Quadratic roots, full range

Average Error: 34.4 → 10.1

Time: 18.2s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b} - b}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -3.6e+101)
   (/ (- (* c (/ a b)) b) a)
   (if (<= b 3.8e-49)
     (/ (- (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 <= -3.6e+101) {
		tmp = ((c * (a / b)) - b) / a;
	} else if (b <= 3.8e-49) {
		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 <= (-3.6d+101)) then
        tmp = ((c * (a / b)) - b) / a
    else if (b <= 3.8d-49) 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 <= -3.6e+101) {
		tmp = ((c * (a / b)) - b) / a;
	} else if (b <= 3.8e-49) {
		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 <= -3.6e+101:
		tmp = ((c * (a / b)) - b) / a
	elif b <= 3.8e-49:
		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 <= -3.6e+101)
		tmp = Float64(Float64(Float64(c * Float64(a / b)) - b) / a);
	elseif (b <= 3.8e-49)
		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 <= -3.6e+101)
		tmp = ((c * (a / b)) - b) / a;
	elseif (b <= 3.8e-49)
		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, -3.6e+101], N[(N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.8e-49], 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 < -3.60000000000000029e101

   1. Initial program 47.9

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

   2. Simplified 47.9

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

      [Start] 47.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 47.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around -inf 9.7

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

   4. Simplified 3.9

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

      [Start] 9.7	\[ \frac{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}{a \cdot 2} \]
      fma-def [=>] 9.7	\[ \frac{\color{blue}{\mathsf{fma}\left(2, \frac{c \cdot a}{b}, -2 \cdot b\right)}}{a \cdot 2} \]
      associate-/l* [=>] 3.9	\[ \frac{\mathsf{fma}\left(2, \color{blue}{\frac{c}{\frac{b}{a}}}, -2 \cdot b\right)}{a \cdot 2} \]
      *-commutative [=>] 3.9	\[ \frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot -2}\right)}{a \cdot 2} \]

   5. Applied egg-rr 4.2

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(b \cdot -2\right) + \frac{0.5}{a} \cdot \left(\frac{c}{\frac{b}{a}} \cdot 2\right)} \]

   6. Simplified 3.8

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

      [Start] 4.2	\[ \frac{0.5}{a} \cdot \left(b \cdot -2\right) + \frac{0.5}{a} \cdot \left(\frac{c}{\frac{b}{a}} \cdot 2\right) \]
      distribute-lft-out [=>] 4.2	\[ \color{blue}{\frac{0.5}{a} \cdot \left(b \cdot -2 + \frac{c}{\frac{b}{a}} \cdot 2\right)} \]
      +-commutative [<=] 4.2	\[ \frac{0.5}{a} \cdot \color{blue}{\left(\frac{c}{\frac{b}{a}} \cdot 2 + b \cdot -2\right)} \]
      *-commutative [=>] 4.2	\[ \frac{0.5}{a} \cdot \left(\color{blue}{2 \cdot \frac{c}{\frac{b}{a}}} + b \cdot -2\right) \]
      fma-def [=>] 4.2	\[ \frac{0.5}{a} \cdot \color{blue}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)} \]
      associate-*l/ [=>] 3.9	\[ \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{a}} \]

   if -3.60000000000000029e101 < b < 3.7999999999999997e-49

   1. Initial program 13.7

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

   2. Simplified 13.7

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

      [Start] 13.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 13.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 13.7

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

   if 3.7999999999999997e-49 < b

   1. Initial program 54.2

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

   2. Simplified 54.2

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

      [Start] 54.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      *-commutative [=>] 54.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 8.2

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

   4. Simplified 8.2

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

      [Start] 8.2	\[ -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 8.2	\[ \color{blue}{-\frac{c}{b}} \]
      distribute-neg-frac [=>] 8.2	\[ \color{blue}{\frac{-c}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b} - b}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b} - b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-52}:\\ \;\;\;\;\left(\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b} - b}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(c \cdot a\right) \cdot -4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Error	13.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b} - b}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Error	13.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-63}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b} - b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Error	40.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	22.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 10^{-227}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Error	62.3
Cost	192

\[\frac{b}{a} \]

Alternative 8
Error	56.7
Cost	192

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

Reproduce

herbie shell --seed 2023031 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))