Quadratic roots, full range

Average Error: 34.5 → 10.6

Time: 18.9s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 0.00146:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{c}{\frac{\frac{{b}^{3}}{a}}{c}} + \frac{c}{b}\right)\\ \end{array} \]

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 -1.35e+154)
   (/ (- b) a)
   (if (<= b 0.00146)
     (/ (- (sqrt (+ (* b b) (* c (* a -4.0)))) b) (* a 2.0))
     (- (+ (/ c (/ (/ (pow b 3.0) a) c)) (/ 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 <= -1.35e+154) {
		tmp = -b / a;
	} else if (b <= 0.00146) {
		tmp = (sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -((c / ((pow(b, 3.0) / a) / c)) + (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 <= (-1.35d+154)) then
        tmp = -b / a
    else if (b <= 0.00146d0) then
        tmp = (sqrt(((b * b) + (c * (a * (-4.0d0))))) - b) / (a * 2.0d0)
    else
        tmp = -((c / (((b ** 3.0d0) / a) / c)) + (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 <= -1.35e+154) {
		tmp = -b / a;
	} else if (b <= 0.00146) {
		tmp = (Math.sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -((c / ((Math.pow(b, 3.0) / a) / c)) + (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 <= -1.35e+154:
		tmp = -b / a
	elif b <= 0.00146:
		tmp = (math.sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0)
	else:
		tmp = -((c / ((math.pow(b, 3.0) / a) / c)) + (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 <= -1.35e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 0.00146)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-Float64(Float64(c / Float64(Float64((b ^ 3.0) / a) / c)) + 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 <= -1.35e+154)
		tmp = -b / a;
	elseif (b <= 0.00146)
		tmp = (sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a * 2.0);
	else
		tmp = -((c / (((b ^ 3.0) / a) / c)) + (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, -1.35e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 0.00146], 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[(N[(c / N[(N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      /-rgt-identity [<=] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      *-commutative [=>] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
      associate-/l* [=>] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
      associate-/l* [<=] 64.0	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
      associate-*r/ [<=] 64.0	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
      /-rgt-identity [<=] 64.0	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
      metadata-eval [<=] 64.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

   3. Taylor expanded in b around -inf 2.8

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

   4. Simplified 2.8

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

      [Start] 2.8	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 2.8	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 2.8	\[ \frac{\color{blue}{-b}}{a} \]

   if -1.35000000000000003e154 < b < 0.0014599999999999999

   1. Initial program 15.3

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

   if 0.0014599999999999999 < b

   1. Initial program 56.1

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

   2. Simplified 56.1

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

      [Start] 56.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 56.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 18.7

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]

   4. Simplified 5.5

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}} \]
      Proof

      [Start] 18.7	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b} \]
      +-commutative [=>] 18.7	\[ \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
      mul-1-neg [=>] 18.7	\[ -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
      unsub-neg [=>] 18.7	\[ \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
      mul-1-neg [=>] 18.7	\[ \color{blue}{\left(-\frac{c}{b}\right)} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      distribute-neg-frac [=>] 18.7	\[ \color{blue}{\frac{-c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      associate-/l* [=>] 16.6	\[ \frac{-c}{b} - \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}} \]
      unpow2 [=>] 16.6	\[ \frac{-c}{b} - \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}} \]
      associate-/l* [=>] 5.5	\[ \frac{-c}{b} - \color{blue}{\frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 0.00146:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{c}{\frac{\frac{{b}^{3}}{a}}{c}} + \frac{c}{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.8
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 0.00122:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{c}{\frac{\frac{{b}^{3}}{a}}{c}} + \frac{c}{b}\right)\\ \end{array} \]

Alternative 2
Error	14.1
Cost	7496

\[\begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 0.00122:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{c}{\frac{\frac{{b}^{3}}{a}}{c}} + \frac{c}{b}\right)\\ \end{array} \]

Alternative 3
Error	14.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 0.008:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Error	14.0
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 0.00122:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Error	39.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 350000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	22.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-258}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Error	56.1
Cost	192

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

Reproduce

herbie shell --seed 2023039 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))