quad2m (problem 3.2.1, negative)

Average Error: 53.7% → 14.21%

Time: 16.8s

Precision: binary64

Cost: 7824

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ t_1 := -0.5 \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -2 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -1.55 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - t_0}}{a}\\ \mathbf{elif}\;b_2 \leq -2.3 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 2.15 \cdot 10^{+147}:\\ \;\;\;\;\frac{-t_0}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

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)))) (t_1 (* -0.5 (/ c b_2))))
   (if (<= b_2 -2e-30)
     t_1
     (if (<= b_2 -1.55e-116)
       (/ (/ (* c (- a)) (- b_2 t_0)) a)
       (if (<= b_2 -2.3e-141)
         t_1
         (if (<= b_2 2.15e+147)
           (- (/ (- t_0) a) (/ b_2 a))
           (+ (* (/ b_2 a) -2.0) (* (/ 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 t_1 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -2e-30) {
		tmp = t_1;
	} else if (b_2 <= -1.55e-116) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= -2.3e-141) {
		tmp = t_1;
	} else if (b_2 <= 2.15e+147) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = ((b_2 / a) * -2.0) + ((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) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    t_1 = (-0.5d0) * (c / b_2)
    if (b_2 <= (-2d-30)) then
        tmp = t_1
    else if (b_2 <= (-1.55d-116)) then
        tmp = ((c * -a) / (b_2 - t_0)) / a
    else if (b_2 <= (-2.3d-141)) then
        tmp = t_1
    else if (b_2 <= 2.15d+147) then
        tmp = (-t_0 / a) - (b_2 / a)
    else
        tmp = ((b_2 / a) * (-2.0d0)) + ((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 t_1 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -2e-30) {
		tmp = t_1;
	} else if (b_2 <= -1.55e-116) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= -2.3e-141) {
		tmp = t_1;
	} else if (b_2 <= 2.15e+147) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = ((b_2 / a) * -2.0) + ((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)))
	t_1 = -0.5 * (c / b_2)
	tmp = 0
	if b_2 <= -2e-30:
		tmp = t_1
	elif b_2 <= -1.55e-116:
		tmp = ((c * -a) / (b_2 - t_0)) / a
	elif b_2 <= -2.3e-141:
		tmp = t_1
	elif b_2 <= 2.15e+147:
		tmp = (-t_0 / a) - (b_2 / a)
	else:
		tmp = ((b_2 / a) * -2.0) + ((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)))
	t_1 = Float64(-0.5 * Float64(c / b_2))
	tmp = 0.0
	if (b_2 <= -2e-30)
		tmp = t_1;
	elseif (b_2 <= -1.55e-116)
		tmp = Float64(Float64(Float64(c * Float64(-a)) / Float64(b_2 - t_0)) / a);
	elseif (b_2 <= -2.3e-141)
		tmp = t_1;
	elseif (b_2 <= 2.15e+147)
		tmp = Float64(Float64(Float64(-t_0) / a) - Float64(b_2 / a));
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + 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)));
	t_1 = -0.5 * (c / b_2);
	tmp = 0.0;
	if (b_2 <= -2e-30)
		tmp = t_1;
	elseif (b_2 <= -1.55e-116)
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	elseif (b_2 <= -2.3e-141)
		tmp = t_1;
	elseif (b_2 <= 2.15e+147)
		tmp = (-t_0 / a) - (b_2 / a);
	else
		tmp = ((b_2 / a) * -2.0) + ((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]}, Block[{t$95$1 = N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -2e-30], t$95$1, If[LessEqual[b$95$2, -1.55e-116], N[(N[(N[(c * (-a)), $MachinePrecision] / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -2.3e-141], t$95$1, If[LessEqual[b$95$2, 2.15e+147], N[(N[((-t$95$0) / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2e-30 or -1.55000000000000009e-116 < b_2 < -2.29999999999999995e-141

   1. Initial program 83.71

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

   2. Taylor expanded in b_2 around -inf 12.84

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

   if -2e-30 < b_2 < -1.55000000000000009e-116

   1. Initial program 49.94

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

   2. Applied egg-rr 26.16

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

   3. Simplified 26.16

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

      [Start] 26.16	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [=>] 26.16	\[ \frac{\frac{\color{blue}{0 - \left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-commutative [=>] 26.16	\[ \frac{\frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-inverses [=>] 26.16	\[ \frac{\frac{0 - \left(\color{blue}{0} + a \cdot c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      associate--r+ [=>] 26.16	\[ \frac{\frac{\color{blue}{\left(0 - 0\right) - a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      metadata-eval [=>] 26.16	\[ \frac{\frac{\color{blue}{0} - a \cdot c}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [<=] 26.16	\[ \frac{\frac{\color{blue}{-a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      distribute-lft-neg-in [=>] 26.16	\[ \frac{\frac{\color{blue}{\left(-a\right) \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 26.16	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 26.16	\[ \frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}}{a} \]

   if -2.29999999999999995e-141 < b_2 < 2.1499999999999999e147

   1. Initial program 16.55

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

   2. Applied egg-rr 16.54

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]

   3. Simplified 16.54

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

      [Start] 16.54	\[ \frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      div0 [=>] 16.54	\[ \color{blue}{0} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      +-commutative [=>] 16.54	\[ 0 - \color{blue}{\left(\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
      associate--r+ [=>] 16.54	\[ \color{blue}{\left(0 - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]
      neg-sub0 [<=] 16.54	\[ \color{blue}{\left(-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} - \frac{b_2}{a} \]
      distribute-neg-frac [=>] 16.54	\[ \color{blue}{\frac{-\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} - \frac{b_2}{a} \]
      *-commutative [=>] 16.54	\[ \frac{-\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}{a} - \frac{b_2}{a} \]

   if 2.1499999999999999e147 < b_2

   1. Initial program 95.49

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

   2. Taylor expanded in b_2 around inf 3.14

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

4. Final simplification 14.21

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-30}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.55 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq -2.3 \cdot 10^{-141}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.15 \cdot 10^{+147}:\\ \;\;\;\;\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	15.81%
Cost	7956

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ t_1 := -0.5 \cdot \frac{c}{b_2}\\ t_2 := \frac{-t_0}{a} - \frac{b_2}{a}\\ \mathbf{if}\;b_2 \leq -3.1 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -1.55 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b_2 \leq -2.3 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -6.5 \cdot 10^{-190}:\\ \;\;\;\;\frac{1}{a} \cdot \left(b_2 - t_0\right)\\ \mathbf{elif}\;b_2 \leq 10^{+146}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 2
Error	15.7%
Cost	7828

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ t_1 := -0.5 \cdot \frac{c}{b_2}\\ t_2 := \frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{if}\;b_2 \leq -4.4 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -4.8 \cdot 10^{-115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b_2 \leq -2.3 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -1.15 \cdot 10^{-189}:\\ \;\;\;\;\frac{1}{a} \cdot \left(b_2 - t_0\right)\\ \mathbf{elif}\;b_2 \leq 10^{+146}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3
Error	21.57%
Cost	7504

\[\begin{array}{l} t_0 := \frac{b_2}{a} \cdot -2\\ t_1 := \frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{if}\;b_2 \leq -2.3 \cdot 10^{-141}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.65 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 1.12 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 1.3 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4
Error	15.21%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{-34}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+146}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 5
Error	61.55%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.9 \cdot 10^{-189}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array} \]

Alternative 6
Error	35.29%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.5 \cdot 10^{-235}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array} \]

Alternative 7
Error	87.29%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023125 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))