quad2p (problem 3.2.1, positive)

Average Error: 34.0 → 10.3

Time: 7.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.3484291912101877 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b_2, -2, \frac{0.5}{\frac{b_2}{c}} \cdot a\right)}{a}\\ \mathbf{elif}\;b_2 \leq 1.4308690840704123 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

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 -1.3484291912101877e+151)
   (/ (fma b_2 -2.0 (* (/ 0.5 (/ b_2 c)) a)) a)
   (if (<= b_2 1.4308690840704123e-132)
     (- (/ (sqrt (- (* b_2 b_2) (* c a))) a) (/ b_2 a))
     (* -0.5 (/ c b_2)))))

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 <= -1.3484291912101877e+151) {
		tmp = fma(b_2, -2.0, ((0.5 / (b_2 / c)) * a)) / a;
	} else if (b_2 <= 1.4308690840704123e-132) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / a);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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 <= -1.3484291912101877e+151)
		tmp = Float64(fma(b_2, -2.0, Float64(Float64(0.5 / Float64(b_2 / c)) * a)) / a);
	elseif (b_2 <= 1.4308690840704123e-132)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) / a) - Float64(b_2 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	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, -1.3484291912101877e+151], N[(N[(b$95$2 * -2.0 + N[(N[(0.5 / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.4308690840704123e-132], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $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 < -1.3484291912101877e151

   1. Initial program 63.1

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

   2. Simplified 63.1

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

   3. Taylor expanded in b_2 around -inf 10.6

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

   4. Simplified 2.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b_2, -2, \frac{0.5}{\frac{b_2}{c}} \cdot a\right)}}{a} \]

   if -1.3484291912101877e151 < b_2 < 1.430869084070412e-132

   1. Initial program 10.4

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

   2. Simplified 10.4

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

   3. Applied egg-rr 10.4

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

   4. Applied egg-rr 10.4

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

   if 1.430869084070412e-132 < b_2

   1. Initial program 50.5

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

   2. Simplified 50.5

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

   3. Taylor expanded in b_2 around inf 12.1

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.3484291912101877 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b_2, -2, \frac{0.5}{\frac{b_2}{c}} \cdot a\right)}{a}\\ \mathbf{elif}\;b_2 \leq 1.4308690840704123 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022151 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))