quad2m (problem 3.2.1, negative)

Average Error: 33.4 → 10.3

Time: 6.2s

Precision: binary64

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.5 \cdot 10^{-22}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c}{b_2} \cdot 0.5\right)\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.5e-22)
   (* -0.5 (/ c b_2))
   (if (<= b_2 2.5e+78)
     (/ (- (- b_2) (sqrt (+ (- (* b_2 b_2) (* c a)) (fma (- c) a (* c a))))) a)
     (fma (/ b_2 a) -2.0 (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.5e-22) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.5e+78) {
		tmp = (-b_2 - sqrt((((b_2 * b_2) - (c * a)) + fma(-c, a, (c * a))))) / a;
	} else {
		tmp = fma((b_2 / a), -2.0, ((c / b_2) * 0.5));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

↓

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.5e-22)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 2.5e+78)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(Float64(b_2 * b_2) - Float64(c * a)) + fma(Float64(-c), a, Float64(c * a))))) / a);
	else
		tmp = fma(Float64(b_2 / a), -2.0, Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.5e-22], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.5e+78], N[(N[((-b$95$2) - N[Sqrt[N[(N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision] + N[((-c) * a + N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0 + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.49999999999999977e-22

   1. Initial program 55.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 6.6

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -2.49999999999999977e-22 < b_2 < 2.49999999999999992e78

   1. Initial program 14.9

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Applied egg-rr 15.0

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, a \cdot c\right) + \mathsf{fma}\left(-c, a, a \cdot c\right)\right)}}}{a} \]

   3. Taylor expanded in c around 0 14.9

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{0} + \mathsf{fma}\left(-c, a, a \cdot c\right)\right)}}{a} \]

   if 2.49999999999999992e78 < b_2

   1. Initial program 41.2

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Applied egg-rr 41.3

      \[\leadsto \color{blue}{\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a}} \]

   3. Taylor expanded in b_2 around inf 4.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}} \]

   4. Simplified 4.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b_2}{a}, -2, 0.5 \cdot \frac{c}{b_2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.5 \cdot 10^{-22}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c}{b_2} \cdot 0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))