quad2p (problem 3.2.1, positive)

Average Error: 52.83% → 10.31%

Time: 13.9s

Precision: binary64

Cost: 7564

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - a \cdot c}\\ \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{+111}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.2 \cdot 10^{-285}:\\ \;\;\;\;\frac{t_0 - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 5.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{-c}{b_2 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* a c)))))
   (if (<= b_2 -3.8e+111)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
     (if (<= b_2 -1.2e-285)
       (/ (- t_0 b_2) a)
       (if (<= b_2 5.2e+99) (/ (- c) (+ b_2 t_0)) (/ (* c -0.5) 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 t_0 = sqrt(((b_2 * b_2) - (a * c)));
	double tmp;
	if (b_2 <= -3.8e+111) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= -1.2e-285) {
		tmp = (t_0 - b_2) / a;
	} else if (b_2 <= 5.2e+99) {
		tmp = -c / (b_2 + t_0);
	} else {
		tmp = (c * -0.5) / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (a * c)))
    if (b_2 <= (-3.8d+111)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= (-1.2d-285)) then
        tmp = (t_0 - b_2) / a
    else if (b_2 <= 5.2d+99) then
        tmp = -c / (b_2 + t_0)
    else
        tmp = (c * (-0.5d0)) / 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 t_0 = Math.sqrt(((b_2 * b_2) - (a * c)));
	double tmp;
	if (b_2 <= -3.8e+111) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= -1.2e-285) {
		tmp = (t_0 - b_2) / a;
	} else if (b_2 <= 5.2e+99) {
		tmp = -c / (b_2 + t_0);
	} else {
		tmp = (c * -0.5) / 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):
	t_0 = math.sqrt(((b_2 * b_2) - (a * c)))
	tmp = 0
	if b_2 <= -3.8e+111:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= -1.2e-285:
		tmp = (t_0 - b_2) / a
	elif b_2 <= 5.2e+99:
		tmp = -c / (b_2 + t_0)
	else:
		tmp = (c * -0.5) / 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)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))
	tmp = 0.0
	if (b_2 <= -3.8e+111)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= -1.2e-285)
		tmp = Float64(Float64(t_0 - b_2) / a);
	elseif (b_2 <= 5.2e+99)
		tmp = Float64(Float64(-c) / Float64(b_2 + t_0));
	else
		tmp = Float64(Float64(c * -0.5) / 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)
	t_0 = sqrt(((b_2 * b_2) - (a * c)));
	tmp = 0.0;
	if (b_2 <= -3.8e+111)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= -1.2e-285)
		tmp = (t_0 - b_2) / a;
	elseif (b_2 <= 5.2e+99)
		tmp = -c / (b_2 + t_0);
	else
		tmp = (c * -0.5) / 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_] := 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, -3.8e+111], 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, -1.2e-285], N[(N[(t$95$0 - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 5.2e+99], N[((-c) / N[(b$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.79999999999999976e111

   1. Initial program 77.01

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

   2. Simplified 77.01

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

      [Start] 77.01	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 77.01	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 77.01	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 5.52

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

   if -3.79999999999999976e111 < b_2 < -1.2e-285

   1. Initial program 14.05

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

   2. Simplified 14.05

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

      [Start] 14.05	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 14.05	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 14.05	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   if -1.2e-285 < b_2 < 5.1999999999999999e99

   1. Initial program 49.11

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

   2. Simplified 49.11

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

      [Start] 49.11	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 49.11	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 49.11	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 49.41

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

   4. Applied egg-rr 55.97

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

   5. Simplified 55.97

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

      [Start] 55.97	\[ \frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, a \cdot c\right)}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      *-commutative [=>] 55.97	\[ \frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot a}\right)}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \]
      *-commutative [=>] 55.97	\[ \frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}\right)} \]

   6. Applied egg-rr 57.26

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

   7. Simplified 24

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

      [Start] 57.26	\[ \frac{b_2}{a} \cdot \frac{b_2}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} + \left(-\frac{\mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\right) \]
      sub-neg [<=] 57.26	\[ \color{blue}{\frac{b_2}{a} \cdot \frac{b_2}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} - \frac{\mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}} \]
      associate-*r/ [=>] 57.65	\[ \color{blue}{\frac{\frac{b_2}{a} \cdot b_2}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}} - \frac{\mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]
      associate-*l/ [=>] 57.6	\[ \frac{\color{blue}{\frac{b_2 \cdot b_2}{a}}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} - \frac{\mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]
      associate-/r* [<=] 56.25	\[ \color{blue}{\frac{b_2 \cdot b_2}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}} - \frac{\mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)} \]
      div-sub [<=] 55.97	\[ \color{blue}{\frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{a \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}} \]
      associate-/r* [=>] 49.14	\[ \color{blue}{\frac{\frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}} \]
      fma-udef [=>] 49.13	\[ \frac{\frac{b_2 \cdot b_2 - \color{blue}{\left(b_2 \cdot b_2 + c \cdot a\right)}}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \]
      associate--r+ [=>] 24	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) - c \cdot a}}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \]
      +-inverses [=>] 24	\[ \frac{\frac{\color{blue}{0} - c \cdot a}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \]
      neg-sub0 [<=] 24	\[ \frac{\frac{\color{blue}{-c \cdot a}}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \]
      distribute-lft-neg-in [=>] 24	\[ \frac{\frac{\color{blue}{\left(-c\right) \cdot a}}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \]
      *-commutative [=>] 24	\[ \frac{\frac{\color{blue}{a \cdot \left(-c\right)}}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \]

   8. Taylor expanded in a around 0 13.38

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

   9. Simplified 13.38

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

      [Start] 13.38	\[ \frac{-1 \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \]
      mul-1-neg [=>] 13.38	\[ \frac{\color{blue}{-c}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \]

   if 5.1999999999999999e99 < b_2

   1. Initial program 92.72

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

   2. Simplified 92.72

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

      [Start] 92.72	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 92.72	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 92.72	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 4.28

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

   4. Simplified 4.26

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

      [Start] 4.28	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 4.26	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      *-commutative [=>] 4.26	\[ \frac{\color{blue}{c \cdot -0.5}}{b_2} \]

3. Recombined 4 regimes into one program.

4. Final simplification 10.31

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{+111}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.2 \cdot 10^{-285}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 5.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.61%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{+111}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.1 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2
Error	22.79%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.6 \cdot 10^{-144}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.5 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	57.85%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.35 \cdot 10^{-229}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]

Alternative 4
Error	36.2%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.05 \cdot 10^{-229}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]

Alternative 5
Error	36.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.62 \cdot 10^{-229}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]

Alternative 6
Error	36%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.35 \cdot 10^{-229}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 7
Error	92.56%
Cost	256

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

Reproduce

herbie shell --seed 2023090 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))