Average Error: 34.3 → 10.4
Time: 13.2s
Precision: binary64
Cost: 14344
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.2 \cdot 10^{+120}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + 2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
(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 -9.2e+120)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 4.4e-52)
     (/
      (- (sqrt (+ (- (* b_2 b_2) (* a c)) (* 2.0 (fma a (- c) (* a c))))) b_2)
      a)
     (/ (* c -0.5) b_2))))
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 <= -9.2e+120) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 4.4e-52) {
		tmp = (sqrt((((b_2 * b_2) - (a * c)) + (2.0 * fma(a, -c, (a * c))))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

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 <= -9.2e+120)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 4.4e-52)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(b_2 * b_2) - Float64(a * c)) + Float64(2.0 * fma(a, Float64(-c), Float64(a * c))))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

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, -9.2e+120], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.4e-52], N[(N[(N[Sqrt[N[(N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(2.0 * 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]]]

\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -9.2 \cdot 10^{+120}:\\
\;\;\;\;-2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \leq 4.4 \cdot 10^{-52}:\\
\;\;\;\;\frac{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + 2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)} - b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot -0.5}{b_2}\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -9.1999999999999997e120

    1. Initial program 52.7

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

    2. Simplified52.7

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in b_2 around -inf 3.3

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

    if -9.1999999999999997e120 < b_2 < 4.40000000000000018e-52

    1. Initial program 14.1

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

    2. Simplified14.1

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr14.1

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

    4. Simplified14.1

      \[\leadsto \frac{\sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}} - b_2}{a} \]
      Proof
      (+.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a)) (*.f64 2 (fma.f64 a (neg.f64 c) (*.f64 c a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (*.f64 b_2 b_2) (Rewrite<= *-commutative_binary64 (*.f64 a c))) (*.f64 2 (fma.f64 a (neg.f64 c) (*.f64 c a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) (*.f64 2 (fma.f64 a (neg.f64 c) (Rewrite<= *-commutative_binary64 (*.f64 a c))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) (Rewrite<= count-2_binary64 (+.f64 (fma.f64 a (neg.f64 c) (*.f64 a c)) (fma.f64 a (neg.f64 c) (*.f64 a c))))): 0 points increase in error, 0 points decrease in error

    if 4.40000000000000018e-52 < b_2

    1. Initial program 54.7

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

    2. Simplified54.7

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in b_2 around inf 8.0

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

    4. Simplified8.0

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
      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)): 2 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 simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9.2 \cdot 10^{+120}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + 2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error10.4
Cost7368
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.6 \cdot 10^{+121}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
Alternative 2
Error13.5
Cost7176
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.2 \cdot 10^{-67}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
Alternative 3
Error23.0
Cost836
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{-290}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
Alternative 4
Error39.5
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
Alternative 5
Error23.1
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.15 \cdot 10^{-248}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]
Alternative 6
Error23.0
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 6 \cdot 10^{-250}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
Alternative 7
Error53.5
Cost388
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
Alternative 8
Error56.6
Cost192
\[\frac{0}{a} \]

Error

Reproduce

herbie shell --seed 2022330 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))