quad2m (problem 3.2.1, negative)

Average Error: 34.3 → 9.8

Time: 17.4s

Precision: binary64

Cost: 7696

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ t_1 := -0.5 \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -9.5 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -9.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - t_0}}{a}\\ \mathbf{elif}\;b_2 \leq -7.5 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 8.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \end{array} \]

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)))) (t_1 (* -0.5 (/ c b_2))))
   (if (<= b_2 -9.5e+79)
     t_1
     (if (<= b_2 -9.5e-68)
       (/ (/ (* c (- a)) (- b_2 t_0)) a)
       (if (<= b_2 -7.5e-102)
         t_1
         (if (<= b_2 8.6e+81) (/ (- (- b_2) t_0) a) (/ b_2 (/ a -2.0))))))))

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 t_1 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -9.5e+79) {
		tmp = t_1;
	} else if (b_2 <= -9.5e-68) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= -7.5e-102) {
		tmp = t_1;
	} else if (b_2 <= 8.6e+81) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = b_2 / (a / -2.0);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    t_1 = (-0.5d0) * (c / b_2)
    if (b_2 <= (-9.5d+79)) then
        tmp = t_1
    else if (b_2 <= (-9.5d-68)) then
        tmp = ((c * -a) / (b_2 - t_0)) / a
    else if (b_2 <= (-7.5d-102)) then
        tmp = t_1
    else if (b_2 <= 8.6d+81) then
        tmp = (-b_2 - t_0) / a
    else
        tmp = b_2 / (a / (-2.0d0))
    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 t_1 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -9.5e+79) {
		tmp = t_1;
	} else if (b_2 <= -9.5e-68) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= -7.5e-102) {
		tmp = t_1;
	} else if (b_2 <= 8.6e+81) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = b_2 / (a / -2.0);
	}
	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)))
	t_1 = -0.5 * (c / b_2)
	tmp = 0
	if b_2 <= -9.5e+79:
		tmp = t_1
	elif b_2 <= -9.5e-68:
		tmp = ((c * -a) / (b_2 - t_0)) / a
	elif b_2 <= -7.5e-102:
		tmp = t_1
	elif b_2 <= 8.6e+81:
		tmp = (-b_2 - t_0) / a
	else:
		tmp = b_2 / (a / -2.0)
	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)))
	t_1 = Float64(-0.5 * Float64(c / b_2))
	tmp = 0.0
	if (b_2 <= -9.5e+79)
		tmp = t_1;
	elseif (b_2 <= -9.5e-68)
		tmp = Float64(Float64(Float64(c * Float64(-a)) / Float64(b_2 - t_0)) / a);
	elseif (b_2 <= -7.5e-102)
		tmp = t_1;
	elseif (b_2 <= 8.6e+81)
		tmp = Float64(Float64(Float64(-b_2) - t_0) / a);
	else
		tmp = Float64(b_2 / Float64(a / -2.0));
	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)));
	t_1 = -0.5 * (c / b_2);
	tmp = 0.0;
	if (b_2 <= -9.5e+79)
		tmp = t_1;
	elseif (b_2 <= -9.5e-68)
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	elseif (b_2 <= -7.5e-102)
		tmp = t_1;
	elseif (b_2 <= 8.6e+81)
		tmp = (-b_2 - t_0) / a;
	else
		tmp = b_2 / (a / -2.0);
	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]}, Block[{t$95$1 = N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -9.5e+79], t$95$1, If[LessEqual[b$95$2, -9.5e-68], N[(N[(N[(c * (-a)), $MachinePrecision] / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -7.5e-102], t$95$1, If[LessEqual[b$95$2, 8.6e+81], N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / a), $MachinePrecision], N[(b$95$2 / N[(a / -2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -9.49999999999999994e79 or -9.4999999999999997e-68 < b_2 < -7.5000000000000008e-102

   1. Initial program 56.4

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

   2. Taylor expanded in b_2 around -inf 6.4

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

   if -9.49999999999999994e79 < b_2 < -9.4999999999999997e-68

   1. Initial program 43.6

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

   2. Applied egg-rr 14.1

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

   3. Simplified 14.1

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

      [Start] 14.1	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [=>] 14.1	\[ \frac{\frac{\color{blue}{0 - \left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-commutative [=>] 14.1	\[ \frac{\frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-inverses [=>] 14.1	\[ \frac{\frac{0 - \left(\color{blue}{0} + a \cdot c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      associate--r+ [=>] 14.1	\[ \frac{\frac{\color{blue}{\left(0 - 0\right) - a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      metadata-eval [=>] 14.1	\[ \frac{\frac{\color{blue}{0} - a \cdot c}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [<=] 14.1	\[ \frac{\frac{\color{blue}{-a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      distribute-lft-neg-in [=>] 14.1	\[ \frac{\frac{\color{blue}{\left(-a\right) \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 14.1	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 14.1	\[ \frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}}{a} \]

   if -7.5000000000000008e-102 < b_2 < 8.6000000000000003e81

   1. Initial program 12.9

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

   if 8.6000000000000003e81 < b_2

   1. Initial program 43.3

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

   2. Applied egg-rr 62.5

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

   3. Simplified 61.7

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

      [Start] 62.5	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [=>] 62.5	\[ \frac{\frac{\color{blue}{0 - \left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-commutative [=>] 62.5	\[ \frac{\frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-inverses [=>] 61.7	\[ \frac{\frac{0 - \left(\color{blue}{0} + a \cdot c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      associate--r+ [=>] 61.7	\[ \frac{\frac{\color{blue}{\left(0 - 0\right) - a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      metadata-eval [=>] 61.7	\[ \frac{\frac{\color{blue}{0} - a \cdot c}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [<=] 61.7	\[ \frac{\frac{\color{blue}{-a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      distribute-lft-neg-in [=>] 61.7	\[ \frac{\frac{\color{blue}{\left(-a\right) \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 61.7	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 61.7	\[ \frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}}{a} \]

   4. Taylor expanded in c around 0 4.6

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   5. Simplified 4.6

      \[\leadsto \color{blue}{\frac{b_2}{\frac{a}{-2}}} \]
      Proof

      [Start] 4.6	\[ -2 \cdot \frac{b_2}{a} \]
      associate-*r/ [=>] 4.7	\[ \color{blue}{\frac{-2 \cdot b_2}{a}} \]
      *-commutative [=>] 4.7	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]
      associate-/l* [=>] 4.6	\[ \color{blue}{\frac{b_2}{\frac{a}{-2}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9.5 \cdot 10^{+79}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -9.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq -7.5 \cdot 10^{-102}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.5
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{-102}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+82}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \end{array} \]

Alternative 2
Error	13.8
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8.5 \cdot 10^{-102}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3
Error	23.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-313}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4
Error	36.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.4 \cdot 10^{-265}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 5
Error	23.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.4 \cdot 10^{-268}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \end{array} \]

Alternative 6
Error	59.3
Cost	256

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

Reproduce

herbie shell --seed 2023054 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))