quadm (p42, negative)

Average Error: 53.66% → 10.09%

Time: 22.0s

Precision: binary64

Cost: 14028

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;-0.5 \cdot \left(\frac{c}{2 \cdot \left(b - \frac{a}{\frac{b}{c}}\right)} \cdot 4\right)\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-305}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - t_0}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{+127}:\\ \;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 (sqrt (fma c (* a -4.0) (* b b)))))
   (if (<= b -2e+117)
     (* -0.5 (* (/ c (* 2.0 (- b (/ a (/ b c))))) 4.0))
     (if (<= b 3.4e-305)
       (* -0.5 (/ (* c 4.0) (- b t_0)))
       (if (<= b 8.4e+127) (* -0.5 (/ (+ b t_0) a)) (- (/ c b) (/ 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 = sqrt(fma(c, (a * -4.0), (b * b)));
	double tmp;
	if (b <= -2e+117) {
		tmp = -0.5 * ((c / (2.0 * (b - (a / (b / c))))) * 4.0);
	} else if (b <= 3.4e-305) {
		tmp = -0.5 * ((c * 4.0) / (b - t_0));
	} else if (b <= 8.4e+127) {
		tmp = -0.5 * ((b + t_0) / a);
	} else {
		tmp = (c / b) - (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 = sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))
	tmp = 0.0
	if (b <= -2e+117)
		tmp = Float64(-0.5 * Float64(Float64(c / Float64(2.0 * Float64(b - Float64(a / Float64(b / c))))) * 4.0));
	elseif (b <= 3.4e-305)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b - t_0)));
	elseif (b <= 8.4e+127)
		tmp = Float64(-0.5 * Float64(Float64(b + t_0) / a));
	else
		tmp = Float64(Float64(c / b) - 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[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2e+117], N[(-0.5 * N[(N[(c / N[(2.0 * N[(b - N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.4e-305], N[(-0.5 * N[(N[(c * 4.0), $MachinePrecision] / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.4e+127], N[(-0.5 * N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	53.66%
Target	33.1%
Herbie	10.09%

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

Derivation

1. Split input into 4 regimes

2. if b < -2.0000000000000001e117

   1. Initial program 94.91

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

   2. Simplified 94.91

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

      [Start] 94.91	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 94.91	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 94.91	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 94.91	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 94.91	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 94.91	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 94.91	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 94.91	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 94.91	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 94.91	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 94.91	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 94.91	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 94.91	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 94.91	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 94.91	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Applied egg-rr 94.99

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

   4. Simplified 84.77

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

      [Start] 94.99	\[ \frac{-0.5}{\frac{a \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
      *-commutative [<=] 94.99	\[ \frac{-0.5}{\frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a}}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
      associate-/l* [<=] 94.99	\[ \color{blue}{\frac{-0.5 \cdot \left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a}} \]
      *-commutative [<=] 94.99	\[ \frac{\color{blue}{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot -0.5}}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a} \]
      associate-*l/ [<=] 94.99	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a} \cdot -0.5} \]
      *-commutative [=>] 94.99	\[ \color{blue}{-0.5 \cdot \frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a}} \]
      *-commutative [=>] 94.99	\[ -0.5 \cdot \frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\color{blue}{a \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}} \]
      associate-/r* [=>] 94.9	\[ -0.5 \cdot \color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{a}}{b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]

   5. Taylor expanded in b around -inf 85

      \[\leadsto -0.5 \cdot \frac{\frac{\left(c \cdot a\right) \cdot 4 + 0 \cdot \left(b \cdot b\right)}{a}}{b - \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + -1 \cdot b\right)}} \]

   6. Simplified 85

      \[\leadsto -0.5 \cdot \frac{\frac{\left(c \cdot a\right) \cdot 4 + 0 \cdot \left(b \cdot b\right)}{a}}{b - \color{blue}{\left(\frac{a}{\frac{b}{c}} \cdot 2 - b\right)}} \]
      Proof

      [Start] 85	\[ -0.5 \cdot \frac{\frac{\left(c \cdot a\right) \cdot 4 + 0 \cdot \left(b \cdot b\right)}{a}}{b - \left(2 \cdot \frac{c \cdot a}{b} + -1 \cdot b\right)} \]
      mul-1-neg [=>] 85	\[ -0.5 \cdot \frac{\frac{\left(c \cdot a\right) \cdot 4 + 0 \cdot \left(b \cdot b\right)}{a}}{b - \left(2 \cdot \frac{c \cdot a}{b} + \color{blue}{\left(-b\right)}\right)} \]
      unsub-neg [=>] 85	\[ -0.5 \cdot \frac{\frac{\left(c \cdot a\right) \cdot 4 + 0 \cdot \left(b \cdot b\right)}{a}}{b - \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}} \]
      *-commutative [=>] 85	\[ -0.5 \cdot \frac{\frac{\left(c \cdot a\right) \cdot 4 + 0 \cdot \left(b \cdot b\right)}{a}}{b - \left(\color{blue}{\frac{c \cdot a}{b} \cdot 2} - b\right)} \]
      *-commutative [=>] 85	\[ -0.5 \cdot \frac{\frac{\left(c \cdot a\right) \cdot 4 + 0 \cdot \left(b \cdot b\right)}{a}}{b - \left(\frac{\color{blue}{a \cdot c}}{b} \cdot 2 - b\right)} \]
      associate-/l* [=>] 85	\[ -0.5 \cdot \frac{\frac{\left(c \cdot a\right) \cdot 4 + 0 \cdot \left(b \cdot b\right)}{a}}{b - \left(\color{blue}{\frac{a}{\frac{b}{c}}} \cdot 2 - b\right)} \]

   7. Applied egg-rr 57.48

      \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a \cdot \left(c \cdot 4\right)}{a \cdot \left(b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)\right)}\right)} - 1\right)} \]

   8. Simplified 2.67

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{c}{2 \cdot \left(b - \frac{a}{\frac{b}{c}}\right)} \cdot 4\right)} \]
      Proof

      [Start] 57.48	\[ -0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{a \cdot \left(c \cdot 4\right)}{a \cdot \left(b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)\right)}\right)} - 1\right) \]
      expm1-def [=>] 26.55	\[ -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \left(c \cdot 4\right)}{a \cdot \left(b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 22.19	\[ -0.5 \cdot \color{blue}{\frac{a \cdot \left(c \cdot 4\right)}{a \cdot \left(b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)\right)}} \]
      *-lft-identity [<=] 22.19	\[ -0.5 \cdot \color{blue}{\left(1 \cdot \frac{a \cdot \left(c \cdot 4\right)}{a \cdot \left(b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)\right)}\right)} \]
      associate-*r/ [=>] 22.19	\[ -0.5 \cdot \color{blue}{\frac{1 \cdot \left(a \cdot \left(c \cdot 4\right)\right)}{a \cdot \left(b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)\right)}} \]
      associate-*l/ [<=] 22.23	\[ -0.5 \cdot \color{blue}{\left(\frac{1}{a \cdot \left(b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)\right)} \cdot \left(a \cdot \left(c \cdot 4\right)\right)\right)} \]
      associate-/r* [=>] 21.42	\[ -0.5 \cdot \left(\color{blue}{\frac{\frac{1}{a}}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)}} \cdot \left(a \cdot \left(c \cdot 4\right)\right)\right) \]
      associate-*l/ [=>] 12.32	\[ -0.5 \cdot \color{blue}{\frac{\frac{1}{a} \cdot \left(a \cdot \left(c \cdot 4\right)\right)}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)}} \]
      associate-/r/ [<=] 12.33	\[ -0.5 \cdot \frac{\color{blue}{\frac{1}{\frac{a}{a \cdot \left(c \cdot 4\right)}}}}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)} \]
      associate-/r* [=>] 3.03	\[ -0.5 \cdot \frac{\frac{1}{\color{blue}{\frac{\frac{a}{a}}{c \cdot 4}}}}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)} \]
      *-commutative [=>] 3.03	\[ -0.5 \cdot \frac{\frac{1}{\frac{\frac{a}{a}}{\color{blue}{4 \cdot c}}}}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)} \]
      associate-/r* [=>] 3.03	\[ -0.5 \cdot \frac{\frac{1}{\color{blue}{\frac{\frac{\frac{a}{a}}{4}}{c}}}}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)} \]
      *-inverses [=>] 3.03	\[ -0.5 \cdot \frac{\frac{1}{\frac{\frac{\color{blue}{1}}{4}}{c}}}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)} \]
      associate-/r/ [=>] 2.89	\[ -0.5 \cdot \frac{\color{blue}{\frac{1}{\frac{1}{4}} \cdot c}}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)} \]
      metadata-eval [=>] 2.89	\[ -0.5 \cdot \frac{\frac{1}{\color{blue}{0.25}} \cdot c}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)} \]
      metadata-eval [=>] 2.89	\[ -0.5 \cdot \frac{\color{blue}{4} \cdot c}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)} \]
      *-commutative [<=] 2.89	\[ -0.5 \cdot \frac{\color{blue}{c \cdot 4}}{b + \left(b - \left(a \cdot 2\right) \cdot \frac{c}{b}\right)} \]

   if -2.0000000000000001e117 < b < 3.4000000000000001e-305

   1. Initial program 52.01

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

   2. Simplified 52.09

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

      [Start] 52.01	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 52.01	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 52.01	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 52.01	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 52.07	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 52.07	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 52.07	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 52.07	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 52.03	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 52.03	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 52.03	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 52.03	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 52.03	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 52.03	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 52.03	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Applied egg-rr 58.97

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

   4. Simplified 25.37

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

      [Start] 58.97	\[ \frac{-0.5}{\frac{a \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
      *-commutative [<=] 58.97	\[ \frac{-0.5}{\frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a}}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
      associate-/l* [<=] 58.94	\[ \color{blue}{\frac{-0.5 \cdot \left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a}} \]
      *-commutative [<=] 58.94	\[ \frac{\color{blue}{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot -0.5}}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a} \]
      associate-*l/ [<=] 58.93	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a} \cdot -0.5} \]
      *-commutative [=>] 58.93	\[ \color{blue}{-0.5 \cdot \frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a}} \]
      *-commutative [=>] 58.93	\[ -0.5 \cdot \frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\color{blue}{a \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}} \]
      associate-/r* [=>] 52	\[ -0.5 \cdot \color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{a}}{b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]

   5. Taylor expanded in c around 0 13.48

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

   if 3.4000000000000001e-305 < b < 8.39999999999999967e127

   1. Initial program 13.49

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

   2. Simplified 13.59

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

      [Start] 13.49	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 13.49	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 13.49	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 13.66	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      distribute-neg-frac [<=] 13.66	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2}{\color{blue}{-\frac{-1}{a}}}} \]
      associate-/r/ [=>] 13.74	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \left(-\frac{-1}{a}\right)} \]
      neg-mul-1 [=>] 13.74	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \color{blue}{\left(-1 \cdot \frac{-1}{a}\right)} \]
      *-commutative [<=] 13.74	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \]
      associate-/r/ [<=] 13.74	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \color{blue}{\frac{-1}{\frac{a}{-1}}} \]
      times-frac [<=] 13.49	\[ \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot -1}{2 \cdot \frac{a}{-1}}} \]
      *-commutative [<=] 13.49	\[ \frac{\color{blue}{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot \frac{a}{-1}} \]
      times-frac [=>] 13.53	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 13.53	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 13.53	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 13.53	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      neg-mul-1 [<=] 13.53	\[ -0.5 \cdot \color{blue}{\left(-\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      distribute-frac-neg [<=] 13.53	\[ -0.5 \cdot \color{blue}{\frac{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{a}} \]

   if 8.39999999999999967e127 < b

   1. Initial program 86.11

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

   2. Simplified 86.17

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

      [Start] 86.11	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 86.11	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 86.11	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 86.11	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 86.17	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 86.17	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 86.17	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 86.17	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 86.17	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 86.17	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 86.17	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 86.17	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 86.17	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 86.17	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 86.17	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Taylor expanded in a around 0 5.33

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

   4. Simplified 5.33

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

      [Start] 5.33	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 5.33	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 5.33	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 10.09

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;-0.5 \cdot \left(\frac{c}{2 \cdot \left(b - \frac{a}{\frac{b}{c}}\right)} \cdot 4\right)\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-305}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{+127}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.27%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-14}:\\ \;\;\;\;-0.5 \cdot \left(\frac{c}{2 \cdot \left(b - \frac{a}{\frac{b}{c}}\right)} \cdot 4\right)\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+127}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	16.23%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-14}:\\ \;\;\;\;-0.5 \cdot \left(\frac{c}{2 \cdot \left(b - \frac{a}{\frac{b}{c}}\right)} \cdot 4\right)\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	16.36%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-14}:\\ \;\;\;\;-0.5 \cdot \left(\frac{c}{2 \cdot \left(b - \frac{a}{\frac{b}{c}}\right)} \cdot 4\right)\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	21.24%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{-11}:\\ \;\;\;\;-0.5 \cdot \left(\frac{c}{2 \cdot \left(b - \frac{a}{\frac{b}{c}}\right)} \cdot 4\right)\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-81}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{-4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	22.13%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.04 \cdot 10^{-13}:\\ \;\;\;\;-0.5 \cdot \left(\frac{c}{2 \cdot \left(b - \frac{a}{\frac{b}{c}}\right)} \cdot 4\right)\\ \mathbf{elif}\;b \leq 1.52 \cdot 10^{-24}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	34.61%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-304}:\\ \;\;\;\;-0.5 \cdot \left(\frac{c}{2 \cdot \left(b - \frac{a}{\frac{b}{c}}\right)} \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 7
Error	34.91%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 8
Error	62.06%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -52000000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 9
Error	34.94%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 10
Error	97.41%
Cost	192

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

Alternative 11
Error	88.53%
Cost	192

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

Reproduce

herbie shell --seed 2023089 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))