Average Error: 34.2 → 10.3
Time: 15.8s
Precision: binary64
Cost: 7624
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, -0.5 \cdot \frac{b}{c}\right)}\\ \end{array} \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e+141)
   (- (/ c b) (/ b a))
   (if (<= b 5.2e-105)
     (/ (- (sqrt (+ (* b b) (* c (* a -4.0)))) b) (* a 2.0))
     (/ 0.5 (fma 0.5 (/ a b) (* -0.5 (/ b c)))))))
double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e+141) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.2e-105) {
		tmp = (sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.5 / fma(0.5, (a / b), (-0.5 * (b / c)));
	}
	return tmp;
}

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end
↓
function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e+141)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5.2e-105)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.5 / fma(0.5, Float64(a / b), Float64(-0.5 * Float64(b / c))));
	end
	return tmp
end

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_, c_] := If[LessEqual[b, -3.5e+141], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-105], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(0.5 * N[(a / b), $MachinePrecision] + N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -3.5 \cdot 10^{+141}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 5.2 \cdot 10^{-105}:\\
\;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, -0.5 \cdot \frac{b}{c}\right)}\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if b < -3.5e141

    1. Initial program 59.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

    2. Simplified59.2

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
      Proof
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= *-commutative_binary64 (*.f64 2 a))): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in b around -inf 2.3

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

    4. Simplified2.3

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
      Proof
      (-.f64 (/.f64 c b) (/.f64 b a)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 c b) (neg.f64 (/.f64 b a)))): 3 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 c b) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 b a)))): 3 points increase in error, 0 points decrease in error

    if -3.5e141 < b < 5.1999999999999997e-105

    1. Initial program 11.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

    if 5.1999999999999997e-105 < b

    1. Initial program 51.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

    2. Simplified51.8

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
      Proof
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= *-commutative_binary64 (*.f64 2 a))): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr46.4

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b}}} \]

    4. Taylor expanded in b around inf 64.0

      \[\leadsto \frac{0.5}{\color{blue}{2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}} + 0.5 \cdot \frac{a}{b}}} \]

    5. Simplified10.9

      \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b}, -0.5 \cdot \frac{b}{c}\right)}} \]
      Proof
      (/.f64 1/2 (fma.f64 1/2 (/.f64 a b) (*.f64 -1/2 (/.f64 b c)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/2 (fma.f64 1/2 (/.f64 a b) (*.f64 (Rewrite<= metadata-eval (/.f64 2 -4)) (/.f64 b c)))): 2 points increase in error, 0 points decrease in error
      (/.f64 1/2 (fma.f64 1/2 (/.f64 a b) (*.f64 (/.f64 2 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -4) (sqrt.f64 -4)))) (/.f64 b c)))): 0 points increase in error, 2 points decrease in error
      (/.f64 1/2 (fma.f64 1/2 (/.f64 a b) (*.f64 (/.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -4) 2))) (/.f64 b c)))): 2 points increase in error, 0 points decrease in error
      (/.f64 1/2 (fma.f64 1/2 (/.f64 a b) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 2 b) (*.f64 (pow.f64 (sqrt.f64 -4) 2) c))))): 0 points increase in error, 2 points decrease in error
      (/.f64 1/2 (fma.f64 1/2 (/.f64 a b) (/.f64 (*.f64 2 b) (Rewrite<= *-commutative_binary64 (*.f64 c (pow.f64 (sqrt.f64 -4) 2)))))): 0 points increase in error, 2 points decrease in error
      (/.f64 1/2 (fma.f64 1/2 (/.f64 a b) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 b (*.f64 c (pow.f64 (sqrt.f64 -4) 2))))))): 2 points increase in error, 0 points decrease in error
      (/.f64 1/2 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/2 (/.f64 a b)) (*.f64 2 (/.f64 b (*.f64 c (pow.f64 (sqrt.f64 -4) 2))))))): 0 points increase in error, 2 points decrease in error
      (/.f64 1/2 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 b (*.f64 c (pow.f64 (sqrt.f64 -4) 2)))) (*.f64 1/2 (/.f64 a b))))): 0 points increase in error, 0 points decrease in error
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, -0.5 \cdot \frac{b}{c}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error10.4
Cost7624
\[\begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-104}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, -0.5 \cdot \frac{b}{c}\right)}\\ \end{array} \]
Alternative 2
Error13.8
Cost7368
\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
Alternative 3
Error14.0
Cost7368
\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, -0.5 \cdot \frac{b}{c}\right)}\\ \end{array} \]
Alternative 4
Error22.8
Cost580
\[\begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-268}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
Alternative 5
Error39.6
Cost388
\[\begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
Alternative 6
Error22.8
Cost388
\[\begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-235}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
Alternative 7
Error56.5
Cost192
\[\frac{c}{b} \]

Error

Reproduce

herbie shell --seed 2022340 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))