The quadratic formula (r2)

Average Accuracy: 46.3% → 84.1%

Time: 20.3s

Precision: binary64

Cost: 13896

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-39)
   (/ (- c) b)
   (if (<= b 4e+77)
     (/ (+ b (sqrt (fma b b (* a (* c -4.0))))) (* a -2.0))
     (/ (- 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 tmp;
	if (b <= -4.2e-39) {
		tmp = -c / b;
	} else if (b <= 4e+77) {
		tmp = (b + sqrt(fma(b, b, (a * (c * -4.0))))) / (a * -2.0);
	} 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)
	tmp = 0.0
	if (b <= -4.2e-39)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 4e+77)
		tmp = Float64(Float64(b + sqrt(fma(b, b, Float64(a * Float64(c * -4.0))))) / Float64(a * -2.0));
	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_] := If[LessEqual[b, -4.2e-39], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 4e+77], N[(N[(b + N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Target

Original	46.3%
Target	66.3%
Herbie	84.1%

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

Derivation

1. Split input into 3 regimes

2. if b < -4.19999999999999987e-39

   1. Initial program 15.5%

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

   2. Taylor expanded in b around -inf 87.9%

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

   3. Simplified 87.9%

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

      [Start] 87.9	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 87.9	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 87.9	\[ \frac{\color{blue}{-c}}{b} \]

   if -4.19999999999999987e-39 < b < 3.99999999999999993e77

   1. Initial program 77.1%

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

   2. Applied egg-rr 23.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a \cdot -2}\right)} - 1} \]
      Proof

      No proof available- proof too large to flatten.

   3. Simplified 77.1%

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

      [Start] 23.8	\[ e^{\mathsf{log1p}\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a \cdot -2}\right)} - 1 \]
      expm1-def [=>] 53.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a \cdot -2}\right)\right)} \]
      expm1-log1p [=>] 77.1	\[ \color{blue}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a \cdot -2}} \]
      associate-*l* [=>] 77.1	\[ \frac{b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -4\right)}\right)}}{a \cdot -2} \]

   if 3.99999999999999993e77 < b

   1. Initial program 33.3%

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

   2. Taylor expanded in b around inf 93.5%

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

   3. Simplified 93.5%

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

      [Start] 93.5	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 93.5	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 93.5	\[ \frac{\color{blue}{-b}}{a} \]

3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.2%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2
Accuracy	84.1%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-39}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.35 \cdot 10^{+77}:\\ \;\;\;\;\left(b + \sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3
Accuracy	78.7%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-39}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Accuracy	78.7%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Accuracy	39.1%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Accuracy	65.2%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -6.1 \cdot 10^{-227}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Accuracy	11.5%
Cost	192

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

Reproduce

herbie shell --seed 2023133 
(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)))