quad2m (problem 3.2.1, negative)

Average Error: 33.9 → 10.4

Time: 19.3s

Precision: binary64

Cost: 7824

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -1.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -3.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - t_0}}{a}\\ \mathbf{elif}\;b_2 \leq -2.05 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{expm1}\left(-0.5 \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{+60}:\\ \;\;\;\;\frac{-t_0}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

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 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= b_2 -1.25e+73)
     (/ (* -0.5 c) b_2)
     (if (<= b_2 -3.5e-19)
       (/ (/ (* c (- a)) (- b_2 t_0)) a)
       (if (<= b_2 -2.05e-67)
         (expm1 (* -0.5 (/ c b_2)))
         (if (<= b_2 8e+60)
           (- (/ (- t_0) a) (/ b_2 a))
           (+ (* (/ b_2 a) -2.0) (* (/ c b_2) 0.5))))))))

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 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -1.25e+73) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -3.5e-19) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= -2.05e-67) {
		tmp = expm1((-0.5 * (c / b_2)));
	} else if (b_2 <= 8e+60) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -1.25e+73) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -3.5e-19) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= -2.05e-67) {
		tmp = Math.expm1((-0.5 * (c / b_2)));
	} else if (b_2 <= 8e+60) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

↓

def code(a, b_2, c):
	t_0 = math.sqrt(((b_2 * b_2) - (c * a)))
	tmp = 0
	if b_2 <= -1.25e+73:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= -3.5e-19:
		tmp = ((c * -a) / (b_2 - t_0)) / a
	elif b_2 <= -2.05e-67:
		tmp = math.expm1((-0.5 * (c / b_2)))
	elif b_2 <= 8e+60:
		tmp = (-t_0 / a) - (b_2 / a)
	else:
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5)
	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 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -1.25e+73)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -3.5e-19)
		tmp = Float64(Float64(Float64(c * Float64(-a)) / Float64(b_2 - t_0)) / a);
	elseif (b_2 <= -2.05e-67)
		tmp = expm1(Float64(-0.5 * Float64(c / b_2)));
	elseif (b_2 <= 8e+60)
		tmp = Float64(Float64(Float64(-t_0) / a) - Float64(b_2 / a));
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + Float64(Float64(c / b_2) * 0.5));
	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[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.25e+73], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -3.5e-19], N[(N[(N[(c * (-a)), $MachinePrecision] / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -2.05e-67], N[(Exp[N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[b$95$2, 8e+60], N[(N[((-t$95$0) / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b_2 < -1.24999999999999994e73

   1. Initial program 58.0

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

   2. Taylor expanded in b_2 around -inf 3.5

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

   3. Applied egg-rr 3.5

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

   if -1.24999999999999994e73 < b_2 < -3.50000000000000015e-19

   1. Initial program 45.6

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

   2. Applied egg-rr 15.8

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

   3. Simplified 15.8

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

      [Start] 15.8	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [=>] 15.8	\[ \frac{\frac{\color{blue}{0 - \left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-commutative [=>] 15.8	\[ \frac{\frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-inverses [=>] 15.8	\[ \frac{\frac{0 - \left(\color{blue}{0} + a \cdot c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      associate--r+ [=>] 15.8	\[ \frac{\frac{\color{blue}{\left(0 - 0\right) - a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      metadata-eval [=>] 15.8	\[ \frac{\frac{\color{blue}{0} - a \cdot c}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [<=] 15.8	\[ \frac{\frac{\color{blue}{-a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      distribute-lft-neg-in [=>] 15.8	\[ \frac{\frac{\color{blue}{\left(-a\right) \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 15.8	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 15.8	\[ \frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}}{a} \]

   if -3.50000000000000015e-19 < b_2 < -2.0499999999999999e-67

   1. Initial program 38.3

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

   2. Taylor expanded in b_2 around -inf 26.3

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

   3. Applied egg-rr 31.0

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

   4. Taylor expanded in c around 0 34.4

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

   if -2.0499999999999999e-67 < b_2 < 7.9999999999999996e60

   1. Initial program 13.9

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

   2. Applied egg-rr 13.9

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]

   3. Simplified 13.9

      \[\leadsto \color{blue}{\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}} \]
      Proof

      [Start] 13.9	\[ \frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      div0 [=>] 13.9	\[ \color{blue}{0} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      +-commutative [=>] 13.9	\[ 0 - \color{blue}{\left(\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
      associate--r+ [=>] 13.9	\[ \color{blue}{\left(0 - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]
      neg-sub0 [<=] 13.9	\[ \color{blue}{\left(-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} - \frac{b_2}{a} \]
      distribute-neg-frac [=>] 13.9	\[ \color{blue}{\frac{-\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} - \frac{b_2}{a} \]
      *-commutative [=>] 13.9	\[ \frac{-\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}{a} - \frac{b_2}{a} \]

   if 7.9999999999999996e60 < b_2

   1. Initial program 39.2

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

   2. Taylor expanded in b_2 around inf 4.6

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -3.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq -2.05 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{expm1}\left(-0.5 \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{+60}:\\ \;\;\;\;\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	15.7
Cost	7897

\[\begin{array}{l} t_0 := \sqrt{c \cdot \left(-a\right)}\\ t_1 := \frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1600000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -5 \cdot 10^{-71}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.3 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 8.6 \cdot 10^{-114} \lor \neg \left(b_2 \leq 1.7 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{b_2}{a} + \frac{t_0}{a}\right)\\ \end{array} \]

Alternative 2
Error	15.7
Cost	7897

\[\begin{array}{l} t_0 := \sqrt{c \cdot \left(-a\right)}\\ \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1800000000000:\\ \;\;\;\;\frac{-1}{\frac{a}{t_0}} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -1.3 \cdot 10^{-67}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.9 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{elif}\;b_2 \leq 9 \cdot 10^{-114} \lor \neg \left(b_2 \leq 6.8 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{b_2}{a} + \frac{t_0}{a}\right)\\ \end{array} \]

Alternative 3
Error	11.9
Cost	7824

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -480000:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{elif}\;b_2 \leq -9.5 \cdot 10^{-68}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{+60}:\\ \;\;\;\;\frac{-t_0}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4
Error	11.9
Cost	7824

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1700000000000:\\ \;\;\;\;\frac{-1}{\frac{a}{t_0}} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -4.5 \cdot 10^{-69}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{+60}:\\ \;\;\;\;\frac{-t_0}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 5
Error	15.8
Cost	7769

\[\begin{array}{l} t_0 := \frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{if}\;b_2 \leq -3.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -450000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -2.9 \cdot 10^{-69}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.3 \cdot 10^{-136} \lor \neg \left(b_2 \leq 8.6 \cdot 10^{-114}\right) \land b_2 \leq 5.2 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 6
Error	11.9
Cost	7696

\[\begin{array}{l} t_0 := \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{if}\;b_2 \leq -4.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1800000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -4.8 \cdot 10^{-68}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 7
Error	22.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 8
Error	36.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.5 \cdot 10^{-194}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 9
Error	36.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.5 \cdot 10^{-194}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 10
Error	22.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.5 \cdot 10^{-194}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 11
Error	53.4
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-303}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 12
Error	56.4
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023039 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))