quad2p (problem 3.2.1, positive)

Average Error: 34.2 → 10.7

Time: 22.8s

Precision: binary64

Cost: 7432

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot c}{b_2}\\ \mathbf{if}\;b_2 \leq -4.4 \cdot 10^{+128}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \mathbf{elif}\;b_2 \leq 5.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \leq 1.92 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 3.9 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (/ (* -0.5 c) b_2)))
   (if (<= b_2 -4.4e+128)
     (/ (* -2.0 b_2) a)
     (if (<= b_2 5.5e-156)
       (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
       (if (<= b_2 1.92e-44)
         t_0
         (if (<= b_2 3.9e-35) (/ (sqrt (- (* c a))) a) t_0))))))

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 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -4.4e+128) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 5.5e-156) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else if (b_2 <= 1.92e-44) {
		tmp = t_0;
	} else if (b_2 <= 3.9e-35) {
		tmp = sqrt(-(c * a)) / a;
	} else {
		tmp = t_0;
	}
	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 = ((-0.5d0) * c) / b_2
    if (b_2 <= (-4.4d+128)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 5.5d-156) then
        tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
    else if (b_2 <= 1.92d-44) then
        tmp = t_0
    else if (b_2 <= 3.9d-35) then
        tmp = sqrt(-(c * a)) / a
    else
        tmp = t_0
    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 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -4.4e+128) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 5.5e-156) {
		tmp = (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else if (b_2 <= 1.92e-44) {
		tmp = t_0;
	} else if (b_2 <= 3.9e-35) {
		tmp = Math.sqrt(-(c * a)) / a;
	} else {
		tmp = t_0;
	}
	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 = (-0.5 * c) / b_2
	tmp = 0
	if b_2 <= -4.4e+128:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 5.5e-156:
		tmp = (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a
	elif b_2 <= 1.92e-44:
		tmp = t_0
	elif b_2 <= 3.9e-35:
		tmp = math.sqrt(-(c * a)) / a
	else:
		tmp = t_0
	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(-0.5 * c) / b_2)
	tmp = 0.0
	if (b_2 <= -4.4e+128)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 5.5e-156)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	elseif (b_2 <= 1.92e-44)
		tmp = t_0;
	elseif (b_2 <= 3.9e-35)
		tmp = Float64(sqrt(Float64(-Float64(c * a))) / a);
	else
		tmp = t_0;
	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 = (-0.5 * c) / b_2;
	tmp = 0.0;
	if (b_2 <= -4.4e+128)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 5.5e-156)
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	elseif (b_2 <= 1.92e-44)
		tmp = t_0;
	elseif (b_2 <= 3.9e-35)
		tmp = sqrt(-(c * a)) / a;
	else
		tmp = t_0;
	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[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]}, If[LessEqual[b$95$2, -4.4e+128], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 5.5e-156], 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], If[LessEqual[b$95$2, 1.92e-44], t$95$0, If[LessEqual[b$95$2, 3.9e-35], N[(N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision] / a), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -4.40000000000000033e128

   1. Initial program 54.2

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

   2. Taylor expanded in b_2 around -inf 3.1

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

   if -4.40000000000000033e128 < b_2 < 5.4999999999999998e-156

   1. Initial program 10.5

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

   if 5.4999999999999998e-156 < b_2 < 1.91999999999999992e-44 or 3.8999999999999998e-35 < b_2

   1. Initial program 50.3

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

   2. Taylor expanded in b_2 around inf 12.6

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

   3. Simplified 12.5

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

   if 1.91999999999999992e-44 < b_2 < 3.8999999999999998e-35

   1. Initial program 41.3

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

   2. Taylor expanded in b_2 around 0 44.4

      \[\leadsto \frac{\color{blue}{\sqrt{-c \cdot a} + -1 \cdot b_2}}{a} \]

   3. Simplified 44.4

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

   4. Taylor expanded in b_2 around 0 43.9

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

Alternatives

Alternative 1
Error	14.2
Cost	7312

\[\begin{array}{l} t_0 := \frac{\sqrt{-c \cdot a}}{a}\\ t_1 := \frac{-0.5 \cdot c}{b_2}\\ \mathbf{if}\;b_2 \leq -2.15 \cdot 10^{-66}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.5 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 3.25 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.0
Cost	7312

\[\begin{array}{l} t_0 := \sqrt{-c \cdot a}\\ t_1 := \frac{-0.5 \cdot c}{b_2}\\ \mathbf{if}\;b_2 \leq -9.5 \cdot 10^{-66}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{t_0 - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.04 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 3.9 \cdot 10^{-35}:\\ \;\;\;\;\frac{t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	22.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 6.8 \cdot 10^{-251}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \end{array} \]

Alternative 4
Error	44.9
Cost	320

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

Alternative 5
Error	44.8
Cost	320

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

Alternative 6
Error	59.2
Cost	256

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

Reproduce

herbie shell --seed 2023010 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))