quad2p (problem 3.2.1, positive)

Average Error: 34.0 → 9.8

Time: 14.3s

Precision: binary64

Cost: 20552

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.26 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \mathsf{fma}\left(a, -c, a \cdot c\right)\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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 -1.26e+142)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 2.7e-48)
     (/ (- (sqrt (+ (* b_2 b_2) (fma a (- c) (fma a (- c) (* a c))))) b_2) a)
     (/ (* c -0.5) 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 <= -1.26e+142) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 2.7e-48) {
		tmp = (sqrt(((b_2 * b_2) + fma(a, -c, fma(a, -c, (a * c))))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / 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 <= -1.26e+142)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 2.7e-48)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) + fma(a, Float64(-c), fma(a, Float64(-c), Float64(a * c))))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / 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, -1.26e+142], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.7e-48], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] + N[(a * (-c) + N[(a * (-c) + N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.26000000000000005e142

   1. Initial program 59.2

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

   2. Taylor expanded in b_2 around -inf 2.5

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

   if -1.26000000000000005e142 < b_2 < 2.70000000000000011e-48

   1. Initial program 12.9

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

   2. Applied egg-rr 12.9

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

   if 2.70000000000000011e-48 < b_2

   1. Initial program 53.9

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

   2. Taylor expanded in b_2 around inf 8.1

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

   3. Simplified 8.1

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
      Proof
      (/.f64 (*.f64 c -1/2) b_2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 c b_2) -1/2)): 1 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (/.f64 c b_2))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.26 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \mathsf{fma}\left(a, -c, a \cdot c\right)\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.26 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-65}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	36.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4
Error	22.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.6 \cdot 10^{-226}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 5
Error	22.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.6 \cdot 10^{-226}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 6
Error	53.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.7 \cdot 10^{-286}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 7
Error	56.3
Cost	192

\[\frac{0}{a} \]

Reproduce

herbie shell --seed 2022308 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))