Quadratic roots, full range

Average Error: 33.6 → 10.2

Time: 14.6s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+97}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 -4.6e+97)
   (/ (- b) a)
   (if (<= b 4.3e-69)
     (/ (- (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 <= -4.6e+97) {
		tmp = -b / a;
	} else if (b <= 4.3e-69) {
		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 <= (-4.6d+97)) then
        tmp = -b / a
    else if (b <= 4.3d-69) 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 <= -4.6e+97) {
		tmp = -b / a;
	} else if (b <= 4.3e-69) {
		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 <= -4.6e+97:
		tmp = -b / a
	elif b <= 4.3e-69:
		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 <= -4.6e+97)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.3e-69)
		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 <= -4.6e+97)
		tmp = -b / a;
	elseif (b <= 4.3e-69)
		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, -4.6e+97], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.3e-69], 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 < -4.60000000000000011e97

   1. Initial program 44.9

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

   2. Simplified 45.0

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]
      Proof
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c 4))) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (neg.f64 (Rewrite=> *-commutative_binary64 (*.f64 4 c))) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 4) c)) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 (neg.f64 4) c)) (*.f64 b b)))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a (neg.f64 4)) c)) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (+.f64 (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4))) c) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (+.f64 (*.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 a))) c) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 (neg.f64 (*.f64 4 a)) c)))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) (neg.f64 b))) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (Rewrite<= metadata-eval (/.f64 1 2)) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) 2) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= associate-/r*_binary64 (/.f64 (neg.f64 -1) (*.f64 2 a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (neg.f64 -1)) (*.f64 2 a))): 11 points increase in error, 29 points decrease in error
      (Rewrite=> associate-/l*_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (*.f64 2 a) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (*.f64 2 a) (Rewrite=> metadata-eval 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite=> /-rgt-identity_binary64 (*.f64 2 a))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b around -inf 3.7

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

   4. Simplified 3.7

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
      Proof
      (/.f64 (neg.f64 b) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 b)) a): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 b a))): 0 points increase in error, 0 points decrease in error

   if -4.60000000000000011e97 < b < 4.3e-69

   1. Initial program 13.4

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

   if 4.3e-69 < b

   1. Initial program 53.6

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

   2. Simplified 53.6

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]
      Proof
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (*.f64 c -4) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c 4))) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (neg.f64 (Rewrite=> *-commutative_binary64 (*.f64 4 c))) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 a (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 4) c)) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 (neg.f64 4) c)) (*.f64 b b)))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a (neg.f64 4)) c)) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (+.f64 (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4))) c) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (+.f64 (*.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 a))) c) (*.f64 b b))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 (neg.f64 (*.f64 4 a)) c)))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) b) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) (neg.f64 b))) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (Rewrite<= metadata-eval (/.f64 1 2)) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) 2) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= associate-/r*_binary64 (/.f64 (neg.f64 -1) (*.f64 2 a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (neg.f64 -1)) (*.f64 2 a))): 11 points increase in error, 29 points decrease in error
      (Rewrite=> associate-/l*_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (*.f64 2 a) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (*.f64 2 a) (Rewrite=> metadata-eval 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite=> /-rgt-identity_binary64 (*.f64 2 a))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in a around 0 9.0

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

   4. Simplified 9.0

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Proof
      (/.f64 (neg.f64 c) b): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 c)) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 c b))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+97}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-69}:\\ \;\;\;\;\left(\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	7496

\[\begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{\frac{a}{0.5 \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Error	13.4
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Error	13.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Error	23.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-304}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 6
Error	39.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-304}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	23.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-226}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8
Error	56.1
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022331 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))