Quadratic roots, narrow range

Average Error: 28.7 → 4.9

Time: 6.7s

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

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.3065115648253984:\\ \;\;\;\;\frac{1}{\frac{b + \sqrt{t_0}}{t_0 - b \cdot b}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5 - \mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))))
   (if (<= b 1.3065115648253984)
     (* (/ 1.0 (/ (+ b (sqrt t_0)) (- t_0 (* b b)))) (/ 0.5 a))
     (-
      (* (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0)) -5.0)
      (fma
       2.0
       (/ (* (* a a) (pow c 3.0)) (pow b 5.0))
       (fma (/ (* c c) (pow b 3.0)) 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 t_0 = fma(a, (c * -4.0), (b * b));
	double tmp;
	if (b <= 1.3065115648253984) {
		tmp = (1.0 / ((b + sqrt(t_0)) / (t_0 - (b * b)))) * (0.5 / a);
	} else {
		tmp = (((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0)) * -5.0) - fma(2.0, (((a * a) * pow(c, 3.0)) / pow(b, 5.0)), fma(((c * c) / pow(b, 3.0)), a, (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)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 1.3065115648253984)
		tmp = Float64(Float64(1.0 / Float64(Float64(b + sqrt(t_0)) / Float64(t_0 - Float64(b * b)))) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(Float64((c ^ 4.0) * (a ^ 3.0)) / (b ^ 7.0)) * -5.0) - fma(2.0, Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 5.0)), fma(Float64(Float64(c * c) / (b ^ 3.0)), a, Float64(c / b))));
	end
	return 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_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.3065115648253984], N[(N[(1.0 / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * -5.0), $MachinePrecision] - N[(2.0 * N[(N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * a + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 1.3065115648253984

   1. Initial program 11.5

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

   2. Simplified 11.5

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

   3. Applied egg-rr 10.7

      \[\leadsto \color{blue}{\frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}}} \cdot \frac{0.5}{a} \]

   if 1.3065115648253984 < b

   1. Initial program 32.0

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

   2. Simplified 32.0

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

   3. Taylor expanded in a around 0 4.0

      \[\leadsto \color{blue}{\left(-\left(2 \cdot \frac{c \cdot a}{b} + \left(2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(10 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}} + 4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)\right)\right)} \cdot \frac{0.5}{a} \]

   4. Simplified 4.0

      \[\leadsto \color{blue}{\left(-2 \cdot \mathsf{fma}\left(\frac{c}{b}, a, \frac{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}{{b}^{3}}\right) - \mathsf{fma}\left(4, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, 10 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}}\right)\right)} \cdot \frac{0.5}{a} \]

   5. Taylor expanded in c around 0 3.8

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

   6. Simplified 3.8

      \[\leadsto \color{blue}{\frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5 - \mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3065115648253984:\\ \;\;\;\;\frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} \cdot -5 - \mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))