quad2p (problem 3.2.1, positive)

Average Error: 33.7 → 6.6

Time: 15.9s

Precision: binary64

Cost: 7564

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - a \cdot c}\\ \mathbf{if}\;b_2 \leq -1.6 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -4.2 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_0 - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{-c}{b_2 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

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) (* a c)))))
   (if (<= b_2 -1.6e+101)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
     (if (<= b_2 -4.2e-266)
       (/ (- t_0 b_2) a)
       (if (<= b_2 3.6e+84) (/ (- c) (+ b_2 t_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) - (a * c)));
	double tmp;
	if (b_2 <= -1.6e+101) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= -4.2e-266) {
		tmp = (t_0 - b_2) / a;
	} else if (b_2 <= 3.6e+84) {
		tmp = -c / (b_2 + t_0);
	} else {
		tmp = (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) - (a * c)))
    if (b_2 <= (-1.6d+101)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= (-4.2d-266)) then
        tmp = (t_0 - b_2) / a
    else if (b_2 <= 3.6d+84) then
        tmp = -c / (b_2 + t_0)
    else
        tmp = (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) - (a * c)));
	double tmp;
	if (b_2 <= -1.6e+101) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= -4.2e-266) {
		tmp = (t_0 - b_2) / a;
	} else if (b_2 <= 3.6e+84) {
		tmp = -c / (b_2 + t_0);
	} else {
		tmp = (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) - (a * c)))
	tmp = 0
	if b_2 <= -1.6e+101:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= -4.2e-266:
		tmp = (t_0 - b_2) / a
	elif b_2 <= 3.6e+84:
		tmp = -c / (b_2 + t_0)
	else:
		tmp = (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(a * c)))
	tmp = 0.0
	if (b_2 <= -1.6e+101)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= -4.2e-266)
		tmp = Float64(Float64(t_0 - b_2) / a);
	elseif (b_2 <= 3.6e+84)
		tmp = Float64(Float64(-c) / Float64(b_2 + t_0));
	else
		tmp = 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) - (a * c)));
	tmp = 0.0;
	if (b_2 <= -1.6e+101)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= -4.2e-266)
		tmp = (t_0 - b_2) / a;
	elseif (b_2 <= 3.6e+84)
		tmp = -c / (b_2 + t_0);
	else
		tmp = (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[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.6e+101], 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, -4.2e-266], N[(N[(t$95$0 - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.6e+84], N[((-c) / N[(b$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.60000000000000003e101

   1. Initial program 47.7

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

   2. Simplified 47.7

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

      [Start] 47.7	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 47.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 47.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 3.8

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

   if -1.60000000000000003e101 < b_2 < -4.19999999999999994e-266

   1. Initial program 8.8

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

   2. Simplified 8.8

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

      [Start] 8.8	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 8.8	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 8.8	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   if -4.19999999999999994e-266 < b_2 < 3.5999999999999999e84

   1. Initial program 29.5

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

   2. Simplified 29.5

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

      [Start] 29.5	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 29.5	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 29.5	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 29.5

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

   4. Applied egg-rr 34.5

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

   5. Simplified 29.5

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

      [Start] 34.5	\[ \frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, a \cdot c\right)}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      associate-/r* [=>] 29.5	\[ \color{blue}{\frac{\frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, a \cdot c\right)}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}} \]
      *-commutative [<=] 29.5	\[ \frac{\frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot a}\right)}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}} \]
      *-commutative [<=] 29.5	\[ \frac{\frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}} \]

   6. Taylor expanded in b_2 around 0 8.9

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

   7. Simplified 8.9

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

      [Start] 8.9	\[ \frac{-1 \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \]
      mul-1-neg [=>] 8.9	\[ \frac{\color{blue}{-c}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \]

   if 3.5999999999999999e84 < b_2

   1. Initial program 58.7

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

   2. Simplified 58.7

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

      [Start] 58.7	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 58.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 58.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 2.9

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.6 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -4.2 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	10.6
Cost	7633

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.85 \cdot 10^{+98}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.8 \cdot 10^{-77} \lor \neg \left(b_2 \leq 8 \cdot 10^{-28}\right) \land b_2 \leq 1.65:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 2
Error	14.3
Cost	7569

\[\begin{array}{l} t_0 := \sqrt{a \cdot \left(-c\right)} - b_2\\ \mathbf{if}\;b_2 \leq -7 \cdot 10^{-15}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{t_0}{a}\\ \mathbf{elif}\;b_2 \leq 4.7 \cdot 10^{-28} \lor \neg \left(b_2 \leq 1\right):\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{t_0}}\\ \end{array} \]

Alternative 3
Error	14.3
Cost	7441

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-15}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.7 \cdot 10^{-77} \lor \neg \left(b_2 \leq 8 \cdot 10^{-28}\right) \land b_2 \leq 1:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 4
Error	36.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 3.8 \cdot 10^{-262}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 5
Error	23.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 3 \cdot 10^{-262}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 6
Error	23.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 4.5 \cdot 10^{-262}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 7
Error	59.3
Cost	256

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

Reproduce

herbie shell --seed 2023053 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))