jeff quadratic root 2

Average Accuracy: 69.5% → 88.9%

Time: 26.7s

Precision: binary64

Cost: 38052

Math

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

↓

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, \sqrt{a \cdot -4} \cdot \sqrt{c}\right) \cdot \frac{0.5}{a} + b \cdot \frac{-0.5}{a}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-259}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(a \cdot \frac{c}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (- b) b))
        (t_1 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_2
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_1)) (/ (- t_1 b) (* 2.0 a)))))
   (if (<= t_2 (- INFINITY))
     (if (>= b 0.0)
       (* c (/ -2.0 (+ b (sqrt (fma b b (* c (* a -4.0)))))))
       (+
        (* (hypot b (* (sqrt (* a -4.0)) (sqrt c))) (/ 0.5 a))
        (* b (/ -0.5 a))))
     (if (<= t_2 -5e-259)
       t_2
       (if (<= t_2 0.0)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- (* 2.0 (* a (/ c b))) b) b))
           (/ t_0 (* 2.0 a)))
         (if (<= t_2 5e+262)
           t_2
           (if (>= b 0.0) (/ (* 2.0 c) t_0) (- (/ c b) (/ b a)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c))));
	} else {
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	}
	return tmp;
}

↓

double code(double a, double b, double c) {
	double t_0 = -b - b;
	double t_1 = sqrt(((b * b) - (c * (4.0 * a))));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_1);
	} else {
		tmp = (t_1 - b) / (2.0 * a);
	}
	double t_2 = tmp;
	double tmp_2;
	if (t_2 <= -((double) INFINITY)) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (b + sqrt(fma(b, b, (c * (a * -4.0))))));
		} else {
			tmp_3 = (hypot(b, (sqrt((a * -4.0)) * sqrt(c))) * (0.5 / a)) + (b * (-0.5 / a));
		}
		tmp_2 = tmp_3;
	} else if (t_2 <= -5e-259) {
		tmp_2 = t_2;
	} else if (t_2 <= 0.0) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (((2.0 * (a * (c / b))) - b) - b);
		} else {
			tmp_4 = t_0 / (2.0 * a);
		}
		tmp_2 = tmp_4;
	} else if (t_2 <= 5e+262) {
		tmp_2 = t_2;
	} else if (b >= 0.0) {
		tmp_2 = (2.0 * c) / t_0;
	} else {
		tmp_2 = (c / b) - (b / a);
	}
	return tmp_2;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))));
	else
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
	end
	return tmp
end

↓

function code(a, b, c)
	t_0 = Float64(Float64(-b) - b)
	t_1 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_1));
	else
		tmp = Float64(Float64(t_1 - b) / Float64(2.0 * a));
	end
	t_2 = tmp
	tmp_2 = 0.0
	if (t_2 <= Float64(-Inf))
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(-2.0 / Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))))));
		else
			tmp_3 = Float64(Float64(hypot(b, Float64(sqrt(Float64(a * -4.0)) * sqrt(c))) * Float64(0.5 / a)) + Float64(b * Float64(-0.5 / a)));
		end
		tmp_2 = tmp_3;
	elseif (t_2 <= -5e-259)
		tmp_2 = t_2;
	elseif (t_2 <= 0.0)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(Float64(2.0 * Float64(a * Float64(c / b))) - b) - b));
		else
			tmp_4 = Float64(t_0 / Float64(2.0 * a));
		end
		tmp_2 = tmp_4;
	elseif (t_2 <= 5e+262)
		tmp_2 = t_2;
	elseif (b >= 0.0)
		tmp_2 = Float64(Float64(2.0 * c) / t_0);
	else
		tmp_2 = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp_2
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[t$95$2, (-Infinity)], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[b ^ 2 + N[(N[Sqrt[N[(a * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] + N[(b * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$2, -5e-259], t$95$2, If[LessEqual[t$95$2, 0.0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(N[(N[(2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$2, 5e+262], t$95$2, If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < -inf.0

   1. Initial program 0.0%

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

   2. Simplified 0.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ } \end{array}} \]
      Proof

      [Start] 0.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   3. Applied egg-rr 32.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{0.5}{a} + \left(-b\right) \cdot \frac{0.5}{a}\\ \end{array} \]
      Proof

      [Start] 0.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \end{array} \]
      *-commutative [=>] 0.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b\right)\\ \end{array} \]
      sub-neg [=>] 0.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} + \left(-b\right)\right)}\\ \end{array} \]
      distribute-rgt-in [=>] 0.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \frac{0.5}{a} + \left(-b\right) \cdot \frac{0.5}{a}\\ \end{array} \]
      fma-udef [=>] 0.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \frac{0.5}{a} + \left(-b\right) \cdot \frac{0.5}{a}\\ \end{array} \]
      add-sqr-sqrt [=>] 0.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}} \cdot \frac{0.5}{a} + \left(-b\right) \cdot \frac{0.5}{a}\\ \end{array} \]
      hypot-def [=>] 32.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{0.5}{a} + \left(-b\right) \cdot \frac{0.5}{a}\\ \end{array} \]

   4. Applied egg-rr 66.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, \sqrt{a \cdot -4} \cdot \sqrt{c}\right) \cdot \frac{0.5}{a} + \left(-b\right) \cdot \frac{0.5}{a}\\ \end{array} \]
      Proof

      [Start] 32.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{0.5}{a} + \left(-b\right) \cdot \frac{0.5}{a}\\ \end{array} \]
      sqrt-prod [=>] 66.6	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, \sqrt{c} \cdot \sqrt{a \cdot -4}\right) \cdot \frac{0.5}{a} + \left(-b\right) \cdot \frac{0.5}{a}\\ \end{array} \]
      *-commutative [=>] 66.6	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, \sqrt{a \cdot -4} \cdot \sqrt{c}\right) \cdot \frac{0.5}{a} + \left(-b\right) \cdot \frac{0.5}{a}\\ \end{array} \]

   if -inf.0 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < -4.99999999999999977e-259 or -0.0 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < 5.00000000000000008e262

   1. Initial program 95.5%

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

   if -4.99999999999999977e-259 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < -0.0

   1. Initial program 44.9%

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

   2. Taylor expanded in b around -inf 44.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]

   3. Taylor expanded in b around inf 80.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{c \cdot a}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]

   4. Simplified 83.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]
      Proof

      [Start] 80.1	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \frac{c \cdot a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]
      associate-/l* [=>] 83.5	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\frac{c}{\frac{b}{a}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]
      associate-/r/ [=>] 83.5	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]

   if 5.00000000000000008e262 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)))

   1. Initial program 9.2%

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

   2. Taylor expanded in b around inf 15.6%

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

   3. Taylor expanded in b around -inf 75.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]

   4. Simplified 75.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
      Proof

      [Start] 75.6	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
      mul-1-neg [=>] 75.6	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]
      unsub-neg [=>] 75.6	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

3. Recombined 4 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, \sqrt{a \cdot -4} \cdot \sqrt{c}\right) \cdot \frac{0.5}{a} + b \cdot \frac{-0.5}{a}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq -5 \cdot 10^{-259}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(a \cdot \frac{c}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.3%
Cost	38052

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{2 \cdot c}{t_0}\\ t_2 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-259}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(a \cdot \frac{c}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Accuracy	89.5%
Cost	7952

\[\begin{array}{l} t_0 := \frac{c}{b} - \frac{b}{a}\\ t_1 := \left(-b\right) - b\\ t_2 := \frac{2 \cdot c}{t_1}\\ t_3 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{+151}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(a \cdot \frac{c}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{2 \cdot a}\\ \end{array} \]

Alternative 3
Accuracy	84.0%
Cost	7756

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{2 \cdot c}{t_0}\\ t_2 := \frac{t_0}{2 \cdot a}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.85 \cdot 10^{-90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(a \cdot \frac{c}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	78.7%
Cost	7696

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{t_0}{2 \cdot a}\\ t_2 := \frac{2 \cdot c}{t_0}\\ t_3 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -5.6 \cdot 10^{-39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(a \cdot \frac{c}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	71.0%
Cost	7368

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{-39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \end{array} \]

Alternative 6
Accuracy	64.4%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \]

Alternative 7
Accuracy	64.4%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \]

Alternative 8
Accuracy	64.6%
Cost	1092

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

Alternative 9
Accuracy	64.3%
Cost	644

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

Reproduce

herbie shell --seed 2023135 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))