quad2m (problem 3.2.1, negative)

Average Error: 35.1 → 10.8

Time: 16.3s

Precision: binary64

Cost: 7696

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot c}{b_2}\\ \mathbf{if}\;b_2 \leq -1.08 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -7 \cdot 10^{-124}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -1.4 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 10^{+120}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

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
 (let* ((t_0 (/ (* -0.5 c) b_2)))
   (if (<= b_2 -1.08e-103)
     t_0
     (if (<= b_2 -7e-124)
       (/ (- (- b_2) (sqrt (* c (- a)))) a)
       (if (<= b_2 -1.4e-156)
         t_0
         (if (<= b_2 1e+120)
           (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
           (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c 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 t_0 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -1.08e-103) {
		tmp = t_0;
	} else if (b_2 <= -7e-124) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else if (b_2 <= -1.4e-156) {
		tmp = t_0;
	} else if (b_2 <= 1e+120) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	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 = ((-0.5d0) * c) / b_2
    if (b_2 <= (-1.08d-103)) then
        tmp = t_0
    else if (b_2 <= (-7d-124)) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else if (b_2 <= (-1.4d-156)) then
        tmp = t_0
    else if (b_2 <= 1d+120) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    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 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -1.08e-103) {
		tmp = t_0;
	} else if (b_2 <= -7e-124) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else if (b_2 <= -1.4e-156) {
		tmp = t_0;
	} else if (b_2 <= 1e+120) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	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 = (-0.5 * c) / b_2
	tmp = 0
	if b_2 <= -1.08e-103:
		tmp = t_0
	elif b_2 <= -7e-124:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	elif b_2 <= -1.4e-156:
		tmp = t_0
	elif b_2 <= 1e+120:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / 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)
	t_0 = Float64(Float64(-0.5 * c) / b_2)
	tmp = 0.0
	if (b_2 <= -1.08e-103)
		tmp = t_0;
	elseif (b_2 <= -7e-124)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	elseif (b_2 <= -1.4e-156)
		tmp = t_0;
	elseif (b_2 <= 1e+120)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	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 = (-0.5 * c) / b_2;
	tmp = 0.0;
	if (b_2 <= -1.08e-103)
		tmp = t_0;
	elseif (b_2 <= -7e-124)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	elseif (b_2 <= -1.4e-156)
		tmp = t_0;
	elseif (b_2 <= 1e+120)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	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[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]}, If[LessEqual[b$95$2, -1.08e-103], t$95$0, If[LessEqual[b$95$2, -7e-124], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -1.4e-156], t$95$0, If[LessEqual[b$95$2, 1e+120], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.0799999999999999e-103 or -6.9999999999999997e-124 < b_2 < -1.4000000000000001e-156

   1. Initial program 50.9

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

   2. Taylor expanded in b_2 around -inf 22.8

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

   3. Simplified 20.0

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a}{\frac{b_2}{c}}}}{a} \]
      Proof
      (*.f64 -1/2 (/.f64 a (/.f64 b_2 c))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a c) b_2))): 43 points increase in error, 46 points decrease in error
      (*.f64 -1/2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 c a)) b_2)): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 19.6

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

   5. Applied egg-rr 12.0

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

   if -1.0799999999999999e-103 < b_2 < -6.9999999999999997e-124

   1. Initial program 29.6

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

   2. Taylor expanded in b_2 around 0 31.8

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

   3. Simplified 31.8

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]
      Proof
      (*.f64 c (neg.f64 a)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c a))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 c a))): 0 points increase in error, 0 points decrease in error

   if -1.4000000000000001e-156 < b_2 < 9.9999999999999998e119

   1. Initial program 11.3

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

   if 9.9999999999999998e119 < b_2

   1. Initial program 52.6

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

   2. Applied egg-rr 36.6

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

   3. Taylor expanded in b_2 around inf 64.0

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

   4. Simplified 4.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(b_2, 2, \frac{c \cdot -0.5}{\frac{b_2}{a}}\right)} \cdot \frac{1}{-a} \]
      Proof
      (fma.f64 b_2 2 (/.f64 (*.f64 c -1/2) (/.f64 b_2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b_2 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/2 c)) (/.f64 b_2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b_2 2 (/.f64 (*.f64 (Rewrite<= metadata-eval (*.f64 1/2 -1)) c) (/.f64 b_2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b_2 2 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 -1 c))) (/.f64 b_2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b_2 2 (/.f64 (*.f64 1/2 (Rewrite<= *-commutative_binary64 (*.f64 c -1))) (/.f64 b_2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b_2 2 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (*.f64 c -1) (/.f64 b_2 a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b_2 2 (*.f64 1/2 (/.f64 (*.f64 c (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))) (/.f64 b_2 a)))): 209 points increase in error, 0 points decrease in error
      (fma.f64 b_2 2 (*.f64 1/2 (/.f64 (*.f64 c (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2))) (/.f64 b_2 a)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b_2 2 (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 c (pow.f64 (sqrt.f64 -1) 2)) a) b_2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b_2 2 (*.f64 1/2 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 c (*.f64 (pow.f64 (sqrt.f64 -1) 2) a))) b_2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 b_2 2) (*.f64 1/2 (/.f64 (*.f64 c (*.f64 (pow.f64 (sqrt.f64 -1) 2) a)) b_2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 b_2)) (*.f64 1/2 (/.f64 (*.f64 c (*.f64 (pow.f64 (sqrt.f64 -1) 2) a)) b_2))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in b_2 around 0 3.7

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.08 \cdot 10^{-103}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -7 \cdot 10^{-124}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -1.4 \cdot 10^{-156}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+120}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.1
Cost	7504

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot c}{b_2}\\ t_1 := \frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{if}\;b_2 \leq -1.08 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -7 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -1.4 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 1.95 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 2
Error	39.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 3
Error	22.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.22 \cdot 10^{-249}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 4
Error	53.4
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 5
Error	56.4
Cost	192

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

Reproduce

herbie shell --seed 2022318 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))