quad2p (problem 3.2.1, positive)

Average Error: 33.7 → 8.4

Time: 16.7s

Precision: binary64

Cost: 7820

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - a \cdot c}\\ \mathbf{if}\;b_2 \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8.5 \cdot 10^{-269}:\\ \;\;\;\;\frac{t_0 - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 9.6 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 + t_0}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* a c)))))
   (if (<= b_2 -1.25e+154)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
     (if (<= b_2 8.5e-269)
       (/ (- t_0 b_2) a)
       (if (<= b_2 9.6e+40)
         (/ (/ (* a (- c)) (+ b_2 t_0)) 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 t_0 = sqrt(((b_2 * b_2) - (a * c)));
	double tmp;
	if (b_2 <= -1.25e+154) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 8.5e-269) {
		tmp = (t_0 - b_2) / a;
	} else if (b_2 <= 9.6e+40) {
		tmp = ((a * -c) / (b_2 + t_0)) / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (a * c)))
    if (b_2 <= (-1.25d+154)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 8.5d-269) then
        tmp = (t_0 - b_2) / a
    else if (b_2 <= 9.6d+40) then
        tmp = ((a * -c) / (b_2 + t_0)) / 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 t_0 = Math.sqrt(((b_2 * b_2) - (a * c)));
	double tmp;
	if (b_2 <= -1.25e+154) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 8.5e-269) {
		tmp = (t_0 - b_2) / a;
	} else if (b_2 <= 9.6e+40) {
		tmp = ((a * -c) / (b_2 + t_0)) / 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):
	t_0 = math.sqrt(((b_2 * b_2) - (a * c)))
	tmp = 0
	if b_2 <= -1.25e+154:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 8.5e-269:
		tmp = (t_0 - b_2) / a
	elif b_2 <= 9.6e+40:
		tmp = ((a * -c) / (b_2 + t_0)) / 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)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))
	tmp = 0.0
	if (b_2 <= -1.25e+154)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 8.5e-269)
		tmp = Float64(Float64(t_0 - b_2) / a);
	elseif (b_2 <= 9.6e+40)
		tmp = Float64(Float64(Float64(a * Float64(-c)) / Float64(b_2 + t_0)) / 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)
	t_0 = sqrt(((b_2 * b_2) - (a * c)));
	tmp = 0.0;
	if (b_2 <= -1.25e+154)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 8.5e-269)
		tmp = (t_0 - b_2) / a;
	elseif (b_2 <= 9.6e+40)
		tmp = ((a * -c) / (b_2 + t_0)) / 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_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.25e+154], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 8.5e-269], N[(N[(t$95$0 - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 9.6e+40], N[(N[(N[(a * (-c)), $MachinePrecision] / N[(b$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.25000000000000001e154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 64.0	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 64.0	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 2.0

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -1.25000000000000001e154 < b_2 < 8.5e-269

   1. Initial program 8.8

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

   2. Simplified 8.8

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

      [Start] 8.8	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 8.8	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 8.8	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   if 8.5e-269 < b_2 < 9.5999999999999999e40

   1. Initial program 29.3

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

   2. Simplified 29.3

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

      [Start] 29.3	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 29.3	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 29.3	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 29.4

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

   4. Simplified 29.4

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

      [Start] 29.4	\[ \frac{\frac{\left(b_2 \cdot b_2 - a \cdot c\right) - \left(-b_2\right) \cdot \left(-b_2\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      associate--l- [=>] 29.4	\[ \frac{\frac{\color{blue}{b_2 \cdot b_2 - \left(a \cdot c + \left(-b_2\right) \cdot \left(-b_2\right)\right)}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      sqr-neg [=>] 29.4	\[ \frac{\frac{b_2 \cdot b_2 - \left(a \cdot c + \color{blue}{b_2 \cdot b_2}\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      *-commutative [=>] 29.4	\[ \frac{\frac{b_2 \cdot b_2 - \left(\color{blue}{c \cdot a} + b_2 \cdot b_2\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      *-commutative [=>] 29.4	\[ \frac{\frac{b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)}{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} - \left(-b_2\right)}}{a} \]

   5. Applied egg-rr 29.4

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

   6. Simplified 16.3

      \[\leadsto \frac{\color{blue}{\frac{0 - c \cdot a}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]
      Proof

      [Start] 29.4	\[ \frac{\left(b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)\right) \cdot \frac{1}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      associate-*r/ [=>] 29.4	\[ \frac{\color{blue}{\frac{\left(b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)\right) \cdot 1}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]
      *-rgt-identity [=>] 29.4	\[ \frac{\frac{\color{blue}{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      fma-udef [=>] 29.4	\[ \frac{\frac{b_2 \cdot b_2 - \color{blue}{\left(b_2 \cdot b_2 + c \cdot a\right)}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      associate--r+ [=>] 16.3	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) - c \cdot a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]
      +-inverses [=>] 16.3	\[ \frac{\frac{\color{blue}{0} - c \cdot a}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]

   if 9.5999999999999999e40 < b_2

   1. Initial program 56.7

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

   2. Simplified 56.7

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

      [Start] 56.7	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 56.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 56.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 4.4

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

   4. Applied egg-rr 4.4

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8.5 \cdot 10^{-269}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 9.6 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2
Error	13.1
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-63}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	13.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-63}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.9 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4
Error	36.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 10^{-267}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 5
Error	36.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.6 \cdot 10^{-269}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 6
Error	22.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.2 \cdot 10^{-268}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 7
Error	50.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.1 \cdot 10^{-267}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2}\\ \end{array} \]

Alternative 8
Error	59.2
Cost	256

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

Reproduce

herbie shell --seed 2023047 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))