quadm (p42, negative)

Average Error: 33.8 → 10.9

Time: 16.9s

Precision: binary64

Cost: 7688

Math

\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 2}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-14)
   (/ (- c) b)
   (if (<= b 5e+109)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (/ (+ (* 2.0 (* c (/ a b))) (* b -2.0)) (* a 2.0)))))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-14) {
		tmp = -c / b;
	} else if (b <= 5e+109) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = ((2.0 * (c * (a / b))) + (b * -2.0)) / (a * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.8d-14)) then
        tmp = -c / b
    else if (b <= 5d+109) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    else
        tmp = ((2.0d0 * (c * (a / b))) + (b * (-2.0d0))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-14) {
		tmp = -c / b;
	} else if (b <= 5e+109) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = ((2.0 * (c * (a / b))) + (b * -2.0)) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

↓

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-14:
		tmp = -c / b
	elif b <= 5e+109:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = ((2.0 * (c * (a / b))) + (b * -2.0)) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-14)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 5e+109)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(c * Float64(a / b))) + Float64(b * -2.0)) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e-14)
		tmp = -c / b;
	elseif (b <= 5e+109)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = ((2.0 * (c * (a / b))) + (b * -2.0)) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := If[LessEqual[b, -3.8e-14], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 5e+109], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	33.8
Target	21.0
Herbie	10.9

\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if b < -3.8000000000000002e-14

   1. Initial program 55.5

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 6.3

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   3. Simplified 6.3

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Proof

      [Start] 6.3	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 6.3	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 6.3	\[ \frac{\color{blue}{-c}}{b} \]

   if -3.8000000000000002e-14 < b < 5.0000000000000001e109

   1. Initial program 15.8

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   if 5.0000000000000001e109 < b

   1. Initial program 47.9

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 10.0

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

   3. Applied egg-rr 4.1

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)} + -2 \cdot b}{2 \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 2}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.0
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+114}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 2}\\ \end{array} \]

Alternative 2
Error	14.8
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	14.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	23.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	40.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+27}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	23.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-290}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	62.3
Cost	192

\[\frac{b}{a} \]

Alternative 8
Error	56.6
Cost	192

\[\frac{c}{b} \]

Reproduce

herbie shell --seed 2023053 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))