Quadratic roots, full range

Average Error: 34.0 → 10.9

Time: 16.0s

Precision: binary64

Cost: 7889

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-143} \lor \neg \left(b \leq 1.65 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+149)
   (/ (- b) a)
   (if (<= b 6.4e-176)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (if (or (<= b 4.5e-143) (not (<= b 1.65e-79)))
       (/ (- c) b)
       (* (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (/ 0.5 a))))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+149) {
		tmp = -b / a;
	} else if (b <= 6.4e-176) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else if ((b <= 4.5e-143) || !(b <= 1.65e-79)) {
		tmp = -c / b;
	} else {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d+149)) then
        tmp = -b / a
    else if (b <= 6.4d-176) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else if ((b <= 4.5d-143) .or. (.not. (b <= 1.65d-79))) then
        tmp = -c / b
    else
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) * (0.5d0 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+149) {
		tmp = -b / a;
	} else if (b <= 6.4e-176) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else if ((b <= 4.5e-143) || !(b <= 1.65e-79)) {
		tmp = -c / b;
	} else {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

↓

def code(a, b, c):
	tmp = 0
	if b <= -2e+149:
		tmp = -b / a
	elif b <= 6.4e-176:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	elif (b <= 4.5e-143) or not (b <= 1.65e-79):
		tmp = -c / b
	else:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a)
	return tmp

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+149)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6.4e-176)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	elseif ((b <= 4.5e-143) || !(b <= 1.65e-79))
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) * Float64(0.5 / a));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+149)
		tmp = -b / a;
	elseif (b <= 6.4e-176)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	elseif ((b <= 4.5e-143) || ~((b <= 1.65e-79)))
		tmp = -c / b;
	else
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := If[LessEqual[b, -2e+149], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6.4e-176], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 4.5e-143], N[Not[LessEqual[b, 1.65e-79]], $MachinePrecision]], N[((-c) / b), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0000000000000001e149

   1. Initial program 62.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. Simplified 62.1

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
      Proof

      [Start] 62.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 62.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 62.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 62.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 62.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 62.1	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 62.1	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 62.1	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 62.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Taylor expanded in b around -inf 3.0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]

   4. Simplified 3.0

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
      Proof

      [Start] 3.0	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 3.0	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 3.0	\[ \frac{\color{blue}{-b}}{a} \]

   if -2.0000000000000001e149 < b < 6.39999999999999969e-176

   1. Initial program 11.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   if 6.39999999999999969e-176 < b < 4.5e-143 or 1.6499999999999999e-79 < b

   1. Initial program 51.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. Simplified 51.0

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
      Proof

      [Start] 51.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 51.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 11.4

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   4. Simplified 11.4

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Proof

      [Start] 11.4	\[ -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 11.4	\[ \color{blue}{-\frac{c}{b}} \]
      distribute-neg-frac [=>] 11.4	\[ \color{blue}{\frac{-c}{b}} \]

   if 4.5e-143 < b < 1.6499999999999999e-79

   1. Initial program 25.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. Simplified 25.1

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
      Proof

      [Start] 25.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 25.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 25.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 25.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 25.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 25.1	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 25.1	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 25.1	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 25.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Applied egg-rr 25.1

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right) \cdot \frac{0.5}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-143} \lor \neg \left(b \leq 1.65 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.9
Cost	7889

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+147}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-176} \lor \neg \left(b \leq 4 \cdot 10^{-142}\right) \land b \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Error	14.6
Cost	7633

\[\begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-176} \lor \neg \left(b \leq 4.5 \cdot 10^{-143}\right) \land b \leq 8.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Error	14.5
Cost	7633

\[\begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -4\right)} - b\\ \mathbf{if}\;b \leq -6.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-176}:\\ \;\;\;\;\frac{t_0}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-143} \lor \neg \left(b \leq 9.5 \cdot 10^{-82}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot t_0\\ \end{array} \]

Alternative 4
Error	40.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 5
Error	22.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-251}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Error	56.9
Cost	192

\[\frac{c}{b} \]

Reproduce

herbie shell --seed 2023088 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))