quad2m (problem 3.2.1, negative)

Average Error: 33.8 → 9.9

Time: 18.6s

Precision: binary64

Cost: 14092

Math

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

↓

\[\begin{array}{l} t_0 := b_2 \cdot b_2 - c \cdot a\\ \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-39}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.75 \cdot 10^{-70}:\\ \;\;\;\;\frac{\left(-b_2\right) - {\left({t_0}^{0.25}\right)}^{2}}{a}\\ \mathbf{elif}\;b_2 \leq -1.08 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{c}{\frac{{b_2}^{3}}{c \cdot a}}, \frac{-0.5}{\frac{b_2}{c}}\right)\\ \mathbf{elif}\;b_2 \leq 10^{+145}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{t_0}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \left(\left(\frac{c}{b_2} \cdot 0.5 + 1\right) + -1\right)\\ \end{array} \]

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
 (let* ((t_0 (- (* b_2 b_2) (* c a))))
   (if (<= b_2 -2.2e-39)
     (* -0.5 (/ c b_2))
     (if (<= b_2 -1.75e-70)
       (/ (- (- b_2) (pow (pow t_0 0.25) 2.0)) a)
       (if (<= b_2 -1.08e-94)
         (fma -0.125 (/ c (/ (pow b_2 3.0) (* c a))) (/ -0.5 (/ b_2 c)))
         (if (<= b_2 1e+145)
           (/ (- (- b_2) (sqrt t_0)) a)
           (+ (* -2.0 (/ b_2 a)) (+ (+ (* (/ c b_2) 0.5) 1.0) -1.0))))))))

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 t_0 = (b_2 * b_2) - (c * a);
	double tmp;
	if (b_2 <= -2.2e-39) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -1.75e-70) {
		tmp = (-b_2 - pow(pow(t_0, 0.25), 2.0)) / a;
	} else if (b_2 <= -1.08e-94) {
		tmp = fma(-0.125, (c / (pow(b_2, 3.0) / (c * a))), (-0.5 / (b_2 / c)));
	} else if (b_2 <= 1e+145) {
		tmp = (-b_2 - sqrt(t_0)) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((((c / b_2) * 0.5) + 1.0) + -1.0);
	}
	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)
	t_0 = Float64(Float64(b_2 * b_2) - Float64(c * a))
	tmp = 0.0
	if (b_2 <= -2.2e-39)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -1.75e-70)
		tmp = Float64(Float64(Float64(-b_2) - ((t_0 ^ 0.25) ^ 2.0)) / a);
	elseif (b_2 <= -1.08e-94)
		tmp = fma(-0.125, Float64(c / Float64((b_2 ^ 3.0) / Float64(c * a))), Float64(-0.5 / Float64(b_2 / c)));
	elseif (b_2 <= 1e+145)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(t_0)) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(Float64(Float64(c / b_2) * 0.5) + 1.0) + -1.0));
	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_] := Block[{t$95$0 = N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -2.2e-39], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -1.75e-70], N[(N[((-b$95$2) - N[Power[N[Power[t$95$0, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -1.08e-94], N[(-0.125 * N[(c / N[(N[Power[b$95$2, 3.0], $MachinePrecision] / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1e+145], N[(N[((-b$95$2) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b_2 < -2.20000000000000001e-39

   1. Initial program 54.1

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

   2. Taylor expanded in b_2 around -inf 6.9

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

   if -2.20000000000000001e-39 < b_2 < -1.74999999999999987e-70

   1. Initial program 37.1

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

   2. Applied egg-rr 37.7

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}}}{a} \]

   3. Applied egg-rr 37.3

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   if -1.74999999999999987e-70 < b_2 < -1.08e-94

   1. Initial program 32.7

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

   2. Applied egg-rr 34.0

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

   3. Taylor expanded in b_2 around -inf 64.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} + -0.125 \cdot \frac{{c}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{4} \cdot a\right)}{{b_2}^{3}}} \]

   4. Simplified 35.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{c}{\frac{{b_2}^{3}}{c \cdot a}}, \frac{-0.5}{\frac{b_2}{c}}\right)} \]
      Proof
      (fma.f64 -1/8 (/.f64 c (/.f64 (pow.f64 b_2 3) (*.f64 c a))) (/.f64 -1/2 (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 c (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (pow.f64 b_2 3) a) c))) (/.f64 -1/2 (/.f64 b_2 c))): 17 points increase in error, 8 points decrease in error
      (fma.f64 -1/8 (/.f64 c (/.f64 (/.f64 (pow.f64 b_2 3) (Rewrite<= *-lft-identity_binary64 (*.f64 1 a))) c)) (/.f64 -1/2 (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 c (/.f64 (/.f64 (pow.f64 b_2 3) (*.f64 (Rewrite<= metadata-eval (*.f64 -1 -1)) a)) c)) (/.f64 -1/2 (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 c (/.f64 (/.f64 (pow.f64 b_2 3) (*.f64 (*.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))) -1) a)) c)) (/.f64 -1/2 (/.f64 b_2 c))): 169 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 c (/.f64 (/.f64 (pow.f64 b_2 3) (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)) -1) a)) c)) (/.f64 -1/2 (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 c (/.f64 (/.f64 (pow.f64 b_2 3) (*.f64 (*.f64 (pow.f64 (sqrt.f64 -1) 2) (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))) a)) c)) (/.f64 -1/2 (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 c (/.f64 (/.f64 (pow.f64 b_2 3) (*.f64 (*.f64 (pow.f64 (sqrt.f64 -1) 2) (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2))) a)) c)) (/.f64 -1/2 (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 c (/.f64 (/.f64 (pow.f64 b_2 3) (*.f64 (Rewrite=> pow-sqr_binary64 (pow.f64 (sqrt.f64 -1) (*.f64 2 2))) a)) c)) (/.f64 -1/2 (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 c (/.f64 (/.f64 (pow.f64 b_2 3) (*.f64 (pow.f64 (sqrt.f64 -1) (Rewrite=> metadata-eval 4)) a)) c)) (/.f64 -1/2 (/.f64 b_2 c))): 0 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) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a)))) (/.f64 -1/2 (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (/.f64 (pow.f64 b_2 3) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a))) (/.f64 -1/2 (/.f64 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) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a)) (pow.f64 b_2 3))) (/.f64 -1/2 (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a)) (pow.f64 b_2 3)) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 -1)) (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a)) (pow.f64 b_2 3)) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 -1 (/.f64 b_2 c))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a)) (pow.f64 b_2 3)) (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 c) b_2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a)) (pow.f64 b_2 3)) (*.f64 1/2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 c -1)) b_2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a)) (pow.f64 b_2 3)) (*.f64 1/2 (/.f64 (*.f64 c (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))) b_2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a)) (pow.f64 b_2 3)) (*.f64 1/2 (/.f64 (*.f64 c (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2))) b_2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a)) (pow.f64 b_2 3))) (*.f64 1/2 (/.f64 (*.f64 c (pow.f64 (sqrt.f64 -1) 2)) b_2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (*.f64 c (pow.f64 (sqrt.f64 -1) 2)) b_2)) (*.f64 -1/8 (/.f64 (*.f64 (pow.f64 c 2) (*.f64 (pow.f64 (sqrt.f64 -1) 4) a)) (pow.f64 b_2 3))))): 0 points increase in error, 0 points decrease in error

   if -1.08e-94 < b_2 < 9.9999999999999999e144

   1. Initial program 11.8

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

   if 9.9999999999999999e144 < b_2

   1. Initial program 60.5

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

   2. Taylor expanded in b_2 around inf 2.0

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

   3. Applied egg-rr 5.2

      \[\leadsto -2 \cdot \frac{b_2}{a} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{\frac{b_2}{c}}\right)\right)} \]

   4. Applied egg-rr 2.2

      \[\leadsto -2 \cdot \frac{b_2}{a} + \color{blue}{\left(\left(0.5 \cdot \frac{c}{b_2} + 1\right) - 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-39}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.75 \cdot 10^{-70}:\\ \;\;\;\;\frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25}\right)}^{2}}{a}\\ \mathbf{elif}\;b_2 \leq -1.08 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{c}{\frac{{b_2}^{3}}{c \cdot a}}, \frac{-0.5}{\frac{b_2}{c}}\right)\\ \mathbf{elif}\;b_2 \leq 10^{+145}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \left(\left(\frac{c}{b_2} \cdot 0.5 + 1\right) + -1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	14.4
Cost	7768

\[\begin{array}{l} t_0 := \sqrt{c \cdot \left(-a\right)}\\ t_1 := \frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-39}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.75 \cdot 10^{-70}:\\ \;\;\;\;\frac{-t_0}{a}\\ \mathbf{elif}\;b_2 \leq -1.08 \cdot 10^{-94}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 3.7 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c \cdot 0.5}{b_2}\\ \end{array} \]

Alternative 2
Error	14.5
Cost	7640

\[\begin{array}{l} t_0 := \frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-39}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.75 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -1.08 \cdot 10^{-94}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 3.7 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c \cdot 0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	9.9
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-39}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+145}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \left(\left(\frac{c}{b_2} \cdot 0.5 + 1\right) + -1\right)\\ \end{array} \]

Alternative 4
Error	39.2
Cost	452

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

Alternative 5
Error	22.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{-259}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 6
Error	22.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{-259}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 7
Error	22.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{-259}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 8
Error	53.3
Cost	388

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

Alternative 9
Error	56.4
Cost	192

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

Reproduce

herbie shell --seed 2022294 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))