quad2m (problem 3.2.1, negative)

Average Error: 53.82% → 15.39%

Time: 17.2s

Precision: binary64

Cost: 7692

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -2.6 \cdot 10^{+59}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -8.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{-c}{a}}{\frac{b_2 - t_0}{a}}\\ \mathbf{elif}\;b_2 \leq 2.25 \cdot 10^{+44}:\\ \;\;\;\;\frac{-t_0}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

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) (* c a)))))
   (if (<= b_2 -2.6e+59)
     (* -0.5 (/ c b_2))
     (if (<= b_2 -8.2e-150)
       (/ (/ (- c) a) (/ (- b_2 t_0) a))
       (if (<= b_2 2.25e+44)
         (- (/ (- t_0) a) (/ b_2 a))
         (+ (* (/ b_2 a) -2.0) (* (/ c b_2) 0.5)))))))

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) - (c * a)));
	double tmp;
	if (b_2 <= -2.6e+59) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -8.2e-150) {
		tmp = (-c / a) / ((b_2 - t_0) / a);
	} else if (b_2 <= 2.25e+44) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	}
	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) - (c * a)))
    if (b_2 <= (-2.6d+59)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= (-8.2d-150)) then
        tmp = (-c / a) / ((b_2 - t_0) / a)
    else if (b_2 <= 2.25d+44) then
        tmp = (-t_0 / a) - (b_2 / a)
    else
        tmp = ((b_2 / a) * (-2.0d0)) + ((c / b_2) * 0.5d0)
    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) - (c * a)));
	double tmp;
	if (b_2 <= -2.6e+59) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -8.2e-150) {
		tmp = (-c / a) / ((b_2 - t_0) / a);
	} else if (b_2 <= 2.25e+44) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	}
	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) - (c * a)))
	tmp = 0
	if b_2 <= -2.6e+59:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= -8.2e-150:
		tmp = (-c / a) / ((b_2 - t_0) / a)
	elif b_2 <= 2.25e+44:
		tmp = (-t_0 / a) - (b_2 / a)
	else:
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5)
	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(c * a)))
	tmp = 0.0
	if (b_2 <= -2.6e+59)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -8.2e-150)
		tmp = Float64(Float64(Float64(-c) / a) / Float64(Float64(b_2 - t_0) / a));
	elseif (b_2 <= 2.25e+44)
		tmp = Float64(Float64(Float64(-t_0) / a) - Float64(b_2 / a));
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + Float64(Float64(c / b_2) * 0.5));
	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) - (c * a)));
	tmp = 0.0;
	if (b_2 <= -2.6e+59)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= -8.2e-150)
		tmp = (-c / a) / ((b_2 - t_0) / a);
	elseif (b_2 <= 2.25e+44)
		tmp = (-t_0 / a) - (b_2 / a);
	else
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	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[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -2.6e+59], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -8.2e-150], N[(N[((-c) / a), $MachinePrecision] / N[(N[(b$95$2 - t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.25e+44], N[(N[((-t$95$0) / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.59999999999999999e59

   1. Initial program 90.27

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

   2. Taylor expanded in b_2 around -inf 5.75

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

   if -2.59999999999999999e59 < b_2 < -8.1999999999999997e-150

   1. Initial program 59.67

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

   2. Applied egg-rr 59.85

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]

   3. Simplified 59.85

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

      [Start] 59.85	\[ \frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      div0 [=>] 59.85	\[ \color{blue}{0} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      +-commutative [=>] 59.85	\[ 0 - \color{blue}{\left(\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
      associate--r+ [=>] 59.85	\[ \color{blue}{\left(0 - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]
      neg-sub0 [<=] 59.85	\[ \color{blue}{\left(-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} - \frac{b_2}{a} \]
      distribute-neg-frac [=>] 59.85	\[ \color{blue}{\frac{-\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} - \frac{b_2}{a} \]
      *-commutative [=>] 59.85	\[ \frac{-\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}{a} - \frac{b_2}{a} \]

   4. Applied egg-rr 79.57

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

   5. Simplified 79.57

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

      [Start] 79.57	\[ \frac{\frac{b_2 \cdot b_2 - c \cdot a}{a \cdot a} - \frac{-b_2}{a} \cdot \frac{-b_2}{a}}{0 - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}} \]
      sub0-neg [=>] 79.57	\[ \frac{\frac{b_2 \cdot b_2 - c \cdot a}{a \cdot a} - \frac{-b_2}{a} \cdot \frac{-b_2}{a}}{\color{blue}{-\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}}} \]

   6. Taylor expanded in b_2 around 0 31.92

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

   7. Simplified 31.92

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

      [Start] 31.92	\[ \frac{-1 \cdot \frac{c}{a}}{-\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}} \]
      mul-1-neg [=>] 31.92	\[ \frac{\color{blue}{-\frac{c}{a}}}{-\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}} \]
      distribute-neg-frac [=>] 31.92	\[ \frac{\color{blue}{\frac{-c}{a}}}{-\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}} \]

   if -8.1999999999999997e-150 < b_2 < 2.25e44

   1. Initial program 18.78

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

   2. Applied egg-rr 18.77

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]

   3. Simplified 18.77

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

      [Start] 18.77	\[ \frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      div0 [=>] 18.77	\[ \color{blue}{0} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      +-commutative [=>] 18.77	\[ 0 - \color{blue}{\left(\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
      associate--r+ [=>] 18.77	\[ \color{blue}{\left(0 - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]
      neg-sub0 [<=] 18.77	\[ \color{blue}{\left(-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} - \frac{b_2}{a} \]
      distribute-neg-frac [=>] 18.77	\[ \color{blue}{\frac{-\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} - \frac{b_2}{a} \]
      *-commutative [=>] 18.77	\[ \frac{-\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}{a} - \frac{b_2}{a} \]

   if 2.25e44 < b_2

   1. Initial program 56.4

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

   2. Taylor expanded in b_2 around inf 9.06

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 15.39

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.6 \cdot 10^{+59}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -8.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{-c}{a}}{\frac{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}}\\ \mathbf{elif}\;b_2 \leq 2.25 \cdot 10^{+44}:\\ \;\;\;\;\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	19.85%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8.5 \cdot 10^{-31}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 2
Error	17.27%
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.5 \cdot 10^{-150}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.25 \cdot 10^{+44}:\\ \;\;\;\;\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3
Error	16.69%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.5 \cdot 10^{-150}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+110}:\\ \;\;\;\;\frac{\left(-\sqrt{b_2 \cdot b_2 - c \cdot a}\right) - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4
Error	21.61%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.5 \cdot 10^{-150}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 5
Error	34.42%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.6 \cdot 10^{-230}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \]

Alternative 6
Error	34.43%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.7 \cdot 10^{-230}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 7
Error	34.33%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7.8 \cdot 10^{-230}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 8
Error	61.08%
Cost	320

\[-0.5 \cdot \frac{c}{b_2} \]

Reproduce

herbie shell --seed 2023089 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))