quad2p (problem 3.2.1, positive)

Average Error: 34.1 → 11.6

Time: 16.1s

Precision: binary64

Cost: 7569

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{+148}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.5 \cdot 10^{-47} \lor \neg \left(b_2 \leq 4.4 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{a \cdot \left(-c\right)} - b_2}}\\ \end{array} \]

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
 (if (<= b_2 -4e+148)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.6e-163)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (if (or (<= b_2 3.5e-47) (not (<= b_2 4.4e-23)))
       (/ (* c -0.5) b_2)
       (/ 1.0 (/ a (- (sqrt (* a (- c))) b_2)))))))

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 <= -4e+148) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.6e-163) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else if ((b_2 <= 3.5e-47) || !(b_2 <= 4.4e-23)) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = 1.0 / (a / (sqrt((a * -c)) - b_2));
	}
	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 <= (-4d+148)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 1.6d-163) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else if ((b_2 <= 3.5d-47) .or. (.not. (b_2 <= 4.4d-23))) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = 1.0d0 / (a / (sqrt((a * -c)) - b_2))
    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 <= -4e+148) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.6e-163) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else if ((b_2 <= 3.5e-47) || !(b_2 <= 4.4e-23)) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = 1.0 / (a / (Math.sqrt((a * -c)) - b_2));
	}
	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 <= -4e+148:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 1.6e-163:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	elif (b_2 <= 3.5e-47) or not (b_2 <= 4.4e-23):
		tmp = (c * -0.5) / b_2
	else:
		tmp = 1.0 / (a / (math.sqrt((a * -c)) - b_2))
	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 <= -4e+148)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.6e-163)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	elseif ((b_2 <= 3.5e-47) || !(b_2 <= 4.4e-23))
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(1.0 / Float64(a / Float64(sqrt(Float64(a * Float64(-c))) - b_2)));
	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 <= -4e+148)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 1.6e-163)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	elseif ((b_2 <= 3.5e-47) || ~((b_2 <= 4.4e-23)))
		tmp = (c * -0.5) / b_2;
	else
		tmp = 1.0 / (a / (sqrt((a * -c)) - b_2));
	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, -4e+148], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.6e-163], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[Or[LessEqual[b$95$2, 3.5e-47], N[Not[LessEqual[b$95$2, 4.4e-23]], $MachinePrecision]], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(1.0 / N[(a / N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -4.0000000000000002e148

   1. Initial program 61.3

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

   2. Simplified 61.3

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

      [Start] 61.3	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 61.3	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 61.3	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 2.3

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -4.0000000000000002e148 < b_2 < 1.59999999999999994e-163

   1. Initial program 10.7

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

   2. Simplified 10.7

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

      [Start] 10.7	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 10.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 10.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   if 1.59999999999999994e-163 < b_2 < 3.4999999999999998e-47 or 4.3999999999999999e-23 < b_2

   1. Initial program 49.7

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

   2. Simplified 49.7

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

      [Start] 49.7	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 49.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 49.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 13.3

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

   4. Simplified 13.3

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

      [Start] 13.3	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 13.3	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      *-commutative [=>] 13.3	\[ \frac{\color{blue}{c \cdot -0.5}}{b_2} \]

   if 3.4999999999999998e-47 < b_2 < 4.3999999999999999e-23

   1. Initial program 38.3

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

   2. Simplified 38.3

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

      [Start] 38.3	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 38.3	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 38.3	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 38.3

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

   4. Applied egg-rr 38.3

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

   5. Taylor expanded in b_2 around 0 42.1

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

   6. Simplified 42.1

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

      [Start] 42.1	\[ \frac{1}{\frac{a}{\sqrt{-1 \cdot \left(c \cdot a\right)} - b_2}} \]
      *-commutative [=>] 42.1	\[ \frac{1}{\frac{a}{\sqrt{\color{blue}{\left(c \cdot a\right) \cdot -1}} - b_2}} \]
      associate-*l* [=>] 42.1	\[ \frac{1}{\frac{a}{\sqrt{\color{blue}{c \cdot \left(a \cdot -1\right)}} - b_2}} \]
      *-commutative [<=] 42.1	\[ \frac{1}{\frac{a}{\sqrt{c \cdot \color{blue}{\left(-1 \cdot a\right)}} - b_2}} \]
      neg-mul-1 [<=] 42.1	\[ \frac{1}{\frac{a}{\sqrt{c \cdot \color{blue}{\left(-a\right)}} - b_2}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{+148}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.5 \cdot 10^{-47} \lor \neg \left(b_2 \leq 4.4 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{a \cdot \left(-c\right)} - b_2}}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.0
Cost	7569

\[\begin{array}{l} t_0 := \sqrt{a \cdot \left(-c\right)} - b_2\\ \mathbf{if}\;b_2 \leq -5.8 \cdot 10^{-31}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{t_0}{a}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{-48} \lor \neg \left(b_2 \leq 4 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{t_0}}\\ \end{array} \]

Alternative 2
Error	15.1
Cost	7441

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.2 \cdot 10^{-27}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{-163} \lor \neg \left(b_2 \leq 2.3 \cdot 10^{-47}\right) \land b_2 \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	22.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{-307}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4
Error	39.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.75 \cdot 10^{+33}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b_2}\\ \end{array} \]

Alternative 5
Error	39.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 8.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b_2}\\ \end{array} \]

Alternative 6
Error	39.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 8.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b_2}\\ \end{array} \]

Alternative 7
Error	22.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 9.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 8
Error	45.3
Cost	320

\[b_2 \cdot \frac{-2}{a} \]

Reproduce

herbie shell --seed 2023088 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))