Quadratic roots, full range

Average Error: 34.4 → 9.7

Time: 17.6s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+101}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

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 -1.1e+101)
   (+ (/ c b) (- (/ b a)))
   (if (<= b 1.25e-80)
     (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* a 2.0))
     (- (/ c b)))))

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 <= -1.1e+101) {
		tmp = (c / b) + -(b / a);
	} else if (b <= 1.25e-80) {
		tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	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 <= (-1.1d+101)) then
        tmp = (c / b) + -(b / a)
    else if (b <= 1.25d-80) then
        tmp = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (a * 2.0d0)
    else
        tmp = -(c / b)
    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 <= -1.1e+101) {
		tmp = (c / b) + -(b / a);
	} else if (b <= 1.25e-80) {
		tmp = (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	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 <= -1.1e+101:
		tmp = (c / b) + -(b / a)
	elif b <= 1.25e-80:
		tmp = (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0)
	else:
		tmp = -(c / b)
	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 <= -1.1e+101)
		tmp = Float64(Float64(c / b) + Float64(-Float64(b / a)));
	elseif (b <= 1.25e-80)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(a * 2.0));
	else
		tmp = Float64(-Float64(c / b));
	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 <= -1.1e+101)
		tmp = (c / b) + -(b / a);
	elseif (b <= 1.25e-80)
		tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	else
		tmp = -(c / b);
	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, -1.1e+101], N[(N[(c / b), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 1.25e-80], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1e101

   1. Initial program 47.2

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

   2. Simplified 47.2

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

      [Start] 47.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 47.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}}{2 \cdot a} \]
      rational.json-simplify-43 [=>] 47.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 47.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 2.8

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

   4. Simplified 2.8

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

      [Start] 2.8	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational.json-simplify-2 [=>] 2.8	\[ \frac{c}{b} + \color{blue}{\frac{b}{a} \cdot -1} \]
      rational.json-simplify-9 [=>] 2.8	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

   if -1.1e101 < b < 1.25e-80

   1. Initial program 12.9

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

   2. Simplified 12.9

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

      [Start] 12.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 12.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}}{2 \cdot a} \]
      rational.json-simplify-43 [=>] 12.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 12.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   if 1.25e-80 < b

   1. Initial program 53.2

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

   2. Simplified 53.2

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

      [Start] 53.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 53.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}}{2 \cdot a} \]
      rational.json-simplify-43 [=>] 53.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 53.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 8.8

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

   4. Simplified 8.8

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

      [Start] 8.8	\[ -1 \cdot \frac{c}{b} \]
      rational.json-simplify-2 [=>] 8.8	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational.json-simplify-9 [=>] 8.8	\[ \color{blue}{-\frac{c}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+101}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{+101}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-85}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 2
Error	9.7
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 3
Error	13.1
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{-88}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-4 \cdot c\right)} + \left(-b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 4
Error	13.4
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-83}:\\ \;\;\;\;\frac{0.5}{a} \cdot \sqrt{\left(c \cdot a\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 5
Error	19.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-91}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	22.3
Cost	644

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-260}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 7
Error	39.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{+17}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 8
Error	22.3
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-239}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 9
Error	56.8
Cost	192

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

Reproduce

herbie shell --seed 2023074 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))