quad2m (problem 3.2.1, negative)

Average Error: 33.9 → 8.6

Time: 14.1s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \mathbf{elif}\;b_2 \leq -1 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - t_0}}{a}\\ \mathbf{elif}\;b_2 \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{\frac{b_2}{a}} \cdot 0.5 - b_2\right) - b_2}{a}\\ \end{array} \]

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
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= b_2 -1.1e+71)
     (/ c (* b_2 -2.0))
     (if (<= b_2 -1e-130)
       (/ (/ (* c (- a)) (- b_2 t_0)) a)
       (if (<= b_2 2e+81)
         (/ (- (- b_2) t_0) a)
         (/ (- (- (* (/ c (/ b_2 a)) 0.5) b_2) b_2) 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 t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -1.1e+71) {
		tmp = c / (b_2 * -2.0);
	} else if (b_2 <= -1e-130) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= 2e+81) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = ((((c / (b_2 / a)) * 0.5) - b_2) - b_2) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

↓

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    if (b_2 <= (-1.1d+71)) then
        tmp = c / (b_2 * (-2.0d0))
    else if (b_2 <= (-1d-130)) then
        tmp = ((c * -a) / (b_2 - t_0)) / a
    else if (b_2 <= 2d+81) then
        tmp = (-b_2 - t_0) / a
    else
        tmp = ((((c / (b_2 / a)) * 0.5d0) - b_2) - b_2) / a
    end if
    code = tmp
end function

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 <= -1.1e+71) {
		tmp = c / (b_2 * -2.0);
	} else if (b_2 <= -1e-130) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= 2e+81) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = ((((c / (b_2 / a)) * 0.5) - b_2) - b_2) / a;
	}
	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 <= -1.1e+71:
		tmp = c / (b_2 * -2.0)
	elif b_2 <= -1e-130:
		tmp = ((c * -a) / (b_2 - t_0)) / a
	elif b_2 <= 2e+81:
		tmp = (-b_2 - t_0) / a
	else:
		tmp = ((((c / (b_2 / a)) * 0.5) - b_2) - b_2) / 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)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -1.1e+71)
		tmp = Float64(c / Float64(b_2 * -2.0));
	elseif (b_2 <= -1e-130)
		tmp = Float64(Float64(Float64(c * Float64(-a)) / Float64(b_2 - t_0)) / a);
	elseif (b_2 <= 2e+81)
		tmp = Float64(Float64(Float64(-b_2) - t_0) / a);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(c / Float64(b_2 / a)) * 0.5) - b_2) - b_2) / a);
	end
	return tmp
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

↓

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt(((b_2 * b_2) - (c * a)));
	tmp = 0.0;
	if (b_2 <= -1.1e+71)
		tmp = c / (b_2 * -2.0);
	elseif (b_2 <= -1e-130)
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	elseif (b_2 <= 2e+81)
		tmp = (-b_2 - t_0) / a;
	else
		tmp = ((((c / (b_2 / a)) * 0.5) - b_2) - b_2) / a;
	end
	tmp_2 = 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, -1.1e+71], N[(c / N[(b$95$2 * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -1e-130], N[(N[(N[(c * (-a)), $MachinePrecision] / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2e+81], N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - b$95$2), $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.09999999999999997e71

   1. Initial program 58.7

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

   2. Taylor expanded in b_2 around -inf 14.6

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

   3. Applied egg-rr 17.8

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

   4. Taylor expanded in a around 0 3.0

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

   5. Simplified 3.0

      \[\leadsto \color{blue}{\frac{c}{b_2 \cdot -2}} \]
      Proof
      (/.f64 c (*.f64 b_2 -2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 c -2) b_2)): 10 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 c -2))) b_2): 0 points increase in error, 10 points decrease in error
      (/.f64 (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 a a)) (/.f64 c -2)) b_2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a c) (*.f64 a -2))) b_2): 10 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 a c) (*.f64 (*.f64 a -2) b_2))): 10 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 a (*.f64 a -2)) (/.f64 c b_2))): 0 points increase in error, 10 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 a a) -2)) (/.f64 c b_2)): 10 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite=> *-inverses_binary64 1) -2) (/.f64 c b_2)): 9 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> metadata-eval -1/2) (/.f64 c b_2)): 0 points increase in error, 10 points decrease in error

   if -1.09999999999999997e71 < b_2 < -1.0000000000000001e-130

   1. Initial program 39.0

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

   2. Applied egg-rr 15.1

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

   3. Simplified 15.1

      \[\leadsto \frac{\color{blue}{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]
      Proof
      (/.f64 (/.f64 (*.f64 c (neg.f64 a)) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 a) c)) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 0 points increase in error, 10 points decrease in error
      (/.f64 (/.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 a c))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 10 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 a c))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 4 points increase in error, 6 points decrease in error
      (/.f64 (/.f64 (-.f64 (Rewrite<= metadata-eval (-.f64 0 0)) (*.f64 a c)) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 6 points increase in error, 4 points decrease in error
      (/.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 0 (+.f64 0 (*.f64 a c)))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 10 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (-.f64 0 (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 b_2 b_2) (*.f64 b_2 b_2))) (*.f64 a c))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 0 points increase in error, 10 points decrease in error
      (/.f64 (/.f64 (-.f64 0 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 a c) (-.f64 (*.f64 b_2 b_2) (*.f64 b_2 b_2))))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 0 points increase in error, 10 points decrease in error
      (/.f64 (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (+.f64 (*.f64 a c) (-.f64 (*.f64 b_2 b_2) (*.f64 b_2 b_2))))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (neg.f64 (+.f64 (*.f64 a c) (-.f64 (*.f64 b_2 b_2) (*.f64 b_2 b_2)))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (Rewrite<= *-commutative_binary64 (*.f64 a c)))))) a): 10 points increase in error, 0 points decrease in error

   if -1.0000000000000001e-130 < b_2 < 1.99999999999999984e81

   1. Initial program 11.3

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

   if 1.99999999999999984e81 < b_2

   1. Initial program 42.5

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

   2. Taylor expanded in b_2 around inf 10.9

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

   3. Simplified 4.8

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}}{a} \]
      Proof
      (/.f64 (/.f64 (*.f64 c (neg.f64 a)) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 a) c)) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 0 points increase in error, 10 points decrease in error
      (/.f64 (/.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 a c))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 10 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 a c))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 4 points increase in error, 6 points decrease in error
      (/.f64 (/.f64 (-.f64 (Rewrite<= metadata-eval (-.f64 0 0)) (*.f64 a c)) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 6 points increase in error, 4 points decrease in error
      (/.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 0 (+.f64 0 (*.f64 a c)))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 10 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (-.f64 0 (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 b_2 b_2) (*.f64 b_2 b_2))) (*.f64 a c))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 0 points increase in error, 10 points decrease in error
      (/.f64 (/.f64 (-.f64 0 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 a c) (-.f64 (*.f64 b_2 b_2) (*.f64 b_2 b_2))))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 0 points increase in error, 10 points decrease in error
      (/.f64 (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (+.f64 (*.f64 a c) (-.f64 (*.f64 b_2 b_2) (*.f64 b_2 b_2))))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 c a))))) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (neg.f64 (+.f64 (*.f64 a c) (-.f64 (*.f64 b_2 b_2) (*.f64 b_2 b_2)))) (-.f64 b_2 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (Rewrite<= *-commutative_binary64 (*.f64 a c)))))) a): 10 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \mathbf{elif}\;b_2 \leq -1 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{\frac{b_2}{a}} \cdot 0.5 - b_2\right) - b_2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.5
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \mathbf{elif}\;b_2 \leq 3 \cdot 10^{+86}:\\ \;\;\;\;\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{\frac{b_2}{a}} \cdot 0.5 - b_2\right) - b_2}{a}\\ \end{array} \]

Alternative 2
Error	9.5
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.95 \cdot 10^{-62}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \mathbf{elif}\;b_2 \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{\frac{b_2}{a}} \cdot 0.5 - b_2\right) - b_2}{a}\\ \end{array} \]

Alternative 3
Error	13.2
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \mathbf{elif}\;b_2 \leq 1.02 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4
Error	36.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 5
Error	36.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.3 \cdot 10^{-207}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 6
Error	22.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{-207}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 7
Error	59.2
Cost	256

\[\frac{-b_2}{a} \]

Reproduce

herbie shell --seed 2022343 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))