Quadratic roots, full range

Average Error: 34.5 → 10.9

Time: 17.9s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \frac{-0.5}{\frac{c}{b}}\right)}\\ \end{array} \]

FPCore

(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 -5.8e+93)
   (- (/ c b) (/ b a))
   (if (<= b 6e-84)
     (/ (- (sqrt (+ (* b b) (* c (* a -4.0)))) b) (* a 2.0))
     (/ 0.5 (fma 0.5 (/ a b) (/ -0.5 (/ c 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 <= -5.8e+93) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6e-84) {
		tmp = (sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.5 / fma(0.5, (a / b), (-0.5 / (c / b)));
	}
	return tmp;
}

Julia

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 <= -5.8e+93)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6e-84)
		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(c / b))));
	end
	return tmp
end

Wolfram

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, -5.8e+93], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-84], 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[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.7999999999999997e93

   1. Initial program 45.5

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

   2. Simplified 45.5

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

      [Start] 45.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 45.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 3.9

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

   4. Simplified 3.9

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
      Proof

      [Start] 3.9	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 3.9	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 3.9	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -5.7999999999999997e93 < b < 6.0000000000000002e-84

   1. Initial program 13.6

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

   if 6.0000000000000002e-84 < b

   1. Initial program 52.5

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

   2. Simplified 52.5

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

      [Start] 52.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 52.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 52.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 52.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 52.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 52.5	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 52.5	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      +-commutative [=>] 52.5	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{\frac{--1}{2}}{a} \]
      unsub-neg [=>] 52.5	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{\frac{--1}{2}}{a} \]
      fma-neg [=>] 52.5	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      associate-*l* [=>] 52.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      distribute-lft-neg-in [=>] 52.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      *-commutative [=>] 52.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      associate-*l* [=>] 52.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [=>] 52.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)} - b\right) \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [=>] 52.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
      metadata-eval [=>] 52.5	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   3. Applied egg-rr 46.6

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

   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. Simplified 11.0

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

      [Start] 64.0	\[ \frac{0.5}{2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}} + 0.5 \cdot \frac{a}{b}} \]
      +-commutative [=>] 64.0	\[ \frac{0.5}{\color{blue}{0.5 \cdot \frac{a}{b} + 2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}}} \]
      fma-def [=>] 64.0	\[ \frac{0.5}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b}, 2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}} \]
      associate-*r/ [=>] 64.0	\[ \frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \color{blue}{\frac{2 \cdot b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}}\right)} \]
      *-commutative [=>] 64.0	\[ \frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \frac{2 \cdot b}{\color{blue}{{\left(\sqrt{-4}\right)}^{2} \cdot c}}\right)} \]
      times-frac [=>] 64.0	\[ \frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \color{blue}{\frac{2}{{\left(\sqrt{-4}\right)}^{2}} \cdot \frac{b}{c}}\right)} \]
      unpow2 [=>] 64.0	\[ \frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \frac{2}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c}\right)} \]
      rem-square-sqrt [=>] 11.0	\[ \frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \frac{2}{\color{blue}{-4}} \cdot \frac{b}{c}\right)} \]
      metadata-eval [=>] 11.0	\[ \frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \color{blue}{-0.5} \cdot \frac{b}{c}\right)} \]

   6. Applied egg-rr 10.9

      \[\leadsto \frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \color{blue}{\frac{-0.5}{\frac{c}{b}}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \frac{-0.5}{\frac{c}{b}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.0
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-85}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \frac{-0.5}{\frac{c}{b}}\right)}\\ \end{array} \]

Alternative 2
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{a}{b} \cdot -0.5 + 0.5 \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 3
Error	14.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-26}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \frac{-0.5}{\frac{c}{b}}\right)}\\ \end{array} \]

Alternative 4
Error	14.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-29}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(0.5, \frac{a}{b}, \frac{-0.5}{\frac{c}{b}}\right)}\\ \end{array} \]

Alternative 5
Error	23.2
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-216}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{a}{b} \cdot -0.5 + 0.5 \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 6
Error	39.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-307}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	23.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-278}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8
Error	56.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023083 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))