The quadratic formula (r1)

Average Error: 33.7 → 10.2

Time: 17.8s

Precision: binary64

Cost: 20936

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, a \cdot \left(c \cdot 4\right)\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -1.45e+97)
   (- (/ c b) (/ b a))
   (if (<= b 8.5e-52)
     (/
      (-
       (sqrt (+ (* b b) (fma c (* a -4.0) (fma c (* a -4.0) (* a (* c 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 <= -1.45e+97) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.5e-52) {
		tmp = (sqrt(((b * b) + fma(c, (a * -4.0), fma(c, (a * -4.0), (a * (c * 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 <= -1.45e+97)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 8.5e-52)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + fma(c, Float64(a * -4.0), fma(c, Float64(a * -4.0), Float64(a * Float64(c * 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, -1.45e+97], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-52], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision] + N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Target

Original	33.7
Target	20.9
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 < -1.44999999999999994e97

   1. Initial program 44.8

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

   2. Simplified 44.8

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

   3. Applied egg-rr 35.0

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b\right) \cdot 0.5}}} \]

   4. Applied egg-rr 47.7

      \[\leadsto \frac{1}{\frac{a}{\left(\mathsf{hypot}\left(b, \color{blue}{\sqrt{c \cdot -4} \cdot \sqrt{a}}\right) - b\right) \cdot 0.5}} \]

   5. Taylor expanded in b around -inf 64.0

      \[\leadsto \color{blue}{-0.25 \cdot \frac{c \cdot {\left(\sqrt{-4}\right)}^{2}}{b} + -1 \cdot \frac{b}{a}} \]

   6. Simplified 3.4

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

   if -1.44999999999999994e97 < b < 8.50000000000000006e-52

   1. Initial program 14.0

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

   2. Applied egg-rr 14.1

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

   if 8.50000000000000006e-52 < b

   1. Initial program 54.3

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

   2. Simplified 54.3

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

   3. Applied egg-rr 59.3

      \[\leadsto \left(\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{1.5}}} - b\right) \cdot \frac{0.5}{a} \]

   4. Taylor expanded in a around 0 8.0

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

   5. Simplified 8.0

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, a \cdot \left(c \cdot 4\right)\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+89}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Error	10.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Error	22.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-302}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Error	23.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{-278}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Error	40.1
Cost	256

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

Alternative 7
Error	56.4
Cost	192

\[\frac{c}{b} \]

Reproduce

herbie shell --seed 2022284 
(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)))