quad2p (problem 3.2.1, positive)

Average Error: 34.6 → 10.4

Time: 18.3s

Precision: binary64

Cost: 13960

Math

\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.4 \cdot 10^{+147}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, c \cdot \frac{c}{\frac{{b_2}^{3}}{a}}, c \cdot \frac{-0.5}{b_2}\right)\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.4e+147)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 3.8e-33)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (fma -0.125 (* c (/ c (/ (pow b_2 3.0) a))) (* c (/ -0.5 b_2))))))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.4e+147) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3.8e-33) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = fma(-0.125, (c * (c / (pow(b_2, 3.0) / a))), (c * (-0.5 / b_2)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

↓

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.4e+147)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 3.8e-33)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = fma(-0.125, Float64(c * Float64(c / Float64((b_2 ^ 3.0) / a))), Float64(c * Float64(-0.5 / b_2)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.4e+147], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.8e-33], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.125 * N[(c * N[(c / N[(N[Power[b$95$2, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.39999999999999958e147

   1. Initial program 61.6

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 1.8

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   if -6.39999999999999958e147 < b_2 < 3.79999999999999994e-33

   1. Initial program 14.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 3.79999999999999994e-33 < b_2

   1. Initial program 55.1

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 19.6

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}} + -0.5 \cdot \frac{c}{b_2}} \]

   3. Simplified 7.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{c}{\frac{{b_2}^{3}}{a}} \cdot c, \frac{-0.5}{b_2} \cdot c\right)} \]
      Proof
      (fma.f64 -1/8 (*.f64 (/.f64 c (/.f64 (pow.f64 b_2 3) a)) c) (*.f64 (/.f64 -1/2 b_2) c)): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (Rewrite<= associate-/r/_binary64 (/.f64 c (/.f64 (/.f64 (pow.f64 b_2 3) a) c))) (*.f64 (/.f64 -1/2 b_2) c)): 2 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c c) (/.f64 (pow.f64 b_2 3) a))) (*.f64 (/.f64 -1/2 b_2) c)): 41 points increase in error, 1 points decrease in error
      (fma.f64 -1/8 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (/.f64 (pow.f64 b_2 3) a)) (*.f64 (/.f64 -1/2 b_2) c)): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b_2 3))) (*.f64 (/.f64 -1/2 b_2) c)): 9 points increase in error, 7 points decrease in error
      (fma.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b_2 3)) (Rewrite<= associate-/r/_binary64 (/.f64 -1/2 (/.f64 b_2 c)))): 16 points increase in error, 14 points decrease in error
      (fma.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b_2 3)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1/2 c) b_2))): 5 points increase in error, 13 points decrease in error
      (fma.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b_2 3)) (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.f64 c b_2)))): 1 points increase in error, 1 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) a) (pow.f64 b_2 3))) (*.f64 -1/2 (/.f64 c b_2)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.4 \cdot 10^{+147}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, c \cdot \frac{c}{\frac{{b_2}^{3}}{a}}, c \cdot \frac{-0.5}{b_2}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.4 \cdot 10^{+147}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \end{array} \]

Alternative 2
Error	13.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.5 \cdot 10^{-61}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \end{array} \]

Alternative 3
Error	22.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.15 \cdot 10^{-247}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \end{array} \]

Alternative 4
Error	22.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.15 \cdot 10^{-247}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \end{array} \]

Alternative 5
Error	21.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.15 \cdot 10^{-247}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \end{array} \]

Alternative 6
Error	45.0
Cost	320

\[b_2 \cdot \frac{-2}{a} \]

Alternative 7
Error	59.2
Cost	256

\[\frac{-b_2}{a} \]

Reproduce

herbie shell --seed 2022318 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))