Quadratic roots, full range

Average Error: 33.6 → 8.5

Time: 16.0s

Precision: binary64

Cost: 13964

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-190}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3200:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot -4}{\frac{b}{c}}\right)}\\ \end{array} \]

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 -1.65e+59)
   (- (/ c b) (/ b a))
   (if (<= b -3.8e-190)
     (/ (- (sqrt (+ (* b b) (* c (* a -4.0)))) b) (* a 2.0))
     (if (<= b 3200.0)
       (/ (* c -2.0) (+ b (hypot b (sqrt (* a (* c -4.0))))))
       (/ (* c -2.0) (fma b 2.0 (* 0.5 (/ (* a -4.0) (/ b c)))))))))

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 <= -1.65e+59) {
		tmp = (c / b) - (b / a);
	} else if (b <= -3.8e-190) {
		tmp = (sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0);
	} else if (b <= 3200.0) {
		tmp = (c * -2.0) / (b + hypot(b, sqrt((a * (c * -4.0)))));
	} else {
		tmp = (c * -2.0) / fma(b, 2.0, (0.5 * ((a * -4.0) / (b / c))));
	}
	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 <= -1.65e+59)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -3.8e-190)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	elseif (b <= 3200.0)
		tmp = Float64(Float64(c * -2.0) / Float64(b + hypot(b, sqrt(Float64(a * Float64(c * -4.0))))));
	else
		tmp = Float64(Float64(c * -2.0) / fma(b, 2.0, Float64(0.5 * Float64(Float64(a * -4.0) / Float64(b / c)))));
	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, -1.65e+59], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.8e-190], 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], If[LessEqual[b, 3200.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(b * 2.0 + N[(0.5 * N[(N[(a * -4.0), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.65e59

   1. Initial program 38.8

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

   2. Simplified 38.8

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

      [Start] 38.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 38.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 5.0

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

   4. Simplified 5.0

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

      [Start] 5.0	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 5.0	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 5.0	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -1.65e59 < b < -3.7999999999999998e-190

   1. Initial program 6.6

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

   if -3.7999999999999998e-190 < b < 3200

   1. Initial program 24.0

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

   2. Simplified 24.0

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

      [Start] 24.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 24.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 24.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 24.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 24.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 24.0	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 24.0	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 24.0	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 24.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Applied egg-rr 24.8

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

   4. Taylor expanded in a around 0 15.3

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

   5. Simplified 15.3

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

      [Start] 15.3	\[ \frac{-2 \cdot c}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]
      *-commutative [=>] 15.3	\[ \frac{\color{blue}{c \cdot -2}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]

   if 3200 < b

   1. Initial program 55.0

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

   2. Simplified 55.0

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

      [Start] 55.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 55.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 55.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 55.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 55.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 55.0	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 55.0	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 55.0	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 55.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Applied egg-rr 56.8

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

   4. Taylor expanded in a around 0 28.8

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

   5. Simplified 28.8

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

      [Start] 28.8	\[ \frac{-2 \cdot c}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]
      *-commutative [=>] 28.8	\[ \frac{\color{blue}{c \cdot -2}}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]

   6. Taylor expanded in b around inf 64.0

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

   7. Simplified 5.5

      \[\leadsto \frac{c \cdot -2}{\color{blue}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot -4}{\frac{b}{c}}\right)}} \]
      Proof

      [Start] 64.0	\[ \frac{c \cdot -2}{0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b} + 2 \cdot b} \]
      +-commutative [=>] 64.0	\[ \frac{c \cdot -2}{\color{blue}{2 \cdot b + 0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}}} \]
      *-commutative [=>] 64.0	\[ \frac{c \cdot -2}{\color{blue}{b \cdot 2} + 0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}} \]
      fma-def [=>] 64.0	\[ \frac{c \cdot -2}{\color{blue}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}\right)}} \]
      *-commutative [=>] 64.0	\[ \frac{c \cdot -2}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{\color{blue}{\left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right) \cdot c}}{b}\right)} \]
      associate-/l* [=>] 64.0	\[ \frac{c \cdot -2}{\mathsf{fma}\left(b, 2, 0.5 \cdot \color{blue}{\frac{a \cdot {\left(\sqrt{-4}\right)}^{2}}{\frac{b}{c}}}\right)} \]
      unpow2 [=>] 64.0	\[ \frac{c \cdot -2}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}{\frac{b}{c}}\right)} \]
      rem-square-sqrt [=>] 5.5	\[ \frac{c \cdot -2}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot \color{blue}{-4}}{\frac{b}{c}}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-190}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3200:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot -4}{\frac{b}{c}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-78}:\\ \;\;\;\;\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-b}{b - \frac{c}{b} \cdot a}\\ \end{array} \]

Alternative 2
Error	10.5
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.66 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-101}:\\ \;\;\;\;\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot -4}{\frac{b}{c}}\right)}\\ \end{array} \]

Alternative 3
Error	10.4
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.66 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, 0.5 \cdot \frac{a \cdot -4}{\frac{b}{c}}\right)}\\ \end{array} \]

Alternative 4
Error	13.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-b}{b - \frac{c}{b} \cdot a}\\ \end{array} \]

Alternative 5
Error	13.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-b}{b - \frac{c}{b} \cdot a}\\ \end{array} \]

Alternative 6
Error	22.8
Cost	1028

\[\begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-259}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-b}{b - \frac{c}{b} \cdot a}\\ \end{array} \]

Alternative 7
Error	22.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-240}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8
Error	45.5
Cost	256

\[\frac{-b}{a} \]

Alternative 9
Error	62.3
Cost	192

\[\frac{b}{a} \]

Reproduce

herbie shell --seed 2023060 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))