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

   4. Simplified 2.5

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

   if -4.3000000000000002e148 < b < 3.0500000000000001e-55

   1. Initial program 12.5

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

   2. Applied egg-rr 12.7

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}}}{2 \cdot a} \]

   3. Applied egg-rr 12.5

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

   if 3.0500000000000001e-55 < 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 8.3

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

   3. Simplified 8.3

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

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{+148}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \mathbf{elif}\;b \leq 3.05 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(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)))