The quadratic formula (r2)

Average Error: 34.5 → 10.5

Time: 11.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -8.430934381176968 \cdot 10^{-103}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.30762203111794 \cdot 10^{+53}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array} \]

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 -8.430934381176968e-103)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b 1.30762203111794e+53)
     (* (/ (+ b (sqrt (fma b b (* c (* a -4.0))))) (- a)) 0.5)
     (* -0.5 (* 2.0 (- (/ b 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 <= -8.430934381176968e-103) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= 1.30762203111794e+53) {
		tmp = ((b + sqrt(fma(b, b, (c * (a * -4.0))))) / -a) * 0.5;
	} else {
		tmp = -0.5 * (2.0 * ((b / a) - (c / b)));
	}
	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)
	tmp = 0.0
	if (b <= -8.430934381176968e-103)
		tmp = Float64(-0.5 * Float64(2.0 * Float64(c / b)));
	elseif (b <= 1.30762203111794e+53)
		tmp = Float64(Float64(Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))) / Float64(-a)) * 0.5);
	else
		tmp = Float64(-0.5 * Float64(2.0 * Float64(Float64(b / a) - Float64(c / b))));
	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_] := If[LessEqual[b, -8.430934381176968e-103], N[(-0.5 * N[(2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.30762203111794e+53], N[(N[(N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision] * 0.5), $MachinePrecision], N[(-0.5 * N[(2.0 * N[(N[(b / a), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.5
Target	21.5
Herbie	10.5

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

Derivation

1. Split input into 3 regimes

2. if b < -8.4309343811769682e-103

   1. Initial program 52.2

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

   2. Simplified 52.2

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

   3. Taylor expanded in b around -inf 10.5

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{c}{b}\right)} \]

   if -8.4309343811769682e-103 < b < 1.30762203111794e53

   1. Initial program 13.3

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

   2. Simplified 13.3

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

   3. Applied egg-rr 13.4

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

   4. Applied egg-rr 13.3

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

   5. Taylor expanded in a around 0 13.3

      \[\leadsto -0.5 \cdot \frac{-\left(b + \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(c \cdot a\right)}}\right)}{-a} \]

   6. Simplified 13.3

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

   if 1.30762203111794e53 < b

   1. Initial program 39.4

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

   2. Simplified 39.4

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

   3. Taylor expanded in b around inf 5.2

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{b}{a} - 2 \cdot \frac{c}{b}\right)} \]

   4. Simplified 5.2

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.430934381176968 \cdot 10^{-103}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.30762203111794 \cdot 10^{+53}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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