quad2m (problem 3.2.1, negative)

Average Error: 33.7 → 8.9

Time: 15.2s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -1.25 \cdot 10^{+52}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -3 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - t_0}}{a}\\ \mathbf{elif}\;b_2 \leq 2.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

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
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= b_2 -1.25e+52)
     (/ (* -0.5 c) b_2)
     (if (<= b_2 -3e-260)
       (/ (/ (* c (- a)) (- b_2 t_0)) a)
       (if (<= b_2 2.2e+72) (/ (- (- b_2) t_0) a) (/ (* b_2 -2.0) 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 t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -1.25e+52) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -3e-260) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= 2.2e+72) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	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 <= (-1.25d+52)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= (-3d-260)) then
        tmp = ((c * -a) / (b_2 - t_0)) / a
    else if (b_2 <= 2.2d+72) then
        tmp = (-b_2 - t_0) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    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 <= -1.25e+52) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -3e-260) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= 2.2e+72) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	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 <= -1.25e+52:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= -3e-260:
		tmp = ((c * -a) / (b_2 - t_0)) / a
	elif b_2 <= 2.2e+72:
		tmp = (-b_2 - t_0) / a
	else:
		tmp = (b_2 * -2.0) / 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)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -1.25e+52)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -3e-260)
		tmp = Float64(Float64(Float64(c * Float64(-a)) / Float64(b_2 - t_0)) / a);
	elseif (b_2 <= 2.2e+72)
		tmp = Float64(Float64(Float64(-b_2) - t_0) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	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 <= -1.25e+52)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= -3e-260)
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	elseif (b_2 <= 2.2e+72)
		tmp = (-b_2 - t_0) / a;
	else
		tmp = (b_2 * -2.0) / a;
	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, -1.25e+52], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -3e-260], N[(N[(N[(c * (-a)), $MachinePrecision] / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.2e+72], N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.25e52

   1. Initial program 57.8

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

   2. Taylor expanded in b_2 around -inf 3.0

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

   3. Applied egg-rr 3.0

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

   if -1.25e52 < b_2 < -3.0000000000000001e-260

   1. Initial program 31.2

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

   2. Applied egg-rr 16.3

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

   3. Simplified 16.3

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

      [Start] 16.3	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [=>] 16.3	\[ \frac{\frac{\color{blue}{0 - \left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-commutative [=>] 16.3	\[ \frac{\frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-inverses [=>] 16.3	\[ \frac{\frac{0 - \left(\color{blue}{0} + a \cdot c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      associate--r+ [=>] 16.3	\[ \frac{\frac{\color{blue}{\left(0 - 0\right) - a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      metadata-eval [=>] 16.3	\[ \frac{\frac{\color{blue}{0} - a \cdot c}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [<=] 16.3	\[ \frac{\frac{\color{blue}{-a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      distribute-lft-neg-in [=>] 16.3	\[ \frac{\frac{\color{blue}{\left(-a\right) \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 16.3	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 16.3	\[ \frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}}{a} \]

   if -3.0000000000000001e-260 < b_2 < 2.2e72

   1. Initial program 10.8

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

   if 2.2e72 < b_2

   1. Initial program 40.7

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

   2. Taylor expanded in b_2 around inf 4.6

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

   3. Simplified 4.6

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

      [Start] 4.6	\[ \frac{-2 \cdot b_2}{a} \]
      *-commutative [=>] 4.6	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

3. Recombined 4 regimes into one program.

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.25 \cdot 10^{+52}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -3 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq 2.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-85}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.35 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 7 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 3
Error	13.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.25 \cdot 10^{-85}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 4
Error	36.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-260}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 5
Error	36.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{-261}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 6
Error	22.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.55 \cdot 10^{-261}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 7
Error	53.3
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{-307}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 8
Error	56.4
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023018 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))