Quadratic roots, full range

Average Error: 33.9 → 10.0

Time: 15.0s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

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 -3.9e+87)
   (+ (/ c b) (- (/ b a)))
   (if (<= b 2.1e-56)
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
     (- (/ 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 <= -3.9e+87) {
		tmp = (c / b) + -(b / a);
	} else if (b <= 2.1e-56) {
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} 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 <= (-3.9d+87)) then
        tmp = (c / b) + -(b / a)
    else if (b <= 2.1d-56) then
        tmp = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
    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 <= -3.9e+87) {
		tmp = (c / b) + -(b / a);
	} else if (b <= 2.1e-56) {
		tmp = (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} 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 <= -3.9e+87:
		tmp = (c / b) + -(b / a)
	elif b <= 2.1e-56:
		tmp = (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)
	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 <= -3.9e+87)
		tmp = Float64(Float64(c / b) + Float64(-Float64(b / a)));
	elseif (b <= 2.1e-56)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
	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 <= -3.9e+87)
		tmp = (c / b) + -(b / a);
	elseif (b <= 2.1e-56)
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	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, -3.9e+87], N[(N[(c / b), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 2.1e-56], 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], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -3.9000000000000002e87

   1. Initial program 45.1

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

   2. Simplified 45.1

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

      [Start] 45.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 45.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 3.9

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

   4. Simplified 3.9

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

      [Start] 3.9	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational.json-simplify-2 [=>] 3.9	\[ \frac{c}{b} + \color{blue}{\frac{b}{a} \cdot -1} \]
      rational.json-simplify-9 [=>] 3.9	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

   if -3.9000000000000002e87 < b < 2.10000000000000006e-56

   1. Initial program 13.6

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

   if 2.10000000000000006e-56 < b

   1. Initial program 54.4

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

   2. Simplified 54.4

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

      [Start] 54.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational.json-simplify-2 [=>] 54.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 8.0

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

   4. Simplified 8.0

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

      [Start] 8.0	\[ -1 \cdot \frac{c}{b} \]
      rational.json-simplify-2 [=>] 8.0	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational.json-simplify-9 [=>] 8.0	\[ \color{blue}{-\frac{c}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.1
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 2
Error	10.0
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-52}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} + \left(-b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 4
Error	13.7
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{-67}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{c \cdot \left(a \cdot -4\right)} + \left(-b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 5
Error	13.7
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{c}{b} + \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 10^{-51}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	39.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-25}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7
Error	22.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{-249}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 8
Error	56.9
Cost	192

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

Reproduce

herbie shell --seed 2023064 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))