Quadratic roots, full range

Average Error: 34.1 → 10.5

Time: 20.2s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+135}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 430:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

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 -7e+135)
   (- (/ c b) (/ b a))
   (if (<= b 430.0)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 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 <= -7e+135) {
		tmp = (c / b) - (b / a);
	} else if (b <= 430.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (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 <= (-7d+135)) then
        tmp = (c / b) - (b / a)
    else if (b <= 430.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (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 <= -7e+135) {
		tmp = (c / b) - (b / a);
	} else if (b <= 430.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (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 <= -7e+135:
		tmp = (c / b) - (b / a)
	elif b <= 430.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (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 <= -7e+135)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 430.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / 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 <= -7e+135)
		tmp = (c / b) - (b / a);
	elseif (b <= 430.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (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, -7e+135], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 430.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -7.0000000000000005e135

   1. Initial program 56.5

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

   2. Simplified 56.5

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

      [Start] 56.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 56.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 64.0

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\left(4 \cdot \left(a \cdot c\right) - b \cdot b\right) \cdot \left(4 \cdot \left(a \cdot c\right) - b \cdot b\right)}}}{a \cdot 2} \]

   4. Taylor expanded in a around 0 2.4

      \[\leadsto \frac{\color{blue}{\left(\left(-2 \cdot \frac{c}{b} + 4 \cdot \frac{c}{b}\right) \cdot a + -1 \cdot b\right) - b}}{a \cdot 2} \]

   5. Simplified 2.4

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

      [Start] 2.4	\[ \frac{\left(\left(-2 \cdot \frac{c}{b} + 4 \cdot \frac{c}{b}\right) \cdot a + -1 \cdot b\right) - b}{a \cdot 2} \]
      rational.json-simplify-1 [=>] 2.4	\[ \frac{\color{blue}{\left(-1 \cdot b + \left(-2 \cdot \frac{c}{b} + 4 \cdot \frac{c}{b}\right) \cdot a\right)} - b}{a \cdot 2} \]
      rational.json-simplify-2 [=>] 2.4	\[ \frac{\left(\color{blue}{b \cdot -1} + \left(-2 \cdot \frac{c}{b} + 4 \cdot \frac{c}{b}\right) \cdot a\right) - b}{a \cdot 2} \]
      rational.json-simplify-8 [<=] 2.4	\[ \frac{\left(\color{blue}{\left(-b\right)} + \left(-2 \cdot \frac{c}{b} + 4 \cdot \frac{c}{b}\right) \cdot a\right) - b}{a \cdot 2} \]
      rational.json-simplify-2 [=>] 2.4	\[ \frac{\left(\left(-b\right) + \color{blue}{a \cdot \left(-2 \cdot \frac{c}{b} + 4 \cdot \frac{c}{b}\right)}\right) - b}{a \cdot 2} \]
      rational.json-simplify-2 [=>] 2.4	\[ \frac{\left(\left(-b\right) + a \cdot \left(\color{blue}{\frac{c}{b} \cdot -2} + 4 \cdot \frac{c}{b}\right)\right) - b}{a \cdot 2} \]
      rational.json-simplify-47 [=>] 2.4	\[ \frac{\left(\left(-b\right) + a \cdot \color{blue}{\left(\frac{c}{b} \cdot \left(4 + -2\right)\right)}\right) - b}{a \cdot 2} \]
      metadata-eval [=>] 2.4	\[ \frac{\left(\left(-b\right) + a \cdot \left(\frac{c}{b} \cdot \color{blue}{2}\right)\right) - b}{a \cdot 2} \]
      rational.json-simplify-43 [=>] 2.4	\[ \frac{\left(\left(-b\right) + \color{blue}{\frac{c}{b} \cdot \left(2 \cdot a\right)}\right) - b}{a \cdot 2} \]

   6. Taylor expanded in b around 0 2.2

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

   7. Simplified 2.2

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

      [Start] 2.2	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational.json-simplify-2 [=>] 2.2	\[ \frac{c}{b} + \color{blue}{\frac{b}{a} \cdot -1} \]
      rational.json-simplify-9 [=>] 2.2	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

   8. Taylor expanded in c around 0 2.2

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

   9. Simplified 2.2

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

      [Start] 2.2	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational.json-simplify-5 [<=] 2.2	\[ \color{blue}{\left(\frac{c}{b} - 0\right)} + -1 \cdot \frac{b}{a} \]
      metadata-eval [<=] 2.2	\[ \left(\frac{c}{b} - \color{blue}{\left(-1 - -1\right)}\right) + -1 \cdot \frac{b}{a} \]
      rational.json-simplify-44 [<=] 2.3	\[ \color{blue}{\left(-1 - \left(-1 - \frac{c}{b}\right)\right)} + -1 \cdot \frac{b}{a} \]
      rational.json-simplify-2 [=>] 2.3	\[ \left(-1 - \left(-1 - \frac{c}{b}\right)\right) + \color{blue}{\frac{b}{a} \cdot -1} \]
      rational.json-simplify-8 [<=] 2.3	\[ \left(-1 - \left(-1 - \frac{c}{b}\right)\right) + \color{blue}{\left(-\frac{b}{a}\right)} \]
      rational.json-simplify-12 [=>] 2.3	\[ \left(-1 - \left(-1 - \frac{c}{b}\right)\right) + \color{blue}{\left(0 - \frac{b}{a}\right)} \]
      metadata-eval [<=] 2.3	\[ \left(-1 - \left(-1 - \frac{c}{b}\right)\right) + \left(\color{blue}{\left(1 - 1\right)} - \frac{b}{a}\right) \]
      rational.json-simplify-46 [<=] 11.4	\[ \left(-1 - \left(-1 - \frac{c}{b}\right)\right) + \color{blue}{\left(1 - \left(1 + \frac{b}{a}\right)\right)} \]
      rational.json-simplify-1 [<=] 11.4	\[ \left(-1 - \left(-1 - \frac{c}{b}\right)\right) + \left(1 - \color{blue}{\left(\frac{b}{a} + 1\right)}\right) \]
      rational.json-simplify-64 [<=] 11.4	\[ \color{blue}{\left(-\left(-1 - \frac{c}{b}\right)\right) - \left(\frac{b}{a} + 1\right)} \]
      rational.json-simplify-13 [<=] 11.4	\[ \color{blue}{\left(0 - \left(-1 - \frac{c}{b}\right)\right)} - \left(\frac{b}{a} + 1\right) \]
      rational.json-simplify-44 [=>] 11.4	\[ \color{blue}{\left(\frac{c}{b} - \left(-1 - 0\right)\right)} - \left(\frac{b}{a} + 1\right) \]
      metadata-eval [=>] 11.4	\[ \left(\frac{c}{b} - \color{blue}{-1}\right) - \left(\frac{b}{a} + 1\right) \]
      rational.json-simplify-42 [<=] 11.4	\[ \color{blue}{\left(\frac{c}{b} - \left(\frac{b}{a} + 1\right)\right) - -1} \]
      rational.json-simplify-46 [=>] 11.4	\[ \color{blue}{\left(\left(\frac{c}{b} - \frac{b}{a}\right) - 1\right)} - -1 \]
      rational.json-simplify-45 [=>] 2.2	\[ \color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right) - \left(1 + -1\right)} \]
      metadata-eval [=>] 2.2	\[ \left(\frac{c}{b} - \frac{b}{a}\right) - \color{blue}{0} \]
      rational.json-simplify-5 [=>] 2.2	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -7.0000000000000005e135 < b < 430

   1. Initial program 15.6

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

   2. Simplified 15.6

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

      [Start] 15.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 15.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 15.6

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

   4. Simplified 15.6

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

      [Start] 15.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2} \]
      rational.json-simplify-2 [=>] 15.6	\[ \frac{\sqrt{b \cdot b - 4 \cdot \color{blue}{\left(c \cdot a\right)}} - b}{a \cdot 2} \]
      rational.json-simplify-43 [=>] 15.6	\[ \frac{\sqrt{b \cdot b - \color{blue}{c \cdot \left(a \cdot 4\right)}} - b}{a \cdot 2} \]

   if 430 < b

   1. Initial program 56.2

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

   2. Simplified 56.2

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

      [Start] 56.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 56.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 16.3

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c \cdot a}{b}}}{a \cdot 2} \]

   4. Taylor expanded in c around 0 5.1

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

   5. Simplified 5.1

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

      [Start] 5.1	\[ -1 \cdot \frac{c}{b} \]
      rational.json-simplify-2 [=>] 5.1	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational.json-simplify-9 [=>] 5.1	\[ \color{blue}{-\frac{c}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+135}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 430:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.3
Cost	7504

\[\begin{array}{l} t_0 := a \cdot \frac{c}{b}\\ t_1 := t_0 - b\\ t_2 := \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.46 \cdot 10^{-46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -2.95 \cdot 10^{-74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.85 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(t_1 \cdot \left(t_1 \cdot 4\right)\right) \cdot \frac{-1}{2 \cdot \left(b - t_0\right)}}{a \cdot 2}\\ \mathbf{elif}\;b \leq 600:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 2
Error	14.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-47}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 320:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 3
Error	19.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-149}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 10^{-123}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 4
Error	22.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 5
Error	39.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1350:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	22.7
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{-291}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 7
Error	56.7
Cost	192

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

Reproduce

herbie shell --seed 2023077 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))