quad2p (problem 3.2.1, positive)

Average Error: 34.3 → 10.2

Time: 13.5s

Precision: binary64

Cost: 14344

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3 \cdot 10^{+61}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + 2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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 -3e+61)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 8e-96)
     (/
      (- (sqrt (+ (- (* b_2 b_2) (* a c)) (* 2.0 (fma a (- c) (* a c))))) b_2)
      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 <= -3e+61) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 8e-96) {
		tmp = (sqrt((((b_2 * b_2) - (a * c)) + (2.0 * fma(a, -c, (a * c))))) - b_2) / a;
	} else {
		tmp = (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 <= -3e+61)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 8e-96)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(b_2 * b_2) - Float64(a * c)) + Float64(2.0 * fma(a, Float64(-c), Float64(a * c))))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -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, -3e+61], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 8e-96], N[(N[(N[Sqrt[N[(N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(a * (-c) + N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3e61

   1. Initial program 40.2

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

   2. Simplified 40.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b_2 around -inf 5.1

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

   4. Simplified 5.1

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

   if -3e61 < b_2 < 7.9999999999999993e-96

   1. Initial program 12.5

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

   2. Simplified 12.5

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 12.6

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

   4. Simplified 12.6

      \[\leadsto \frac{\sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}} - b_2}{a} \]
      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 7.9999999999999993e-96 < b_2

   1. Initial program 52.6

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

   2. Simplified 52.6

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b_2 around inf 10.3

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

   4. Simplified 10.3

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
      Proof
      (/.f64 (*.f64 c -1/2) b_2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 c b_2) -1/2)): 2 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.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 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3 \cdot 10^{+61}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + 2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3 \cdot 10^{+61}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.12 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	39.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.25 \cdot 10^{-299}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	39.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.25 \cdot 10^{-299}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	22.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.2 \cdot 10^{-299}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 6
Error	53.4
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.25 \cdot 10^{-299}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	56.4
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022339 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))