quad2m (problem 3.2.1, negative)

Average Error: 34.1 → 10.8

Time: 7.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.647573847109141 \cdot 10^{-44}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.1115744018599626 \cdot 10^{+78}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c}{b_2} \cdot 0.5\right)\\ \end{array} \]

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 -6.647573847109141e-44)
   (* -0.5 (/ c b_2))
   (if (<= b_2 6.1115744018599626e+78)
     (- (/ (- b_2) a) (/ (sqrt (- (* b_2 b_2) (* 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 <= -6.647573847109141e-44) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6.1115744018599626e+78) {
		tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (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 <= -6.647573847109141e-44)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 6.1115744018599626e+78)
		tmp = Float64(Float64(Float64(-b_2) / a) - Float64(sqrt(Float64(Float64(b_2 * b_2) - 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, -6.647573847109141e-44], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6.1115744018599626e+78], N[(N[((-b$95$2) / a), $MachinePrecision] - N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $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 < -6.6475738471091406e-44

   1. Initial program 54.1

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

   2. Taylor expanded in b_2 around -inf 7.7

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

   if -6.6475738471091406e-44 < b_2 < 6.11157440185996259e78

   1. Initial program 15.2

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

   2. Applied egg-rr 15.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{b_2}{\sqrt[3]{a}}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]

   3. Applied egg-rr 15.2

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

   if 6.11157440185996259e78 < b_2

   1. Initial program 42.2

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

   2. Applied egg-rr 42.3

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

   3. Taylor expanded in b_2 around inf 5.5

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

   4. Simplified 5.5

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

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.647573847109141 \cdot 10^{-44}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.1115744018599626 \cdot 10^{+78}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c}{b_2} \cdot 0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))