The quadratic formula (r2)

Percentage Accurate: 57.4% → 86.6%

Time: 26.0s

Precision: binary64

Cost: 14160

• Unsound rule application detected in e-graph. Results may not be sound.

• could not determine a ground truth

  1. a = 3.5075299573203835e+146
  2. b = -1.2188738977806838e-214
  3. c = -1.1298104663403366e-125

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -1.85 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-37}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{elif}\;b \leq -2.25 \cdot 10^{-119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.0002:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 (/ (- c) b)))
   (if (<= b -1.85e+28)
     t_0
     (if (<= b -1.2e-37)
       (* -0.5 (/ (+ b (sqrt (- (* b b) (* a (* c 4.0))))) a))
       (if (<= b -2.25e-119)
         t_0
         (if (<= b 0.0002)
           (* -0.5 (/ (+ b (sqrt (fma a (* c -4.0) (* b b)))) a))
           (/ (- b) 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 = -c / b;
	double tmp;
	if (b <= -1.85e+28) {
		tmp = t_0;
	} else if (b <= -1.2e-37) {
		tmp = -0.5 * ((b + sqrt(((b * b) - (a * (c * 4.0))))) / a);
	} else if (b <= -2.25e-119) {
		tmp = t_0;
	} else if (b <= 0.0002) {
		tmp = -0.5 * ((b + sqrt(fma(a, (c * -4.0), (b * b)))) / a);
	} else {
		tmp = -b / 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 = Float64(Float64(-c) / b)
	tmp = 0.0
	if (b <= -1.85e+28)
		tmp = t_0;
	elseif (b <= -1.2e-37)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 4.0))))) / a));
	elseif (b <= -2.25e-119)
		tmp = t_0;
	elseif (b <= 0.0002)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) / a));
	else
		tmp = Float64(Float64(-b) / 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[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -1.85e+28], t$95$0, If[LessEqual[b, -1.2e-37], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -2.25e-119], t$95$0, If[LessEqual[b, 0.0002], N[(-0.5 * N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]]

Target

Original	57.4%
Target	79.2%
Herbie	86.6%

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

Derivation

1. Split input into 4 regimes

2. if b < -1.85e28 or -1.19999999999999995e-37 < b < -2.2500000000000001e-119

   1. Initial program 21.6%

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

   2. Taylor expanded in b around -inf 88.8%

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

   3. Simplified 88.8%

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

      [Start] 88.8%	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 88.8%	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 88.8%	\[ \frac{\color{blue}{-c}}{b} \]

   if -1.85e28 < b < -1.19999999999999995e-37

   1. Initial program 74.1%

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

   2. Simplified 74.1%

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

      [Start] 74.1%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 74.1%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 74.1%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 73.8%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 73.8%	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 73.8%	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 73.8%	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 73.8%	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 73.8%	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 73.8%	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 74.1%	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 74.1%	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 74.1%	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 74.1%	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 74.1%	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 74.1%	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 74.0%	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Applied egg-rr 74.1%

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

      [Start] 74.1%	\[ -0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \]
      fma-udef [=>] 74.1%	\[ -0.5 \cdot \frac{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}}{a} \]
      associate-*r* [=>] 74.1%	\[ -0.5 \cdot \frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4} + b \cdot b}}{a} \]
      metadata-eval [<=] 74.1%	\[ -0.5 \cdot \frac{b + \sqrt{\left(a \cdot c\right) \cdot \color{blue}{\left(-4\right)} + b \cdot b}}{a} \]
      distribute-rgt-neg-in [<=] 74.1%	\[ -0.5 \cdot \frac{b + \sqrt{\color{blue}{\left(-\left(a \cdot c\right) \cdot 4\right)} + b \cdot b}}{a} \]
      *-commutative [<=] 74.1%	\[ -0.5 \cdot \frac{b + \sqrt{\left(-\color{blue}{4 \cdot \left(a \cdot c\right)}\right) + b \cdot b}}{a} \]
      +-commutative [=>] 74.1%	\[ -0.5 \cdot \frac{b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}}{a} \]
      sub-neg [<=] 74.1%	\[ -0.5 \cdot \frac{b + \sqrt{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a} \]
      *-commutative [=>] 74.1%	\[ -0.5 \cdot \frac{b + \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 4}}}{a} \]
      associate-*l* [=>] 74.1%	\[ -0.5 \cdot \frac{b + \sqrt{b \cdot b - \color{blue}{a \cdot \left(c \cdot 4\right)}}}{a} \]

   if -2.2500000000000001e-119 < b < 2.0000000000000001e-4

   1. Initial program 81.7%

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

   2. Simplified 81.7%

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

      [Start] 81.7%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 81.7%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 81.7%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 81.5%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 81.5%	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 81.5%	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 81.5%	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 81.5%	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 81.5%	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 81.5%	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 81.7%	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 81.7%	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 81.7%	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 81.7%	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 81.7%	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 81.7%	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 81.6%	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   if 2.0000000000000001e-4 < b

   1. Initial program 86.0%

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

   2. Taylor expanded in b around inf 100.0%

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

   3. Simplified 100.0%

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

      [Start] 100.0%	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 100.0%	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 100.0%	\[ \frac{\color{blue}{-b}}{a} \]

3. Recombined 4 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+28}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-37}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{elif}\;b \leq -2.25 \cdot 10^{-119}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 0.0002:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.6%
Cost	14160

\[\begin{array}{l} t_0 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -1.85 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-37}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{elif}\;b \leq -2.25 \cdot 10^{-119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.0002:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2
Accuracy	86.6%
Cost	7888

\[\begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := -0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{if}\;b \leq -1.85 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.82 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.0002:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3
Accuracy	86.3%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-100}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Accuracy	77.0%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}{a}\\ \end{array} \]

Alternative 5
Accuracy	77.0%
Cost	580

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

Alternative 6
Accuracy	46.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+37}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Accuracy	76.8%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-304}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8
Accuracy	1.9%
Cost	192

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

Alternative 9
Accuracy	9.9%
Cost	192

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

Reproduce

herbie shell --seed 2023277 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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