quad2m (problem 3.2.1, negative)

Average Error: 34.1 → 10.4

Time: 11.4s

Precision: binary64

Cost: 7556

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-110}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2} + \left(c \cdot -0.125\right) \cdot \left(\frac{c}{{b_2}^{3}} \cdot a\right)\\ \mathbf{elif}\;b_2 \leq 2.3 \cdot 10^{+50}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{0.5}{\frac{b_2}{c}}\\ \end{array} \]

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 -1e-110)
   (+ (* -0.5 (/ c b_2)) (* (* c -0.125) (* (/ c (pow b_2 3.0)) a)))
   (if (<= b_2 2.3e+50)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (/ 0.5 (/ b_2 c))))))

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 <= -1e-110) {
		tmp = (-0.5 * (c / b_2)) + ((c * -0.125) * ((c / pow(b_2, 3.0)) * a));
	} else if (b_2 <= 2.3e+50) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 / (b_2 / c));
	}
	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 <= (-1d-110)) then
        tmp = ((-0.5d0) * (c / b_2)) + ((c * (-0.125d0)) * ((c / (b_2 ** 3.0d0)) * a))
    else if (b_2 <= 2.3d+50) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 / (b_2 / c))
    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 <= -1e-110) {
		tmp = (-0.5 * (c / b_2)) + ((c * -0.125) * ((c / Math.pow(b_2, 3.0)) * a));
	} else if (b_2 <= 2.3e+50) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 / (b_2 / c));
	}
	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 <= -1e-110:
		tmp = (-0.5 * (c / b_2)) + ((c * -0.125) * ((c / math.pow(b_2, 3.0)) * a))
	elif b_2 <= 2.3e+50:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 / (b_2 / c))
	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 <= -1e-110)
		tmp = Float64(Float64(-0.5 * Float64(c / b_2)) + Float64(Float64(c * -0.125) * Float64(Float64(c / (b_2 ^ 3.0)) * a)));
	elseif (b_2 <= 2.3e+50)
		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(b_2 / c)));
	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 <= -1e-110)
		tmp = (-0.5 * (c / b_2)) + ((c * -0.125) * ((c / (b_2 ^ 3.0)) * a));
	elseif (b_2 <= 2.3e+50)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 / (b_2 / c));
	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, -1e-110], N[(N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.125), $MachinePrecision] * N[(N[(c / N[Power[b$95$2, 3.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.3e+50], 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[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.0000000000000001e-110

   1. Initial program 51.5

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

   2. Applied egg-rr 46.2

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

   3. Taylor expanded in b_2 around -inf 64.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} + -0.125 \cdot \frac{{c}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{4} \cdot a\right)}{{b_2}^{3}}} \]

   4. Simplified 13.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{c}{\frac{{b_2}^{3}}{c \cdot a}}, \frac{c \cdot -0.5}{b_2}\right)} \]

   5. Applied egg-rr 11.2

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2} + \left(-0.125 \cdot c\right) \cdot \left(\frac{c}{{b_2}^{3}} \cdot a\right)} \]

   if -1.0000000000000001e-110 < b_2 < 2.29999999999999997e50

   1. Initial program 12.5

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

   if 2.29999999999999997e50 < b_2

   1. Initial program 37.2

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

   2. Taylor expanded in b_2 around inf 4.8

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   3. Applied egg-rr 4.8

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-110}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2} + \left(c \cdot -0.125\right) \cdot \left(\frac{c}{{b_2}^{3}} \cdot a\right)\\ \mathbf{elif}\;b_2 \leq 2.3 \cdot 10^{+50}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.3 \cdot 10^{+50}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.16 \cdot 10^{-115}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 3
Error	14.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 4
Error	39.1
Cost	452

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

Alternative 5
Error	22.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.3 \cdot 10^{-191}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \]

Alternative 6
Error	22.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.3 \cdot 10^{-191}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \]

Alternative 7
Error	22.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.3 \cdot 10^{-191}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 8
Error	53.1
Cost	388

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

Alternative 9
Error	56.2
Cost	192

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

Reproduce

herbie shell --seed 2022228 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))