quad2m (problem 3.2.1, negative)

Average Error: 34.2 → 11.9

Time: 15.6s

Precision: binary64

Cost: 20232

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b_2}{a}, -\frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

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 -1.85e-93)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.7e-40)
     (fma -1.0 (/ b_2 a) (- (/ (hypot b_2 (sqrt (* c (- a)))) a)))
     (/ (* b_2 -2.0) a))))

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 <= -1.85e-93) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.7e-40) {
		tmp = fma(-1.0, (b_2 / a), -(hypot(b_2, sqrt((c * -a))) / a));
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	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 <= -1.85e-93)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.7e-40)
		tmp = fma(-1.0, Float64(b_2 / a), Float64(-Float64(hypot(b_2, sqrt(Float64(c * Float64(-a)))) / a)));
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	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, -1.85e-93], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.7e-40], N[(-1.0 * N[(b$95$2 / a), $MachinePrecision] + (-N[(N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] / a), $MachinePrecision])), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.85000000000000001e-93

   1. Initial program 52.4

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

   2. Taylor expanded in b_2 around -inf 9.5

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   3. Applied egg-rr 9.5

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -1.85000000000000001e-93 < b_2 < 2.7e-40

   1. Initial program 15.7

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

   2. Applied egg-rr 16.6

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

   if 2.7e-40 < b_2

   1. Initial program 29.2

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

   2. Taylor expanded in b_2 around inf 10.0

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

   3. Simplified 10.0

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
      Proof
      (*.f64 b_2 -2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -2 b_2)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b_2}{a}, -\frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.1
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 4
Error	39.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 5
Error	23.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.04 \cdot 10^{-190}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 6
Error	23.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.04 \cdot 10^{-190}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \]

Alternative 7
Error	22.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.04 \cdot 10^{-190}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 8
Error	53.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 9
Error	56.2
Cost	192

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

Reproduce

herbie shell --seed 2022308 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))