quad2m (problem 3.2.1, negative)

Average Error: 34.2 → 11.0

Time: 1.4min

Precision: binary64

Cost: 7432

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{c}{-2}}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

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
 (if (<= b_2 -8.5e-19)
   (/ (/ c -2.0) b_2)
   (if (<= b_2 3.6e+71)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* -2.0 (/ 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 tmp;
	if (b_2 <= -8.5e-19) {
		tmp = (c / -2.0) / b_2;
	} else if (b_2 <= 3.6e+71) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = -2.0 * (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) :: tmp
    if (b_2 <= (-8.5d-19)) then
        tmp = (c / (-2.0d0)) / b_2
    else if (b_2 <= 3.6d+71) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = (-2.0d0) * (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 tmp;
	if (b_2 <= -8.5e-19) {
		tmp = (c / -2.0) / b_2;
	} else if (b_2 <= 3.6e+71) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = -2.0 * (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):
	tmp = 0
	if b_2 <= -8.5e-19:
		tmp = (c / -2.0) / b_2
	elif b_2 <= 3.6e+71:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = -2.0 * (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)
	tmp = 0.0
	if (b_2 <= -8.5e-19)
		tmp = Float64(Float64(c / -2.0) / b_2);
	elseif (b_2 <= 3.6e+71)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(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)
	tmp = 0.0;
	if (b_2 <= -8.5e-19)
		tmp = (c / -2.0) / b_2;
	elseif (b_2 <= 3.6e+71)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = -2.0 * (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_] := If[LessEqual[b$95$2, -8.5e-19], N[(N[(c / -2.0), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.6e+71], 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], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.50000000000000003e-19

   1. Initial program 54.3

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

   2. Taylor expanded in b_2 around -inf 18.5

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

   3. Applied egg-rr 6.9

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

   4. Applied egg-rr 6.7

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

   if -8.50000000000000003e-19 < b_2 < 3.6e71

   1. Initial program 16.5

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

   if 3.6e71 < b_2

   1. Initial program 40.6

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

   2. Taylor expanded in b_2 around inf 4.8

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{c}{-2}}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{c}{-2}}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 2
Error	14.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{c}{-2}}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 3
Error	19.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.25 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{c}{-2}}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.1 \cdot 10^{-175}:\\ \;\;\;\;-\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a} + c \cdot \frac{0.5}{b_2}\\ \end{array} \]

Alternative 4
Error	22.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.1 \cdot 10^{-245}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 5
Error	22.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.05 \cdot 10^{-250}:\\ \;\;\;\;\frac{\frac{c}{-2}}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 6
Error	45.5
Cost	320

\[-2 \cdot \frac{b_2}{a} \]

Reproduce

herbie shell --seed 2023099 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))