quad2m (problem 3.2.1, negative)

Average Error: 53.58% → 14.74%

Time: 15.4s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -3.05 \cdot 10^{+60}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - t_0}}{a}\\ \mathbf{elif}\;b_2 \leq 8.3 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

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 -3.05e+60)
     (/ (* -0.5 c) b_2)
     (if (<= b_2 -1e-116)
       (/ (/ (* c (- a)) (- b_2 t_0)) a)
       (if (<= b_2 8.3e+52)
         (/ (- (- b_2) t_0) a)
         (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))))

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 <= -3.05e+60) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -1e-116) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= 8.3e+52) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	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 <= (-3.05d+60)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= (-1d-116)) then
        tmp = ((c * -a) / (b_2 - t_0)) / a
    else if (b_2 <= 8.3d+52) then
        tmp = (-b_2 - t_0) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    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 <= -3.05e+60) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -1e-116) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= 8.3e+52) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	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 <= -3.05e+60:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= -1e-116:
		tmp = ((c * -a) / (b_2 - t_0)) / a
	elif b_2 <= 8.3e+52:
		tmp = (-b_2 - t_0) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	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 <= -3.05e+60)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -1e-116)
		tmp = Float64(Float64(Float64(c * Float64(-a)) / Float64(b_2 - t_0)) / a);
	elseif (b_2 <= 8.3e+52)
		tmp = Float64(Float64(Float64(-b_2) - t_0) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	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 <= -3.05e+60)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= -1e-116)
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	elseif (b_2 <= 8.3e+52)
		tmp = (-b_2 - t_0) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	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, -3.05e+60], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -1e-116], N[(N[(N[(c * (-a)), $MachinePrecision] / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 8.3e+52], N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.05e60

   1. Initial program 88.82

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

   2. Applied egg-rr 88.82

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

   3. Simplified 88.82

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

      [Start] 88.82	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{-1}{a} \]
      *-commutative [=>] 88.82	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}\right) \cdot \frac{-1}{a} \]

   4. Applied egg-rr 89.36

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

   5. Simplified 44.37

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

      [Start] 89.36	\[ \frac{-1}{\frac{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}{b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)}} \]
      associate-/l* [<=] 89.36	\[ \color{blue}{\frac{-1 \cdot \left(b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)\right)}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}} \]
      mul-1-neg [=>] 89.36	\[ \frac{\color{blue}{-\left(b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)\right)}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]
      neg-sub0 [=>] 89.36	\[ \frac{\color{blue}{0 - \left(b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)\right)}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]
      associate--r- [=>] 70.24	\[ \frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + c \cdot a\right)}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]
      associate--r+ [=>] 70.24	\[ \frac{\color{blue}{\left(0 - \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right) - c \cdot a}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]
      +-inverses [=>] 44.37	\[ \frac{\left(0 - \color{blue}{0}\right) - c \cdot a}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]
      metadata-eval [=>] 44.37	\[ \frac{\color{blue}{0} - c \cdot a}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]
      neg-sub0 [<=] 44.37	\[ \frac{\color{blue}{-c \cdot a}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]
      distribute-rgt-neg-out [<=] 44.37	\[ \frac{\color{blue}{c \cdot \left(-a\right)}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]

   6. Taylor expanded in b_2 around -inf 5.97

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

   7. Simplified 5.95

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

      [Start] 5.97	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 5.95	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -3.05e60 < b_2 < -9.9999999999999999e-117

   1. Initial program 63.83

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

   2. Applied egg-rr 25.02

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

   3. Simplified 25.02

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

      [Start] 25.02	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [=>] 25.02	\[ \frac{\frac{\color{blue}{0 - \left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-commutative [=>] 25.02	\[ \frac{\frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-inverses [=>] 25.02	\[ \frac{\frac{0 - \left(\color{blue}{0} + a \cdot c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      associate--r+ [=>] 25.02	\[ \frac{\frac{\color{blue}{\left(0 - 0\right) - a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      metadata-eval [=>] 25.02	\[ \frac{\frac{\color{blue}{0} - a \cdot c}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [<=] 25.02	\[ \frac{\frac{\color{blue}{-a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      distribute-lft-neg-in [=>] 25.02	\[ \frac{\frac{\color{blue}{\left(-a\right) \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 25.02	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 25.02	\[ \frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}}{a} \]

   if -9.9999999999999999e-117 < b_2 < 8.29999999999999994e52

   1. Initial program 19.78

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

   if 8.29999999999999994e52 < b_2

   1. Initial program 60.73

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

   2. Taylor expanded in b_2 around inf 9.32

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

4. Final simplification 14.74

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.05 \cdot 10^{+60}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq 8.3 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.61%
Cost	7688

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.95 \cdot 10^{-83}:\\ \;\;\;\;\frac{c \cdot \left(-a\right)}{a \cdot \left(b_2 - t_0\right)}\\ \mathbf{elif}\;b_2 \leq 8.3 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2
Error	16.54%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3
Error	21.39%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4
Error	34.92%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.1 \cdot 10^{-284}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Error	56.99%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.45 \cdot 10^{-268}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 6
Error	56.98%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6 \cdot 10^{-267}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 7
Error	34.94%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.7 \cdot 10^{-269}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 8
Error	92.51%
Cost	256

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

Reproduce

herbie shell --seed 2023121 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))