quad2m (problem 3.2.1, negative)

Average Error: 33.9 → 9.5

Time: 14.8s

Precision: binary64

Cost: 14408

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-90}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

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 -2.9e-90)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5.5e+111)
     (/
      (-
       (- b_2)
       (sqrt (+ (- (* b_2 b_2) (* c a)) (* 2.0 (fma a (- c) (* c a))))))
      a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c 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 <= -2.9e-90) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.5e+111) {
		tmp = (-b_2 - sqrt((((b_2 * b_2) - (c * a)) + (2.0 * fma(a, -c, (c * a)))))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / 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 <= -2.9e-90)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5.5e+111)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(Float64(b_2 * b_2) - Float64(c * a)) + Float64(2.0 * fma(a, Float64(-c), Float64(c * a)))))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / 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, -2.9e-90], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5.5e+111], N[(N[((-b$95$2) - N[Sqrt[N[(N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(a * (-c) + N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.89999999999999983e-90

   1. Initial program 53.0

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

   2. Applied egg-rr 47.4

      \[\leadsto \color{blue}{\left(b_2 + \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right) \cdot \frac{1}{-a}} \]

   3. Taylor expanded in b_2 around -inf 64.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]

   4. Simplified 8.9

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      Proof
      (/.f64 (*.f64 -1/2 c) b_2): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= metadata-eval (*.f64 1/2 -1)) c) b_2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 -1 c))) b_2): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 1/2 (*.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))) c)) b_2): 232 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 1/2 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)) c)) b_2): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 1/2 (Rewrite<= *-commutative_binary64 (*.f64 c (pow.f64 (sqrt.f64 -1) 2)))) b_2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (*.f64 c (pow.f64 (sqrt.f64 -1) 2)) b_2))): 0 points increase in error, 0 points decrease in error

   if -2.89999999999999983e-90 < b_2 < 5.4999999999999998e111

   1. Initial program 12.0

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

   2. Applied egg-rr 12.0

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]

   3. Simplified 12.0

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]
      Proof
      (+.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a)) (*.f64 2 (fma.f64 a (neg.f64 c) (*.f64 c a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (*.f64 b_2 b_2) (Rewrite<= *-commutative_binary64 (*.f64 a c))) (*.f64 2 (fma.f64 a (neg.f64 c) (*.f64 c a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) (*.f64 2 (fma.f64 a (neg.f64 c) (Rewrite<= *-commutative_binary64 (*.f64 a c))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)) (Rewrite<= count-2_binary64 (+.f64 (fma.f64 a (neg.f64 c) (*.f64 a c)) (fma.f64 a (neg.f64 c) (*.f64 a c))))): 0 points increase in error, 0 points decrease in error

   if 5.4999999999999998e111 < b_2

   1. Initial program 49.9

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

   2. Taylor expanded in b_2 around inf 3.0

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-90}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.5
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.75 \cdot 10^{-90}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2 \cdot 10^{+108}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2
Error	12.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.55 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3
Error	13.2
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.56 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4
Error	36.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.6 \cdot 10^{-254}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 5
Error	22.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.6 \cdot 10^{-253}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \]

Alternative 6
Error	22.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{-253}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \]

Alternative 7
Error	22.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8.5 \cdot 10^{-254}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 8
Error	53.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.4 \cdot 10^{-281}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 9
Error	56.3
Cost	192

\[\frac{0}{a} \]

Reproduce

herbie shell --seed 2022329 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))