quad2m (problem 3.2.1, negative)

Average Error: 33.8 → 9.8

Time: 16.2s

Precision: binary64

Cost: 7560

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{+84}:\\ \;\;\;\;\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

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
 (if (<= b_2 -2.1e-65)
   (* -0.5 (/ c b_2))
   (if (<= b_2 8e+84)
     (- (/ (- (sqrt (- (* b_2 b_2) (* c a)))) a) (/ b_2 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 tmp;
	if (b_2 <= -2.1e-65) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 8e+84) {
		tmp = (-sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / 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) :: tmp
    if (b_2 <= (-2.1d-65)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 8d+84) then
        tmp = (-sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / 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 tmp;
	if (b_2 <= -2.1e-65) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 8e+84) {
		tmp = (-Math.sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / 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):
	tmp = 0
	if b_2 <= -2.1e-65:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 8e+84:
		tmp = (-math.sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / 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)
	tmp = 0.0
	if (b_2 <= -2.1e-65)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 8e+84)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a) - Float64(b_2 / 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)
	tmp = 0.0;
	if (b_2 <= -2.1e-65)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 8e+84)
		tmp = (-sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / 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_] := If[LessEqual[b$95$2, -2.1e-65], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 8e+84], N[(N[((-N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.10000000000000003e-65

   1. Initial program 53.0

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

   2. Taylor expanded in b_2 around -inf 8.4

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

   if -2.10000000000000003e-65 < b_2 < 8.00000000000000046e84

   1. Initial program 13.0

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

   2. Applied egg-rr 13.0

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

   3. Simplified 13.0

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

      [Start] 13.0	\[ \frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      div0 [=>] 13.0	\[ \color{blue}{0} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      +-commutative [=>] 13.0	\[ 0 - \color{blue}{\left(\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
      associate--r+ [=>] 13.0	\[ \color{blue}{\left(0 - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]
      neg-sub0 [<=] 13.0	\[ \color{blue}{\left(-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} - \frac{b_2}{a} \]
      distribute-neg-frac [=>] 13.0	\[ \color{blue}{\frac{-\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} - \frac{b_2}{a} \]
      *-commutative [=>] 13.0	\[ \frac{-\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}{a} - \frac{b_2}{a} \]

   if 8.00000000000000046e84 < b_2

   1. Initial program 44.1

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

   2. Taylor expanded in b_2 around inf 4.9

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

   3. Simplified 4.9

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

      [Start] 4.9	\[ \frac{-2 \cdot b_2}{a} \]
      *-commutative [=>] 4.9	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{+84}:\\ \;\;\;\;\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.9
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6 \cdot 10^{-62}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(-\sqrt{b_2 \cdot b_2 - c \cdot a}\right) - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 2
Error	13.1
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.2 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3
Error	22.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.36 \cdot 10^{-244}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 4
Error	22.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.2 \cdot 10^{-244}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 5
Error	39.7
Cost	320

\[-0.5 \cdot \frac{c}{b_2} \]

Alternative 6
Error	56.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023017 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))