Quadratic roots, full range

Average Error: 34.3 → 10.2

Time: 16.3s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\left(\frac{c \cdot 2}{\frac{b}{a}} - b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -9.5e+95)
   (/ (- (- (/ (* c 2.0) (/ b a)) b) b) (* 2.0 a))
   (if (<= b 1.55e-75)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 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 <= -9.5e+95) {
		tmp = ((((c * 2.0) / (b / a)) - b) - b) / (2.0 * a);
	} else if (b <= 1.55e-75) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (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 <= (-9.5d+95)) then
        tmp = ((((c * 2.0d0) / (b / a)) - b) - b) / (2.0d0 * a)
    else if (b <= 1.55d-75) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (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 <= -9.5e+95) {
		tmp = ((((c * 2.0) / (b / a)) - b) - b) / (2.0 * a);
	} else if (b <= 1.55e-75) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (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 <= -9.5e+95:
		tmp = ((((c * 2.0) / (b / a)) - b) - b) / (2.0 * a)
	elif b <= 1.55e-75:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (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 <= -9.5e+95)
		tmp = Float64(Float64(Float64(Float64(Float64(c * 2.0) / Float64(b / a)) - b) - b) / Float64(2.0 * a));
	elseif (b <= 1.55e-75)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / 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 <= -9.5e+95)
		tmp = ((((c * 2.0) / (b / a)) - b) - b) / (2.0 * a);
	elseif (b <= 1.55e-75)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (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, -9.5e+95], N[(N[(N[(N[(N[(c * 2.0), $MachinePrecision] / N[(b / a), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-75], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.5000000000000004e95

   1. Initial program 47.0

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

   2. Simplified 47.0

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

      [Start] 47.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 47.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 10.7

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

   4. Simplified 4.0

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

      [Start] 10.7	\[ \frac{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} + -1 \cdot b\right)}{a \cdot 2} \]
      mul-1-neg [=>] 10.7	\[ \frac{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} + \color{blue}{\left(-b\right)}\right)}{a \cdot 2} \]
      unsub-neg [=>] 10.7	\[ \frac{\left(-b\right) + \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{a \cdot 2} \]
      *-commutative [=>] 10.7	\[ \frac{\left(-b\right) + \left(\color{blue}{\frac{c \cdot a}{b} \cdot 2} - b\right)}{a \cdot 2} \]
      associate-/l* [=>] 4.0	\[ \frac{\left(-b\right) + \left(\color{blue}{\frac{c}{\frac{b}{a}}} \cdot 2 - b\right)}{a \cdot 2} \]
      associate-*l/ [=>] 4.0	\[ \frac{\left(-b\right) + \left(\color{blue}{\frac{c \cdot 2}{\frac{b}{a}}} - b\right)}{a \cdot 2} \]

   if -9.5000000000000004e95 < b < 1.55000000000000003e-75

   1. Initial program 13.2

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

   if 1.55000000000000003e-75 < b

   1. Initial program 52.9

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

   2. Simplified 52.9

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

      [Start] 52.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 52.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 9.2

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

   4. Simplified 9.2

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

      [Start] 9.2	\[ -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 9.2	\[ \color{blue}{-\frac{c}{b}} \]
      distribute-neg-frac [=>] 9.2	\[ \color{blue}{\frac{-c}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\left(\frac{c \cdot 2}{\frac{b}{a}} - b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(\frac{c \cdot 2}{\frac{b}{a}} - b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Error	13.4
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-61}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Error	22.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Error	39.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 390:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	22.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-283}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Error	56.3
Cost	192

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

Reproduce

herbie shell --seed 2023032 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))