Average Error: 34.2 → 9.7
Time: 17.6s
Precision: binary64
Cost: 13832
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
(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 -4e+151)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.4e-60)
     (- (/ (sqrt (fma b_2 b_2 (* a (- c)))) a) (/ 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 <= -4e+151) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.4e-60) {
		tmp = (sqrt(fma(b_2, b_2, (a * -c))) / a) - (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 <= -4e+151)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.4e-60)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(a * Float64(-c)))) / a) - Float64(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, -4e+151], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.4e-60], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * (-c)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $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 -4 \cdot 10^{+151}:\\
\;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \leq 1.4 \cdot 10^{-60}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)}}{a} - \frac{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 < -4.00000000000000007e151

    1. Initial program 63.3

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

    2. Simplified63.3

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      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 2.7

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

    if -4.00000000000000007e151 < b_2 < 1.4000000000000001e-60

    1. Initial program 12.6

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

    2. Simplified12.6

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      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-rr12.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}}{1}, \frac{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}}{a}, -\frac{b_2}{a}\right)} \]

    4. Simplified12.6

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{a} - \frac{b_2}{a}} \]
      Proof
      (-.f64 (/.f64 (sqrt.f64 (fma.f64 b_2 b_2 (*.f64 c (neg.f64 a)))) a) (/.f64 b_2 a)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (sqrt.f64 (fma.f64 b_2 b_2 (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 c a))))) a) (/.f64 b_2 a)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (sqrt.f64 (fma.f64 b_2 b_2 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 a c))))) a) (/.f64 b_2 a)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) (/.f64 b_2 a)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) 1/2)) a) (/.f64 b_2 a)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (pow.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) (Rewrite<= metadata-eval (*.f64 2 1/4))) a) (/.f64 b_2 a)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) 1/4) (pow.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) 1/4))) a) (/.f64 b_2 a)): 42 points increase in error, 11 points decrease in error
      (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (pow.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) 1/4) (/.f64 (pow.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) 1/4) a))) (/.f64 b_2 a)): 18 points increase in error, 24 points decrease in error
      (Rewrite=> fma-neg_binary64 (fma.f64 (pow.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) 1/4) (/.f64 (pow.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) 1/4) a) (neg.f64 (/.f64 b_2 a)))): 32 points increase in error, 26 points decrease in error
      (fma.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (pow.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) 1/4) 1)) (/.f64 (pow.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) 1/4) a) (neg.f64 (/.f64 b_2 a))): 0 points increase in error, 0 points decrease in error

    if 1.4000000000000001e-60 < b_2

    1. Initial program 54.1

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

    2. Simplified54.1

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      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. Taylor expanded in b_2 around inf 7.8

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

    4. Simplified7.8

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
      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 simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error9.7
Cost7368
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.25 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
Alternative 2
Error13.0
Cost7176
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.9 \cdot 10^{-87}:\\ \;\;\;\;\left(\frac{0.5 \cdot c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
Alternative 3
Error22.6
Cost964
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6 \cdot 10^{-304}:\\ \;\;\;\;\left(\frac{0.5 \cdot c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
Alternative 4
Error22.6
Cost836
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6 \cdot 10^{-304}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
Alternative 5
Error22.8
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.6 \cdot 10^{-292}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]
Alternative 6
Error22.7
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.65 \cdot 10^{-289}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]
Alternative 7
Error22.6
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.65 \cdot 10^{-289}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
Alternative 8
Error45.6
Cost320
\[b_2 \cdot \frac{-2}{a} \]

Error

Reproduce

herbie shell --seed 2022325 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))