quad2p (problem 3.2.1, positive)

Average Error: 34.5 → 10.6

Time: 8.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.6088049906029185 \cdot 10^{+109}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.498456083999789 \cdot 10^{-41}:\\ \;\;\;\;\frac{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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 -5.6088049906029185e+109)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 2.498456083999789e-41)
     (/ (- b_2 (sqrt (- (* b_2 b_2) (* a c)))) (- a))
     (/ (* c -0.5) b_2))))

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 <= -5.6088049906029185e+109) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.498456083999789e-41) {
		tmp = (b_2 - sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	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 <= (-5.6088049906029185d+109)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 2.498456083999789d-41) then
        tmp = (b_2 - sqrt(((b_2 * b_2) - (a * c)))) / -a
    else
        tmp = (c * (-0.5d0)) / b_2
    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 <= -5.6088049906029185e+109) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.498456083999789e-41) {
		tmp = (b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	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 <= -5.6088049906029185e+109:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 2.498456083999789e-41:
		tmp = (b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / -a
	else:
		tmp = (c * -0.5) / b_2
	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 <= -5.6088049906029185e+109)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 2.498456083999789e-41)
		tmp = Float64(Float64(b_2 - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / Float64(-a));
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	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 <= -5.6088049906029185e+109)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 2.498456083999789e-41)
		tmp = (b_2 - sqrt(((b_2 * b_2) - (a * c)))) / -a;
	else
		tmp = (c * -0.5) / b_2;
	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, -5.6088049906029185e+109], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.498456083999789e-41], 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[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.6088049906029185e109

   1. Initial program 49.2

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

   2. Simplified 49.2

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

   3. Taylor expanded in b_2 around -inf 4.3

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

   if -5.6088049906029185e109 < b_2 < 2.49845608399978904e-41

   1. Initial program 14.7

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

   2. Simplified 14.7

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

   3. Applied egg-rr 14.7

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

   if 2.49845608399978904e-41 < b_2

   1. Initial program 54.6

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

   2. Simplified 54.6

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

   3. Taylor expanded in b_2 around inf 7.8

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

   4. Simplified 7.8

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5.6088049906029185 \cdot 10^{+109}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.498456083999789 \cdot 10^{-41}:\\ \;\;\;\;\frac{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))