quadp (p42, positive)

Average Error: 34.2 → 10.3

Time: 21.5s

Precision: binary64

Cost: 28104

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-c, a \cdot 4, \left(c \cdot a\right) \cdot 4\right)\\ \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right) + \left(t_0 + t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \frac{a}{b} + -0.5 \cdot \frac{b}{c}}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (- c) (* a 4.0) (* (* c a) 4.0))))
   (if (<= b -2e+153)
     (- (/ c b) (/ b a))
     (if (<= b 2.2e-117)
       (/ (- (sqrt (+ (fma b b (* (* c a) -4.0)) (+ t_0 t_0))) b) (* a 2.0))
       (/ 0.5 (+ (* 0.5 (/ a b)) (* -0.5 (/ 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 t_0 = fma(-c, (a * 4.0), ((c * a) * 4.0));
	double tmp;
	if (b <= -2e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.2e-117) {
		tmp = (sqrt((fma(b, b, ((c * a) * -4.0)) + (t_0 + t_0))) - b) / (a * 2.0);
	} else {
		tmp = 0.5 / ((0.5 * (a / b)) + (-0.5 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	t_0 = fma(Float64(-c), Float64(a * 4.0), Float64(Float64(c * a) * 4.0))
	tmp = 0.0
	if (b <= -2e+153)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.2e-117)
		tmp = Float64(Float64(sqrt(Float64(fma(b, b, Float64(Float64(c * a) * -4.0)) + Float64(t_0 + t_0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.5 / Float64(Float64(0.5 * Float64(a / b)) + Float64(-0.5 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) * N[(a * 4.0), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+153], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.2e-117], N[(N[(N[Sqrt[N[(N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	34.2
Target	21.2
Herbie	10.3

\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if b < -2e153

   1. Initial program 63.8

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

   2. Simplified 63.8

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

      [Start] 63.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 63.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 63.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 63.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 63.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 63.8	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 63.8	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 63.8	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 63.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Taylor expanded in b around -inf 12.7

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

   4. Simplified 3.4

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

      [Start] 12.7	\[ \left(2 \cdot \frac{c \cdot a}{b} + -2 \cdot b\right) \cdot \frac{0.5}{a} \]
      fma-def [=>] 12.7	\[ \color{blue}{\mathsf{fma}\left(2, \frac{c \cdot a}{b}, -2 \cdot b\right)} \cdot \frac{0.5}{a} \]
      associate-/l* [=>] 3.4	\[ \mathsf{fma}\left(2, \color{blue}{\frac{c}{\frac{b}{a}}}, -2 \cdot b\right) \cdot \frac{0.5}{a} \]
      *-commutative [=>] 3.4	\[ \mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot -2}\right) \cdot \frac{0.5}{a} \]

   5. Taylor expanded in c around 0 3.0

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

   6. Simplified 3.0

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

      [Start] 3.0	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 3.0	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 3.0	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -2e153 < b < 2.2000000000000001e-117

   1. Initial program 10.9

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

   2. Applied egg-rr 10.9

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

   if 2.2000000000000001e-117 < b

   1. Initial program 51.6

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

   2. Simplified 51.6

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

      [Start] 51.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 51.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 51.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 51.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 51.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 51.6	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 51.6	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 51.6	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 51.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Applied egg-rr 46.7

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

   4. Applied egg-rr 46.7

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b} \cdot a}} \]

   5. Taylor expanded in b around inf 64.0

      \[\leadsto \frac{0.5}{\color{blue}{\left(2 \cdot \frac{b}{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)} + 0.5 \cdot \frac{1}{b}\right)} \cdot a} \]

   6. Simplified 20.1

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

      [Start] 64.0	\[ \frac{0.5}{\left(2 \cdot \frac{b}{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)} + 0.5 \cdot \frac{1}{b}\right) \cdot a} \]
      associate-*r/ [=>] 64.0	\[ \frac{0.5}{\left(\color{blue}{\frac{2 \cdot b}{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}} + 0.5 \cdot \frac{1}{b}\right) \cdot a} \]
      *-commutative [=>] 64.0	\[ \frac{0.5}{\left(\frac{2 \cdot b}{\color{blue}{\left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right) \cdot c}} + 0.5 \cdot \frac{1}{b}\right) \cdot a} \]
      times-frac [=>] 64.0	\[ \frac{0.5}{\left(\color{blue}{\frac{2}{a \cdot {\left(\sqrt{-4}\right)}^{2}} \cdot \frac{b}{c}} + 0.5 \cdot \frac{1}{b}\right) \cdot a} \]
      *-commutative [=>] 64.0	\[ \frac{0.5}{\left(\frac{2}{\color{blue}{{\left(\sqrt{-4}\right)}^{2} \cdot a}} \cdot \frac{b}{c} + 0.5 \cdot \frac{1}{b}\right) \cdot a} \]
      associate-/r* [=>] 64.0	\[ \frac{0.5}{\left(\color{blue}{\frac{\frac{2}{{\left(\sqrt{-4}\right)}^{2}}}{a}} \cdot \frac{b}{c} + 0.5 \cdot \frac{1}{b}\right) \cdot a} \]
      unpow2 [=>] 64.0	\[ \frac{0.5}{\left(\frac{\frac{2}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}}}{a} \cdot \frac{b}{c} + 0.5 \cdot \frac{1}{b}\right) \cdot a} \]
      rem-square-sqrt [=>] 20.1	\[ \frac{0.5}{\left(\frac{\frac{2}{\color{blue}{-4}}}{a} \cdot \frac{b}{c} + 0.5 \cdot \frac{1}{b}\right) \cdot a} \]
      metadata-eval [=>] 20.1	\[ \frac{0.5}{\left(\frac{\color{blue}{-0.5}}{a} \cdot \frac{b}{c} + 0.5 \cdot \frac{1}{b}\right) \cdot a} \]
      associate-*r/ [=>] 20.1	\[ \frac{0.5}{\left(\frac{-0.5}{a} \cdot \frac{b}{c} + \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \cdot a} \]
      metadata-eval [=>] 20.1	\[ \frac{0.5}{\left(\frac{-0.5}{a} \cdot \frac{b}{c} + \frac{\color{blue}{0.5}}{b}\right) \cdot a} \]

   7. Taylor expanded in a around 0 11.6

      \[\leadsto \frac{0.5}{\color{blue}{0.5 \cdot \frac{a}{b} + -0.5 \cdot \frac{b}{c}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right) + \left(\mathsf{fma}\left(-c, a \cdot 4, \left(c \cdot a\right) \cdot 4\right) + \mathsf{fma}\left(-c, a \cdot 4, \left(c \cdot a\right) \cdot 4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \frac{a}{b} + -0.5 \cdot \frac{b}{c}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-111}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \frac{a}{b} + -0.5 \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 2
Error	10.3
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \frac{a}{b} + -0.5 \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 3
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -0.012:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{-110}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \frac{a}{b} + -0.5 \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 4
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -0.0085:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \frac{a}{b} + -0.5 \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 5
Error	22.6
Cost	964

\[\begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-217}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \frac{a}{b} + -0.5 \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 6
Error	39.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 420000000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7
Error	22.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-249}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8
Error	62.4
Cost	192

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

Alternative 9
Error	57.1
Cost	192

\[\frac{c}{b} \]

Reproduce

herbie shell --seed 2023187 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))