Quadratic roots, narrow range

Average Error: 28.6 → 5.9

Time: 4.8s

Precision: binary64

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

\[\left(-0.25 \cdot \frac{{\left(-2 \cdot {\left(c \cdot a\right)}^{2}\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} - \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) - \frac{c}{b} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (-
  (+
   (*
    -0.25
    (/
     (+
      (pow (* -2.0 (pow (* c a) 2.0)) 2.0)
      (* 16.0 (* (pow c 4.0) (pow a 4.0))))
     (* a (pow b 7.0))))
   (-
    (* -2.0 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))
    (/ (* a (pow c 2.0)) (pow b 3.0))))
  (/ 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) {
	return ((-0.25 * ((pow((-2.0 * pow((c * a), 2.0)), 2.0) + (16.0 * (pow(c, 4.0) * pow(a, 4.0)))) / (a * pow(b, 7.0)))) + ((-2.0 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0))) - ((a * pow(c, 2.0)) / pow(b, 3.0)))) - (c / b);
}

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
    code = (((-0.25d0) * (((((-2.0d0) * ((c * a) ** 2.0d0)) ** 2.0d0) + (16.0d0 * ((c ** 4.0d0) * (a ** 4.0d0)))) / (a * (b ** 7.0d0)))) + (((-2.0d0) * (((c ** 3.0d0) * (a ** 2.0d0)) / (b ** 5.0d0))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0)))) - (c / b)
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) {
	return ((-0.25 * ((Math.pow((-2.0 * Math.pow((c * a), 2.0)), 2.0) + (16.0 * (Math.pow(c, 4.0) * Math.pow(a, 4.0)))) / (a * Math.pow(b, 7.0)))) + ((-2.0 * ((Math.pow(c, 3.0) * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

↓

def code(a, b, c):
	return ((-0.25 * ((math.pow((-2.0 * math.pow((c * a), 2.0)), 2.0) + (16.0 * (math.pow(c, 4.0) * math.pow(a, 4.0)))) / (a * math.pow(b, 7.0)))) + ((-2.0 * ((math.pow(c, 3.0) * math.pow(a, 2.0)) / math.pow(b, 5.0))) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))) - (c / b)

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)
	return Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(-2.0 * (Float64(c * a) ^ 2.0)) ^ 2.0) + Float64(16.0 * Float64((c ^ 4.0) * (a ^ 4.0)))) / Float64(a * (b ^ 7.0)))) + Float64(Float64(-2.0 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

↓

function tmp = code(a, b, c)
	tmp = ((-0.25 * ((((-2.0 * ((c * a) ^ 2.0)) ^ 2.0) + (16.0 * ((c ^ 4.0) * (a ^ 4.0)))) / (a * (b ^ 7.0)))) + ((-2.0 * (((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) - ((a * (c ^ 2.0)) / (b ^ 3.0)))) - (c / b);
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_] := N[(N[(N[(-0.25 * N[(N[(N[Power[N[(-2.0 * N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(16.0 * N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.6

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

2. Simplified 28.6

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

3. Taylor expanded in b around inf 5.9

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

4. Applied egg-rr 5.9

   \[\leadsto -1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \color{blue}{{\left(c \cdot a\right)}^{2}}\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]

5. Final simplification 5.9

   \[\leadsto \left(-0.25 \cdot \frac{{\left(-2 \cdot {\left(c \cdot a\right)}^{2}\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} - \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) - \frac{c}{b} \]

Reproduce

herbie shell --seed 2022169 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))