quad2p (problem 3.2.1, positive)

Average Error: 53.41% → 15.34%

Time: 17.3s

Precision: binary64

Cost: 7496

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.2 \cdot 10^{+148}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.15 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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 -3.2e+148)
   (- (- (* 0.5 (/ c b_2)) (/ b_2 a)) (/ b_2 a))
   (if (<= b_2 2.15e-107)
     (- (/ (sqrt (- (* b_2 b_2) (* c a))) a) (/ b_2 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 <= -3.2e+148) {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	} else if (b_2 <= 2.15e-107) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / 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 <= (-3.2d+148)) then
        tmp = ((0.5d0 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
    else if (b_2 <= 2.15d-107) then
        tmp = (sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / 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 <= -3.2e+148) {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	} else if (b_2 <= 2.15e-107) {
		tmp = (Math.sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / 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 <= -3.2e+148:
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
	elif b_2 <= 2.15e-107:
		tmp = (math.sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / 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 <= -3.2e+148)
		tmp = Float64(Float64(Float64(0.5 * Float64(c / b_2)) - Float64(b_2 / a)) - Float64(b_2 / a));
	elseif (b_2 <= 2.15e-107)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) / a) - Float64(b_2 / 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 <= -3.2e+148)
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	elseif (b_2 <= 2.15e-107)
		tmp = (sqrt(((b_2 * b_2) - (c * a))) / a) - (b_2 / 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, -3.2e+148], N[(N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.15e-107], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.1999999999999999e148

   1. Initial program 96.88

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

   2. Simplified 96.88

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

      [Start] 96.88	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 96.88	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 96.88	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 96.88

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

   4. Taylor expanded in b_2 around -inf 4.55

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\right)} - \frac{b_2}{a} \]

   5. Simplified 4.55

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right)} - \frac{b_2}{a} \]
      Proof

      [Start] 4.55	\[ \left(-1 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\right) - \frac{b_2}{a} \]
      +-commutative [=>] 4.55	\[ \color{blue}{\left(0.5 \cdot \frac{c}{b_2} + -1 \cdot \frac{b_2}{a}\right)} - \frac{b_2}{a} \]
      mul-1-neg [=>] 4.55	\[ \left(0.5 \cdot \frac{c}{b_2} + \color{blue}{\left(-\frac{b_2}{a}\right)}\right) - \frac{b_2}{a} \]
      unsub-neg [=>] 4.55	\[ \color{blue}{\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right)} - \frac{b_2}{a} \]

   if -3.1999999999999999e148 < b_2 < 2.1499999999999999e-107

   1. Initial program 17.18

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

   2. Simplified 17.18

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

      [Start] 17.18	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 17.18	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 17.18	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 17.18

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

   if 2.1499999999999999e-107 < b_2

   1. Initial program 81.16

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

   2. Simplified 81.16

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

      [Start] 81.16	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 81.16	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 81.16	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 16.31

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

   4. Applied egg-rr 16.31

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

4. Final simplification 15.34

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.2 \cdot 10^{+148}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.15 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.37%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2
Error	21.9%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.05 \cdot 10^{-134}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 5.4 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	57.24%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.05 \cdot 10^{-189}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 4
Error	35.59%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.05 \cdot 10^{-189}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 5
Error	35.58%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.56 \cdot 10^{-189}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 6
Error	35.58%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.05 \cdot 10^{-189}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 7
Error	35.49%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.05 \cdot 10^{-189}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 8
Error	82.92%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	87.67%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023103 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))