quad2m (problem 3.2.1, negative)

Average Accuracy: 47.0% → 87.1%

Time: 15.7s

Precision: binary64

Cost: 14216

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6 \cdot 10^{+34}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -7.4 \cdot 10^{-277}:\\ \;\;\;\;\frac{\frac{-1}{\frac{1}{\frac{c}{\frac{b_2 - \sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{a}}}}}{a}\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c \cdot 0.5}{\frac{b_2}{a}} - b_2\right) - b_2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6e+34)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 -7.4e-277)
     (/
      (/ -1.0 (/ 1.0 (/ c (/ (- b_2 (sqrt (fma b_2 b_2 (* c (- a))))) a))))
      a)
     (if (<= b_2 4e+111)
       (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
       (/ (- (- (/ (* c 0.5) (/ b_2 a)) b_2) b_2) a)))))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e+34) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -7.4e-277) {
		tmp = (-1.0 / (1.0 / (c / ((b_2 - sqrt(fma(b_2, b_2, (c * -a)))) / a)))) / a;
	} else if (b_2 <= 4e+111) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((((c * 0.5) / (b_2 / a)) - b_2) - b_2) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

↓

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6e+34)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -7.4e-277)
		tmp = Float64(Float64(-1.0 / Float64(1.0 / Float64(c / Float64(Float64(b_2 - sqrt(fma(b_2, b_2, Float64(c * Float64(-a))))) / a)))) / a);
	elseif (b_2 <= 4e+111)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(c * 0.5) / Float64(b_2 / a)) - b_2) - b_2) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6e+34], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -7.4e-277], N[(N[(-1.0 / N[(1.0 / N[(c / N[(N[(b$95$2 - N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 4e+111], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(c * 0.5), $MachinePrecision] / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - b$95$2), $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -6.00000000000000037e34

   1. Initial program 11.6%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 92.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   3. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      Proof

      [Start] 92.8	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 92.9	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -6.00000000000000037e34 < b_2 < -7.3999999999999997e-277

   1. Initial program 53.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Applied egg-rr 52.2%

      \[\leadsto \frac{\color{blue}{\frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}}{\frac{1}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]
      Proof

      [Start] 53.2	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      flip-- [=>] 53.1	\[ \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
      frac-2neg [=>] 53.1	\[ \frac{\color{blue}{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}{a} \]
      add-sqr-sqrt [=>] 53.1	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-\left(\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqrt-unprod [=>] 53.0	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-\left(\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqr-neg [=>] 53.0	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-\left(\sqrt{\color{blue}{b_2 \cdot b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqrt-prod [=>] 0.0	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-\left(\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      add-sqr-sqrt [<=] 49.4	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-\left(\color{blue}{b_2} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      distribute-neg-out [<=] 49.4	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{\color{blue}{\left(-b_2\right) + \left(-\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}{a} \]
      sub-neg [<=] 49.4	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{\color{blue}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
      flip-- [=>] 49.3	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]
      div-inv [=>] 49.2	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{\color{blue}{\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]

   3. Taylor expanded in a around 0 53.1%

      \[\leadsto \frac{\frac{\color{blue}{-1}}{\frac{1}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]

   4. Applied egg-rr 73.3%

      \[\leadsto \frac{\frac{-1}{\frac{1}{\color{blue}{\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right) \cdot \frac{1}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}}{a} \]
      Proof

      [Start] 53.1	\[ \frac{\frac{-1}{\frac{1}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
      flip-+ [=>] 53.1	\[ \frac{\frac{-1}{\frac{1}{\color{blue}{\frac{b_2 \cdot b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}}{a} \]
      div-inv [=>] 53.0	\[ \frac{\frac{-1}{\frac{1}{\color{blue}{\left(b_2 \cdot b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}}{a} \]
      add-sqr-sqrt [<=] 53.0	\[ \frac{\frac{-1}{\frac{1}{\left(b_2 \cdot b_2 - \color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}\right) \cdot \frac{1}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]
      associate--r- [=>] 73.3	\[ \frac{\frac{-1}{\frac{1}{\color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right)} \cdot \frac{1}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]
      +-commutative [=>] 73.3	\[ \frac{\frac{-1}{\frac{1}{\color{blue}{\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)} \cdot \frac{1}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]

   5. Simplified 77.6%

      \[\leadsto \frac{\frac{-1}{\frac{1}{\color{blue}{\frac{c}{\frac{b_2 - \sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{a}}}}}}{a} \]
      Proof

      [Start] 73.3	\[ \frac{\frac{-1}{\frac{1}{\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right) \cdot \frac{1}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]
      associate-*r/ [=>] 73.3	\[ \frac{\frac{-1}{\frac{1}{\color{blue}{\frac{\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right) \cdot 1}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}}{a} \]
      *-rgt-identity [=>] 73.3	\[ \frac{\frac{-1}{\frac{1}{\frac{\color{blue}{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]
      +-commutative [=>] 73.3	\[ \frac{\frac{-1}{\frac{1}{\frac{\color{blue}{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]
      +-inverses [=>] 73.3	\[ \frac{\frac{-1}{\frac{1}{\frac{\color{blue}{0} + a \cdot c}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]
      +-lft-identity [=>] 73.3	\[ \frac{\frac{-1}{\frac{1}{\frac{\color{blue}{a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]
      *-commutative [=>] 73.3	\[ \frac{\frac{-1}{\frac{1}{\frac{\color{blue}{c \cdot a}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{a} \]
      associate-/l* [=>] 77.6	\[ \frac{\frac{-1}{\frac{1}{\color{blue}{\frac{c}{\frac{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}}}}}{a} \]
      fma-neg [=>] 77.6	\[ \frac{\frac{-1}{\frac{1}{\frac{c}{\frac{b_2 - \sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}}{a}}}}}{a} \]
      distribute-lft-neg-in [=>] 77.6	\[ \frac{\frac{-1}{\frac{1}{\frac{c}{\frac{b_2 - \sqrt{\mathsf{fma}\left(b_2, b_2, \color{blue}{\left(-a\right) \cdot c}\right)}}{a}}}}}{a} \]
      *-commutative [=>] 77.6	\[ \frac{\frac{-1}{\frac{1}{\frac{c}{\frac{b_2 - \sqrt{\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)}}{a}}}}}{a} \]

   if -7.3999999999999997e-277 < b_2 < 3.99999999999999983e111

   1. Initial program 85.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 3.99999999999999983e111 < b_2

   1. Initial program 22.7%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 82.7%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(b_2 + -0.5 \cdot \frac{c \cdot a}{b_2}\right)}}{a} \]

   3. Simplified 93.4%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(b_2 + \frac{c \cdot -0.5}{\frac{b_2}{a}}\right)}}{a} \]
      Proof

      [Start] 82.7	\[ \frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c \cdot a}{b_2}\right)}{a} \]
      associate-/l* [=>] 93.4	\[ \frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \color{blue}{\frac{c}{\frac{b_2}{a}}}\right)}{a} \]
      associate-*r/ [=>] 93.4	\[ \frac{\left(-b_2\right) - \left(b_2 + \color{blue}{\frac{-0.5 \cdot c}{\frac{b_2}{a}}}\right)}{a} \]
      *-commutative [=>] 93.4	\[ \frac{\left(-b_2\right) - \left(b_2 + \frac{\color{blue}{c \cdot -0.5}}{\frac{b_2}{a}}\right)}{a} \]

3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6 \cdot 10^{+34}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -7.4 \cdot 10^{-277}:\\ \;\;\;\;\frac{\frac{-1}{\frac{1}{\frac{c}{\frac{b_2 - \sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{a}}}}}{a}\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c \cdot 0.5}{\frac{b_2}{a}} - b_2\right) - b_2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.9%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+112}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c \cdot 0.5}{\frac{b_2}{a}} - b_2\right) - b_2}{a}\\ \end{array} \]

Alternative 2
Accuracy	78.6%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{-79}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3
Accuracy	64.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4
Accuracy	42.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-277}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 5
Accuracy	42.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.8 \cdot 10^{-277}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 6
Accuracy	64.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{-277}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 7
Accuracy	7.5%
Cost	256

\[\frac{-b_2}{a} \]

Reproduce

herbie shell --seed 2023131 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))