quadm (p42, negative)

Average Accuracy: 46.2% → 82.8%

Time: 22.7s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-172}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-172)
   (/ -0.5 (fma -0.5 (/ a b) (* 0.5 (/ b c))))
   (if (<= b 9.4e+48)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* a 2.0))
     (- (/ 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 tmp;
	if (b <= -1.9e-172) {
		tmp = -0.5 / fma(-0.5, (a / b), (0.5 * (b / c)));
	} else if (b <= 9.4e+48) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} 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)
	tmp = 0.0
	if (b <= -1.9e-172)
		tmp = Float64(-0.5 / fma(-0.5, Float64(a / b), Float64(0.5 * Float64(b / c))));
	elseif (b <= 9.4e+48)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(a * 2.0));
	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_] := If[LessEqual[b, -1.9e-172], N[(-0.5 / N[(-0.5 * N[(a / b), $MachinePrecision] + N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.4e+48], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	46.2%
Target	67.2%
Herbie	82.8%

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

Derivation

1. Split input into 3 regimes

2. if b < -1.89999999999999993e-172

   1. Initial program 22.5%

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

   2. Simplified 22.5%

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

      [Start] 22.5	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 22.5	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 22.5	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 22.5	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 22.5	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 22.5	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 22.5	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 22.5	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 22.5	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 22.5	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 22.5	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 22.5	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 22.5	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 22.5	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 22.5	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Applied egg-rr 22.5%

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

      [Start] 22.5	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \]
      fma-udef [=>] 22.5	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}\right) \]

   4. Applied egg-rr 29.5%

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

      [Start] 22.5	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}\right) \]
      associate-/r/ [<=] 22.5	\[ \color{blue}{\frac{-0.5}{\frac{a}{b + \sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}}}} \]
      +-commutative [=>] 22.5	\[ \frac{-0.5}{\frac{a}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}} \]
      add-sqr-sqrt [=>] 19.8	\[ \frac{-0.5}{\frac{a}{b + \sqrt{b \cdot b + \color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}}}}} \]
      hypot-def [=>] 29.5	\[ \frac{-0.5}{\frac{a}{b + \color{blue}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}} \]

   5. Taylor expanded in b around -inf 0.0%

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

   6. Simplified 78.8%

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

      [Start] 0.0	\[ \frac{-0.5}{-2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}} + -0.5 \cdot \frac{a}{b}} \]
      +-commutative [=>] 0.0	\[ \frac{-0.5}{\color{blue}{-0.5 \cdot \frac{a}{b} + -2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}}} \]
      fma-def [=>] 0.0	\[ \frac{-0.5}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{a}{b}, -2 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}} \]
      associate-*r/ [=>] 0.0	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{\frac{-2 \cdot b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}}\right)} \]
      *-commutative [=>] 0.0	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{-2 \cdot b}{\color{blue}{{\left(\sqrt{-4}\right)}^{2} \cdot c}}\right)} \]
      times-frac [=>] 0.0	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{\frac{-2}{{\left(\sqrt{-4}\right)}^{2}} \cdot \frac{b}{c}}\right)} \]
      unpow2 [=>] 0.0	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{-2}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c}\right)} \]
      rem-square-sqrt [=>] 78.8	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{-2}{\color{blue}{-4}} \cdot \frac{b}{c}\right)} \]
      metadata-eval [=>] 78.8	\[ \frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{0.5} \cdot \frac{b}{c}\right)} \]

   if -1.89999999999999993e-172 < b < 9.40000000000000025e48

   1. Initial program 82.7%

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

   if 9.40000000000000025e48 < b

   1. Initial program 40.6%

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

   2. Simplified 40.4%

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

      [Start] 40.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      *-lft-identity [<=] 40.6	\[ \color{blue}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]
      metadata-eval [<=] 40.6	\[ \color{blue}{\left(--1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      associate-*r/ [=>] 40.6	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \]
      associate-*l/ [<=] 40.4	\[ \color{blue}{\frac{--1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-neg-frac [<=] 40.4	\[ \color{blue}{\left(-\frac{-1}{2 \cdot a}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]
      distribute-lft-neg-in [<=] 40.4	\[ \color{blue}{-\frac{-1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      distribute-rgt-neg-out [<=] 40.4	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)} \]
      associate-/r* [=>] 40.4	\[ \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      metadata-eval [=>] 40.4	\[ \frac{\color{blue}{-0.5}}{a} \cdot \left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \]
      sub-neg [=>] 40.4	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      distribute-neg-out [=>] 40.4	\[ \frac{-0.5}{a} \cdot \left(-\color{blue}{\left(-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)}\right) \]
      remove-double-neg [=>] 40.4	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]
      sub-neg [=>] 40.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}\right) \]
      +-commutative [=>] 40.4	\[ \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + b \cdot b}}\right) \]

   3. Taylor expanded in a around 0 92.6%

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

   4. Simplified 92.6%

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

      [Start] 92.6	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      +-commutative [=>] 92.6	\[ \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
      associate-*r/ [=>] 92.6	\[ \color{blue}{\frac{-1 \cdot b}{a}} + \frac{c}{b} \]
      mul-1-neg [=>] 92.6	\[ \frac{\color{blue}{-b}}{a} + \frac{c}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-172}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.8%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-172}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{+48}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Accuracy	78.3%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-172}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3
Accuracy	78.5%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6.7 \cdot 10^{-167}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{a}{b} \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4
Accuracy	78.2%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-172}:\\ \;\;\;\;\frac{-0.5}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Accuracy	37.8%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Accuracy	65.8%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-239}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Accuracy	11.5%
Cost	192

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

Reproduce

herbie shell --seed 2023140 
(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)))