quadp (p42, positive)

Average Accuracy: 46.9% → 87.2%

Time: 20.6s

Precision: binary64

Cost: 13964

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-221}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \cdot -0.5\\ \mathbf{elif}\;b \leq 1900000000000:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{-4 \cdot \left(c \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -5e+108)
   (- (/ c b) (/ b a))
   (if (<= b -6e-221)
     (* (/ (- b (sqrt (fma a (* c -4.0) (* b b)))) a) -0.5)
     (if (<= b 1900000000000.0)
       (/ (* c -2.0) (+ b (hypot b (sqrt (* -4.0 (* c a))))))
       (/ (- 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 <= -5e+108) {
		tmp = (c / b) - (b / a);
	} else if (b <= -6e-221) {
		tmp = ((b - sqrt(fma(a, (c * -4.0), (b * b)))) / a) * -0.5;
	} else if (b <= 1900000000000.0) {
		tmp = (c * -2.0) / (b + hypot(b, sqrt((-4.0 * (c * a)))));
	} else {
		tmp = -c / b;
	}
	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)
	tmp = 0.0
	if (b <= -5e+108)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -6e-221)
		tmp = Float64(Float64(Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) / a) * -0.5);
	elseif (b <= 1900000000000.0)
		tmp = Float64(Float64(c * -2.0) / Float64(b + hypot(b, sqrt(Float64(-4.0 * Float64(c * a))))));
	else
		tmp = Float64(Float64(-c) / b);
	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_] := If[LessEqual[b, -5e+108], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6e-221], N[(N[(N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[b, 1900000000000.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Target

Original	46.9%
Target	66.9%
Herbie	87.2%

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

Derivation

1. Split input into 4 regimes

2. if b < -4.99999999999999991e108

   1. Initial program 23.9%

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

   2. Simplified 23.8%

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

      [Start] 23.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 23.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 23.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 23.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 23.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 23.9	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 23.8	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 23.8	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 23.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Taylor expanded in b around -inf 94.4%

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

   4. Simplified 94.4%

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

      [Start] 94.4	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 94.4	\[ \frac{c}{b} + \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 94.4	\[ \frac{c}{b} + \frac{\color{blue}{-b}}{a} \]

   if -4.99999999999999991e108 < b < -6.0000000000000003e-221

   1. Initial program 88.1%

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

   2. Simplified 87.9%

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

      [Start] 88.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-/r* [=>] 88.1	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}} \]
      /-rgt-identity [<=] 88.1	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\color{blue}{\frac{a}{1}}} \]
      metadata-eval [<=] 88.1	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\frac{a}{\color{blue}{-1 \cdot -1}}} \]
      associate-/l/ [<=] 88.1	\[ \frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{\color{blue}{\frac{\frac{a}{-1}}{-1}}} \]
      associate-/l* [<=] 88.1	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot -1}{\frac{a}{-1}}} \]
      associate-*r/ [<=] 87.8	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{-1}{\frac{a}{-1}}} \]
      times-frac [<=] 88.1	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}{2 \cdot \frac{a}{-1}}} \]
      *-commutative [=>] 88.1	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}{\color{blue}{\frac{a}{-1} \cdot 2}} \]
      times-frac [=>] 87.9	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \cdot \frac{-1}{2}} \]

   if -6.0000000000000003e-221 < b < 1.9e12

   1. Initial program 62.3%

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

   2. Simplified 62.2%

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

      [Start] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 62.3	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 62.3	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 62.3	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 62.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Applied egg-rr 52.7%

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

   4. Applied egg-rr 52.7%

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

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{\frac{-0.5 \cdot \left(0 - -4 \cdot \left(c \cdot a\right)\right)}{a}}{-\left(\left(-b\right) - \mathsf{hypot}\left(b, \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\right)}} \]
      Proof

      [Start] 52.7	\[ \frac{-\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b\right)}{-a \cdot \left(-\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\right)} \cdot -0.5 \]
      associate-*l/ [=>] 52.7	\[ \color{blue}{\frac{\left(-\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b\right)\right) \cdot -0.5}{-a \cdot \left(-\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\right)}} \]
      distribute-rgt-neg-in [=>] 52.7	\[ \frac{\left(-\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b\right)\right) \cdot -0.5}{\color{blue}{a \cdot \left(-\left(-\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)}} \]
      associate-/r* [=>] 61.4	\[ \color{blue}{\frac{\frac{\left(-\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - b \cdot b\right)\right) \cdot -0.5}{a}}{-\left(-\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\right)}} \]

   6. Taylor expanded in c around 0 77.1%

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

   if 1.9e12 < b

   1. Initial program 12.6%

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

   2. Taylor expanded in b around inf 91.7%

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

   3. Simplified 91.7%

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

      [Start] 91.7	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 91.7	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 91.7	\[ \frac{\color{blue}{-c}}{b} \]

3. Recombined 4 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-221}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \cdot -0.5\\ \mathbf{elif}\;b \leq 1900000000000:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{-4 \cdot \left(c \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.2%
Cost	13964

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-293}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a}\\ \mathbf{elif}\;b \leq 850000000000:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Accuracy	87.2%
Cost	13964

\[\begin{array}{l} \mathbf{if}\;b \leq -4.65 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.75 \cdot 10^{-218}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a}\\ \mathbf{elif}\;b \leq 800000000000:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Accuracy	87.2%
Cost	13964

\[\begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+103}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -5.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \cdot -0.5\\ \mathbf{elif}\;b \leq 1000000000000:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Accuracy	83.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Accuracy	79.1%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-23}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-58}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Accuracy	79.0%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Accuracy	38.0%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 8
Accuracy	64.1%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 9
Accuracy	11.2%
Cost	192

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

Reproduce

herbie shell --seed 2023129 
(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)))