quad2p (problem 3.2.1, positive)

Average Error: 34.1 → 10.5

Time: 14.5s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \end{array} \]

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 -7.8e+49)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.5e-115)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ c (* b_2 -2.0)))))

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 tmp;
	if (b_2 <= -7.8e+49) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.5e-115) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = c / (b_2 * -2.0);
	}
	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) :: tmp
    if (b_2 <= (-7.8d+49)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 1.5d-115) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = c / (b_2 * (-2.0d0))
    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 <= -7.8e+49) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.5e-115) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = c / (b_2 * -2.0);
	}
	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):
	tmp = 0
	if b_2 <= -7.8e+49:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 1.5e-115:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = c / (b_2 * -2.0)
	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)
	tmp = 0.0
	if (b_2 <= -7.8e+49)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1.5e-115)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(c / Float64(b_2 * -2.0));
	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)
	tmp = 0.0;
	if (b_2 <= -7.8e+49)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 1.5e-115)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = c / (b_2 * -2.0);
	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_] := If[LessEqual[b$95$2, -7.8e+49], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.5e-115], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c / N[(b$95$2 * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.8000000000000002e49

   1. Initial program 37.8

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

   2. Simplified 37.8

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

      [Start] 37.8	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 37.8	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 37.8	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 5.9

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

   4. Simplified 5.9

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

      [Start] 5.9	\[ \frac{-2 \cdot b_2}{a} \]
      *-commutative [=>] 5.9	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if -7.8000000000000002e49 < b_2 < 1.5000000000000001e-115

   1. Initial program 12.1

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

   2. Simplified 12.1

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

      [Start] 12.1	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 12.1	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 12.1	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   if 1.5000000000000001e-115 < b_2

   1. Initial program 51.7

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

   2. Simplified 51.7

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

      [Start] 51.7	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 51.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 51.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 11.1

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

   4. Applied egg-rr 33.2

      \[\leadsto \color{blue}{e^{\log \left(-0.5 \cdot \frac{c}{b_2}\right)}} \]

   5. Taylor expanded in c around 0 11.1

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

   6. Simplified 11.1

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

      [Start] 11.1	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 11.1	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      /-rgt-identity [<=] 11.1	\[ \frac{\color{blue}{\frac{-0.5 \cdot c}{1}}}{b_2} \]
      *-commutative [=>] 11.1	\[ \frac{\frac{\color{blue}{c \cdot -0.5}}{1}}{b_2} \]
      associate-/l* [=>] 11.1	\[ \frac{\color{blue}{\frac{c}{\frac{1}{-0.5}}}}{b_2} \]
      metadata-eval [=>] 11.1	\[ \frac{\frac{c}{\color{blue}{-2}}}{b_2} \]
      metadata-eval [<=] 11.1	\[ \frac{\frac{c}{\color{blue}{-1 + -1}}}{b_2} \]
      associate-/l/ [=>] 11.1	\[ \color{blue}{\frac{c}{b_2 \cdot \left(-1 + -1\right)}} \]
      metadata-eval [=>] 11.1	\[ \frac{c}{b_2 \cdot \color{blue}{-2}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.2
Cost	7440

\[\begin{array}{l} t_0 := \frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ t_1 := -2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -1.2 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -1.4 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -2.15 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 1.35 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \end{array} \]

Alternative 2
Error	22.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \end{array} \]

Alternative 3
Error	36.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.95 \cdot 10^{-230}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 4
Error	22.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 6.5 \cdot 10^{-230}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 5
Error	22.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 8.5 \cdot 10^{-230}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 6
Error	22.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 7.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \end{array} \]

Alternative 7
Error	22.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 6.5 \cdot 10^{-231}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2 \cdot -2}\\ \end{array} \]

Alternative 8
Error	59.2
Cost	256

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

Reproduce

herbie shell --seed 2023011 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))