quad2m (problem 3.2.1, negative)

Average Error: 33.9 → 11.0

Time: 17.6s

Precision: binary64

Cost: 26892

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{if}\;b_2 \leq -2 \cdot 10^{-105}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.4 \cdot 10^{-239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -2.6 \cdot 10^{-274}:\\ \;\;\;\;\frac{\left(-b_2\right) - {\left(e^{0.25 \cdot \left(\log \left(-c\right) - \log \left(\frac{1}{a}\right)\right)}\right)}^{2}}{a}\\ \mathbf{elif}\;b_2 \leq 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

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 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)))
   (if (<= b_2 -2e-105)
     (* -0.5 (/ c b_2))
     (if (<= b_2 -2.4e-239)
       t_0
       (if (<= b_2 -2.6e-274)
         (/
          (- (- b_2) (pow (exp (* 0.25 (- (log (- c)) (log (/ 1.0 a))))) 2.0))
          a)
         (if (<= b_2 1e+60) t_0 (/ (* b_2 -2.0) 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 = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	double tmp;
	if (b_2 <= -2e-105) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -2.4e-239) {
		tmp = t_0;
	} else if (b_2 <= -2.6e-274) {
		tmp = (-b_2 - pow(exp((0.25 * (log(-c) - log((1.0 / a))))), 2.0)) / a;
	} else if (b_2 <= 1e+60) {
		tmp = t_0;
	} else {
		tmp = (b_2 * -2.0) / 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 = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    if (b_2 <= (-2d-105)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= (-2.4d-239)) then
        tmp = t_0
    else if (b_2 <= (-2.6d-274)) then
        tmp = (-b_2 - (exp((0.25d0 * (log(-c) - log((1.0d0 / a))))) ** 2.0d0)) / a
    else if (b_2 <= 1d+60) then
        tmp = t_0
    else
        tmp = (b_2 * (-2.0d0)) / 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 = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	double tmp;
	if (b_2 <= -2e-105) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -2.4e-239) {
		tmp = t_0;
	} else if (b_2 <= -2.6e-274) {
		tmp = (-b_2 - Math.pow(Math.exp((0.25 * (Math.log(-c) - Math.log((1.0 / a))))), 2.0)) / a;
	} else if (b_2 <= 1e+60) {
		tmp = t_0;
	} else {
		tmp = (b_2 * -2.0) / 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 = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	tmp = 0
	if b_2 <= -2e-105:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= -2.4e-239:
		tmp = t_0
	elif b_2 <= -2.6e-274:
		tmp = (-b_2 - math.pow(math.exp((0.25 * (math.log(-c) - math.log((1.0 / a))))), 2.0)) / a
	elif b_2 <= 1e+60:
		tmp = t_0
	else:
		tmp = (b_2 * -2.0) / 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 = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a)
	tmp = 0.0
	if (b_2 <= -2e-105)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -2.4e-239)
		tmp = t_0;
	elseif (b_2 <= -2.6e-274)
		tmp = Float64(Float64(Float64(-b_2) - (exp(Float64(0.25 * Float64(log(Float64(-c)) - log(Float64(1.0 / a))))) ^ 2.0)) / a);
	elseif (b_2 <= 1e+60)
		tmp = t_0;
	else
		tmp = Float64(Float64(b_2 * -2.0) / 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 = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	tmp = 0.0;
	if (b_2 <= -2e-105)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= -2.4e-239)
		tmp = t_0;
	elseif (b_2 <= -2.6e-274)
		tmp = (-b_2 - (exp((0.25 * (log(-c) - log((1.0 / a))))) ^ 2.0)) / a;
	elseif (b_2 <= 1e+60)
		tmp = t_0;
	else
		tmp = (b_2 * -2.0) / 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[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[b$95$2, -2e-105], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -2.4e-239], t$95$0, If[LessEqual[b$95$2, -2.6e-274], N[(N[((-b$95$2) - N[Power[N[Exp[N[(0.25 * N[(N[Log[(-c)], $MachinePrecision] - N[Log[N[(1.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1e+60], t$95$0, N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.99999999999999993e-105

   1. Initial program 51.2

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

   2. Taylor expanded in b_2 around -inf 11.0

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

   if -1.99999999999999993e-105 < b_2 < -2.39999999999999993e-239 or -2.6e-274 < b_2 < 9.9999999999999995e59

   1. Initial program 12.5

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

   if -2.39999999999999993e-239 < b_2 < -2.6e-274

   1. Initial program 15.5

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

   2. Applied egg-rr 15.7

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

   3. Taylor expanded in a around inf 37.7

      \[\leadsto \frac{\left(-b_2\right) - {\color{blue}{\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{1}{a}\right) + \log \left(-c\right)\right)}\right)}}^{2}}{a} \]

   if 9.9999999999999995e59 < b_2

   1. Initial program 39.4

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

   2. Taylor expanded in b_2 around inf 4.9

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

   3. Simplified 4.9

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-105}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.4 \cdot 10^{-239}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{elif}\;b_2 \leq -2.6 \cdot 10^{-274}:\\ \;\;\;\;\frac{\left(-b_2\right) - {\left(e^{0.25 \cdot \left(\log \left(-c\right) - \log \left(\frac{1}{a}\right)\right)}\right)}^{2}}{a}\\ \mathbf{elif}\;b_2 \leq 10^{+60}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.5
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-105}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+60}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 2
Error	13.5
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-105}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 3
Error	36.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.05 \cdot 10^{-306}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 4
Error	22.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.05 \cdot 10^{-306}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 5
Error	53.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.6 \cdot 10^{-252}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 6
Error	56.5
Cost	192

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

Reproduce

herbie shell --seed 2022289 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))