quad2m (problem 3.2.1, negative)

Average Accuracy: 47.6% → 86.4%

Time: 16.3s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{+30}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -4.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - t_0}}{a}\\ \mathbf{elif}\;b_2 \leq 5.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

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
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= b_2 -1.8e+30)
     (* -0.5 (/ c b_2))
     (if (<= b_2 -4.5e-139)
       (/ (/ (* c (- a)) (- b_2 t_0)) a)
       (if (<= b_2 5.5e+110)
         (/ (- (- b_2) t_0) a)
         (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))))

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 t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -1.8e+30) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -4.5e-139) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= 5.5e+110) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

↓

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    if (b_2 <= (-1.8d+30)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= (-4.5d-139)) then
        tmp = ((c * -a) / (b_2 - t_0)) / a
    else if (b_2 <= 5.5d+110) then
        tmp = (-b_2 - t_0) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -1.8e+30) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -4.5e-139) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= 5.5e+110) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

↓

def code(a, b_2, c):
	t_0 = math.sqrt(((b_2 * b_2) - (c * a)))
	tmp = 0
	if b_2 <= -1.8e+30:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= -4.5e-139:
		tmp = ((c * -a) / (b_2 - t_0)) / a
	elif b_2 <= 5.5e+110:
		tmp = (-b_2 - t_0) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	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)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -1.8e+30)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -4.5e-139)
		tmp = Float64(Float64(Float64(c * Float64(-a)) / Float64(b_2 - t_0)) / a);
	elseif (b_2 <= 5.5e+110)
		tmp = Float64(Float64(Float64(-b_2) - t_0) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

↓

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt(((b_2 * b_2) - (c * a)));
	tmp = 0.0;
	if (b_2 <= -1.8e+30)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= -4.5e-139)
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	elseif (b_2 <= 5.5e+110)
		tmp = (-b_2 - t_0) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = 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_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.8e+30], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -4.5e-139], N[(N[(N[(c * (-a)), $MachinePrecision] / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 5.5e+110], N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.8000000000000001e30

   1. Initial program 10.9%

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

   2. Taylor expanded in b_2 around -inf 92.2%

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

   if -1.8000000000000001e30 < b_2 < -4.50000000000000023e-139

   1. Initial program 44.2%

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

   2. Applied egg-rr 74.1%

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

      [Start] 44.2	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      flip-- [=>] 44.1	\[ \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
      frac-2neg [=>] 44.1	\[ \frac{\color{blue}{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}{a} \]
      add-sqr-sqrt [<=] 44.1	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}\right)}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      associate--r- [=>] 74.1	\[ \frac{\frac{-\color{blue}{\left(\left(\left(-b_2\right) \cdot \left(-b_2\right) - b_2 \cdot b_2\right) + a \cdot c\right)}}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      +-commutative [=>] 74.1	\[ \frac{\frac{-\color{blue}{\left(a \cdot c + \left(\left(-b_2\right) \cdot \left(-b_2\right) - b_2 \cdot b_2\right)\right)}}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqr-neg [=>] 74.1	\[ \frac{\frac{-\left(a \cdot c + \left(\color{blue}{b_2 \cdot b_2} - b_2 \cdot b_2\right)\right)}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      add-sqr-sqrt [=>] 74.0	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{-\left(\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqrt-unprod [=>] 74.1	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{-\left(\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqr-neg [=>] 74.1	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{-\left(\sqrt{\color{blue}{b_2 \cdot b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqrt-prod [=>] 0.0	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{-\left(\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      add-sqr-sqrt [<=] 38.1	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{-\left(\color{blue}{b_2} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]

   3. Simplified 74.1%

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

      [Start] 74.1	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [=>] 74.1	\[ \frac{\frac{\color{blue}{0 - \left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-commutative [=>] 74.1	\[ \frac{\frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-inverses [=>] 74.1	\[ \frac{\frac{0 - \left(\color{blue}{0} + a \cdot c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      associate--r+ [=>] 74.1	\[ \frac{\frac{\color{blue}{\left(0 - 0\right) - a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      metadata-eval [=>] 74.1	\[ \frac{\frac{\color{blue}{0} - a \cdot c}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [<=] 74.1	\[ \frac{\frac{\color{blue}{-a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      distribute-lft-neg-in [=>] 74.1	\[ \frac{\frac{\color{blue}{\left(-a\right) \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 74.1	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 74.1	\[ \frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}}{a} \]

   if -4.50000000000000023e-139 < b_2 < 5.49999999999999996e110

   1. Initial program 83.2%

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

   if 5.49999999999999996e110 < b_2

   1. Initial program 21.6%

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

   2. Taylor expanded in b_2 around inf 95.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{+30}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -4.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq 5.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.5%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -190000000:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 2
Accuracy	78.9%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.02 \cdot 10^{-42}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 7.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3
Accuracy	63.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.02 \cdot 10^{-264}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 4
Accuracy	29.1%
Cost	320

\[-2 \cdot \frac{b_2}{a} \]

Reproduce

herbie shell --seed 2023135 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))