quadp (p42, positive)

Percentage Accurate: 52.8% → 84.0%

Time: 12.9s

Precision: binary64

Cost: 14417

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;\frac{b - t_0}{a} \cdot -0.5\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-98} \lor \neg \left(b \leq 1.6 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{-0.5}{a} - t_0 \cdot \frac{-0.5}{a}\\ \end{array} \]

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 (sqrt (fma a (* c -4.0) (* b b)))))
   (if (<= b -2.8e+59)
     (- (/ c b) (/ b a))
     (if (<= b 1.95e-147)
       (* (/ (- b t_0) a) -0.5)
       (if (or (<= b 4.3e-98) (not (<= b 1.6e-16)))
         (/ (- c) b)
         (- (* b (/ -0.5 a)) (* t_0 (/ -0.5 a))))))))

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 = sqrt(fma(a, (c * -4.0), (b * b)));
	double tmp;
	if (b <= -2.8e+59) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-147) {
		tmp = ((b - t_0) / a) * -0.5;
	} else if ((b <= 4.3e-98) || !(b <= 1.6e-16)) {
		tmp = -c / b;
	} else {
		tmp = (b * (-0.5 / a)) - (t_0 * (-0.5 / a));
	}
	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 = sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))
	tmp = 0.0
	if (b <= -2.8e+59)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.95e-147)
		tmp = Float64(Float64(Float64(b - t_0) / a) * -0.5);
	elseif ((b <= 4.3e-98) || !(b <= 1.6e-16))
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(b * Float64(-0.5 / a)) - Float64(t_0 * Float64(-0.5 / a)));
	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[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.8e+59], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-147], N[(N[(N[(b - t$95$0), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], If[Or[LessEqual[b, 4.3e-98], N[Not[LessEqual[b, 1.6e-16]], $MachinePrecision]], N[((-c) / b), $MachinePrecision], N[(N[(b * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	52.8%
Target	71.2%
Herbie	84.0%

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

Derivation

1. Split input into 4 regimes

2. if b < -2.7999999999999998e59

   1. Initial program 57.1%

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

   2. Simplified 57.2%

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

      [Start] 57.1%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 57.1%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 57.1%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 57.1%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 57.1%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 57.1%	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 57.1%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around -inf 96.7%

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

   4. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
      Step-by-step derivation

      [Start] 96.7%	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 96.7%	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 96.7%	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -2.7999999999999998e59 < b < 1.9499999999999999e-147

   1. Initial program 90.9%

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

   2. Simplified 90.9%

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

      [Start] 90.9%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-/r* [=>] 90.9%	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}} \]
      /-rgt-identity [<=] 90.9%	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\color{blue}{\frac{a}{1}}} \]
      metadata-eval [<=] 90.9%	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\frac{a}{\color{blue}{--1}}} \]
      metadata-eval [=>] 90.9%	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\frac{a}{\color{blue}{1}}} \]
      metadata-eval [<=] 90.9%	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\frac{a}{\color{blue}{-1 \cdot -1}}} \]
      associate-/l/ [<=] 90.9%	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\color{blue}{\frac{\frac{a}{-1}}{-1}}} \]
      associate-/l* [<=] 90.9%	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot -1}{\frac{a}{-1}}} \]
      associate-*r/ [<=] 90.8%	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{-1}{\frac{a}{-1}}} \]
      times-frac [<=] 90.9%	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}{2 \cdot \frac{a}{-1}}} \]
      *-commutative [=>] 90.9%	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}{\color{blue}{\frac{a}{-1} \cdot 2}} \]
      times-frac [=>] 90.9%	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \cdot \frac{-1}{2}} \]

   if 1.9499999999999999e-147 < b < 4.29999999999999988e-98 or 1.60000000000000011e-16 < b

   1. Initial program 13.2%

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

   2. Simplified 13.2%

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

      [Start] 13.2%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 13.2%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 13.2%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 13.2%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 13.2%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 13.2%	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 13.2%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Taylor expanded in b around inf 88.2%

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

   4. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Step-by-step derivation

      [Start] 88.2%	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 88.2%	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 88.2%	\[ \frac{\color{blue}{-c}}{b} \]

   if 4.29999999999999988e-98 < b < 1.60000000000000011e-16

   1. Initial program 72.7%

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

   2. Simplified 72.7%

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

      [Start] 72.7%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      neg-sub0 [=>] 72.7%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-+l- [=>] 72.7%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 72.7%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 72.7%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]
      *-commutative [=>] 72.7%	\[ \frac{\color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}}{2 \cdot a} \]
      associate-*r/ [<=] 72.7%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{2 \cdot a}} \]

   3. Applied egg-rr 72.8%

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

      [Start] 72.7%	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 72.7%	\[ \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)} \]
      sub-neg [=>] 72.7%	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\right)} \]
      distribute-lft-in [=>] 72.8%	\[ \color{blue}{\frac{-0.5}{a} \cdot b + \frac{-0.5}{a} \cdot \left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \cdot -0.5\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-98} \lor \neg \left(b \leq 1.6 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{-0.5}{a} - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \frac{-0.5}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.0%
Cost	14417

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;\frac{b - t_0}{a} \cdot -0.5\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-98} \lor \neg \left(b \leq 1.6 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{-0.5}{a} - t_0 \cdot \frac{-0.5}{a}\\ \end{array} \]

Alternative 2
Accuracy	83.9%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \cdot -0.5\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-98} \lor \neg \left(b \leq 3.3 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \end{array} \]

Alternative 3
Accuracy	83.6%
Cost	7889

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-147} \lor \neg \left(b \leq 4.3 \cdot 10^{-98}\right) \land b \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Accuracy	84.0%
Cost	7889

\[\begin{array}{l} t_0 := b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;-0.5 \cdot \frac{t_0}{a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-98} \lor \neg \left(b \leq 1.6 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot t_0\\ \end{array} \]

Alternative 5
Accuracy	77.9%
Cost	7633

\[\begin{array}{l} \mathbf{if}\;b \leq -265000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-147} \lor \neg \left(b \leq 4.3 \cdot 10^{-98}\right) \land b \leq 1.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Accuracy	78.0%
Cost	7633

\[\begin{array}{l} t_0 := b - \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -18500000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;-0.5 \cdot \frac{t_0}{a}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-98} \lor \neg \left(b \leq 2.1 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot t_0\\ \end{array} \]

Alternative 7
Accuracy	67.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8
Accuracy	43.6%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 3.85 \cdot 10^{-268}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Accuracy	67.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 10
Accuracy	11.2%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023165 
(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)))