quad2m (problem 3.2.1, negative)

Average Error: 34.3 → 10.5

Time: 14.2s

Precision: binary64

Cost: 7560

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.15 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.25 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{-a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

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
 (if (<= b_2 -1.15e-90)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.25e+117)
     (- (/ (sqrt (- (* b_2 b_2) (* a c))) (- a)) (/ b_2 a))
     (/ (* 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 tmp;
	if (b_2 <= -1.15e-90) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.25e+117) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) / -a) - (b_2 / a);
	} 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) :: tmp
    if (b_2 <= (-1.15d-90)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.25d+117) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) / -a) - (b_2 / a)
    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 tmp;
	if (b_2 <= -1.15e-90) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.25e+117) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) / -a) - (b_2 / a);
	} 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):
	tmp = 0
	if b_2 <= -1.15e-90:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.25e+117:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) / -a) - (b_2 / a)
	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)
	tmp = 0.0
	if (b_2 <= -1.15e-90)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.25e+117)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) / Float64(-a)) - Float64(b_2 / a));
	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)
	tmp = 0.0;
	if (b_2 <= -1.15e-90)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.25e+117)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) / -a) - (b_2 / a);
	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_] := If[LessEqual[b$95$2, -1.15e-90], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.25e+117], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.1499999999999999e-90

   1. Initial program 51.7

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

   2. Taylor expanded in b_2 around -inf 10.2

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

   if -1.1499999999999999e-90 < b_2 < 1.24999999999999996e117

   1. Initial program 12.8

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

   2. Applied egg-rr 12.8

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

   if 1.24999999999999996e117 < b_2

   1. Initial program 52.1

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

   2. Taylor expanded in b_2 around inf 3.8

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

   3. Simplified 3.8

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

      [Start] 3.8	\[ \frac{-2 \cdot b_2}{a} \]
      rational_best-simplify-2 [<=] 3.8	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.15 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.25 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{-a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.5
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.15 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.15 \cdot 10^{+116}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.15 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3
Error	13.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.15 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.22 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4
Error	20.2
Cost	6984

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.4 \cdot 10^{-203}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.8 \cdot 10^{-178}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Error	22.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6
Error	22.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.4 \cdot 10^{-251}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \]

Alternative 7
Error	22.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.7 \cdot 10^{-249}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 8
Error	39.9
Cost	320

\[-0.5 \cdot \frac{c}{b_2} \]

Reproduce

herbie shell --seed 2023097 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))