quad2m (problem 3.2.1, negative)

Average Error: 34.0 → 6.5

Time: 15.0s

Precision: binary64

Cost: 7564

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -1.32 \cdot 10^{+134}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 7.5 \cdot 10^{-273}:\\ \;\;\;\;c \cdot \frac{-1}{b_2 - t_0}\\ \mathbf{elif}\;b_2 \leq 6.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

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)))))
   (if (<= b_2 -1.32e+134)
     (* -0.5 (/ c b_2))
     (if (<= b_2 7.5e-273)
       (* c (/ -1.0 (- b_2 t_0)))
       (if (<= b_2 6.5e+87)
         (/ (- (- b_2) t_0) a)
         (+ (* -2.0 (/ b_2 a)) (* (/ 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 tmp;
	if (b_2 <= -1.32e+134) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 7.5e-273) {
		tmp = c * (-1.0 / (b_2 - t_0));
	} else if (b_2 <= 6.5e+87) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((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) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    if (b_2 <= (-1.32d+134)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 7.5d-273) then
        tmp = c * ((-1.0d0) / (b_2 - t_0))
    else if (b_2 <= 6.5d+87) then
        tmp = (-b_2 - t_0) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((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 tmp;
	if (b_2 <= -1.32e+134) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 7.5e-273) {
		tmp = c * (-1.0 / (b_2 - t_0));
	} else if (b_2 <= 6.5e+87) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((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)))
	tmp = 0
	if b_2 <= -1.32e+134:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 7.5e-273:
		tmp = c * (-1.0 / (b_2 - t_0))
	elif b_2 <= 6.5e+87:
		tmp = (-b_2 - t_0) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((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)))
	tmp = 0.0
	if (b_2 <= -1.32e+134)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 7.5e-273)
		tmp = Float64(c * Float64(-1.0 / Float64(b_2 - t_0)));
	elseif (b_2 <= 6.5e+87)
		tmp = Float64(Float64(Float64(-b_2) - t_0) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + 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)));
	tmp = 0.0;
	if (b_2 <= -1.32e+134)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 7.5e-273)
		tmp = c * (-1.0 / (b_2 - t_0));
	elseif (b_2 <= 6.5e+87)
		tmp = (-b_2 - t_0) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((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]}, If[LessEqual[b$95$2, -1.32e+134], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7.5e-273], N[(c * N[(-1.0 / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6.5e+87], N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.32e134

   1. Initial program 62.3

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

   2. Taylor expanded in b_2 around -inf 2.1

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

   if -1.32e134 < b_2 < 7.5000000000000007e-273

   1. Initial program 33.1

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

   2. Applied egg-rr 33.1

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

   3. Applied egg-rr 37.5

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

   4. Simplified 20.1

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

      [Start] 37.5	\[ \frac{-1}{\frac{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{b_2 \cdot b_2 - \left(b_2 \cdot b_2 - a \cdot c\right)}} \]
      associate-/l* [<=] 37.5	\[ \color{blue}{\frac{-1 \cdot \left(b_2 \cdot b_2 - \left(b_2 \cdot b_2 - a \cdot c\right)\right)}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}} \]
      mul-1-neg [=>] 37.5	\[ \frac{\color{blue}{-\left(b_2 \cdot b_2 - \left(b_2 \cdot b_2 - a \cdot c\right)\right)}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      neg-sub0 [=>] 37.5	\[ \frac{\color{blue}{0 - \left(b_2 \cdot b_2 - \left(b_2 \cdot b_2 - a \cdot c\right)\right)}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      associate--r- [=>] 20.1	\[ \frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right)}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      associate--r+ [=>] 20.1	\[ \frac{\color{blue}{\left(0 - \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right) - a \cdot c}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      +-inverses [=>] 20.1	\[ \frac{\left(0 - \color{blue}{0}\right) - a \cdot c}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      metadata-eval [=>] 20.1	\[ \frac{\color{blue}{0} - a \cdot c}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      neg-sub0 [<=] 20.1	\[ \frac{\color{blue}{-a \cdot c}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      distribute-lft-neg-in [=>] 20.1	\[ \frac{\color{blue}{\left(-a\right) \cdot c}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      *-commutative [=>] 20.1	\[ \frac{\color{blue}{c \cdot \left(-a\right)}}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      *-commutative [<=] 20.1	\[ \frac{c \cdot \left(-a\right)}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}\right)} \]

   5. Applied egg-rr 50.7

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c \cdot \left(-a\right)}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\right)} - 1} \]

   6. Simplified 8.3

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

      [Start] 50.7	\[ e^{\mathsf{log1p}\left(\frac{c \cdot \left(-a\right)}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\right)} - 1 \]
      expm1-def [=>] 30.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c \cdot \left(-a\right)}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\right)\right)} \]
      expm1-log1p [=>] 20.1	\[ \color{blue}{\frac{c \cdot \left(-a\right)}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}} \]
      *-rgt-identity [<=] 20.1	\[ \color{blue}{\frac{c \cdot \left(-a\right)}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \cdot 1} \]
      associate-*l/ [=>] 20.1	\[ \color{blue}{\frac{\left(c \cdot \left(-a\right)\right) \cdot 1}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}} \]
      associate-*r/ [<=] 20.6	\[ \color{blue}{\left(c \cdot \left(-a\right)\right) \cdot \frac{1}{a \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}} \]
      *-commutative [=>] 20.6	\[ \left(c \cdot \left(-a\right)\right) \cdot \frac{1}{\color{blue}{\left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot a}} \]
      associate-/l/ [<=] 20.2	\[ \left(c \cdot \left(-a\right)\right) \cdot \color{blue}{\frac{\frac{1}{a}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}} \]
      distribute-rgt-neg-out [=>] 20.2	\[ \color{blue}{\left(-c \cdot a\right)} \cdot \frac{\frac{1}{a}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}} \]
      mul-1-neg [<=] 20.2	\[ \color{blue}{\left(-1 \cdot \left(c \cdot a\right)\right)} \cdot \frac{\frac{1}{a}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}} \]
      associate-*l* [=>] 20.2	\[ \color{blue}{-1 \cdot \left(\left(c \cdot a\right) \cdot \frac{\frac{1}{a}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}\right)} \]
      *-commutative [=>] 20.2	\[ -1 \cdot \left(\color{blue}{\left(a \cdot c\right)} \cdot \frac{\frac{1}{a}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}\right) \]
      associate-*r* [<=] 25.5	\[ -1 \cdot \color{blue}{\left(a \cdot \left(c \cdot \frac{\frac{1}{a}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}\right)\right)} \]
      associate-/l/ [=>] 25.6	\[ -1 \cdot \left(a \cdot \left(c \cdot \color{blue}{\frac{1}{\left(b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot a}}\right)\right) \]

   if 7.5000000000000007e-273 < b_2 < 6.5000000000000002e87

   1. Initial program 8.8

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

   if 6.5000000000000002e87 < b_2

   1. Initial program 43.5

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

   2. Taylor expanded in b_2 around inf 4.5

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.32 \cdot 10^{+134}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 7.5 \cdot 10^{-273}:\\ \;\;\;\;c \cdot \frac{-1}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{elif}\;b_2 \leq 6.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	10.1
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.25 \cdot 10^{-59}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.55 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 2
Error	14.8
Cost	7304

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.9 \cdot 10^{-59}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.2 \cdot 10^{-194}:\\ \;\;\;\;\left(b_2 + \sqrt{c \cdot \left(-a\right)}\right) \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3
Error	14.8
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.2 \cdot 10^{-194}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4
Error	22.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.5 \cdot 10^{-234}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \]

Alternative 5
Error	22.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.5 \cdot 10^{-235}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 6
Error	22.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.8 \cdot 10^{-230}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 7
Error	22.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7.5 \cdot 10^{-235}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 8
Error	39.5
Cost	320

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

Reproduce

herbie shell --seed 2023187 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))