quad2p (problem 3.2.1, positive)

Average Error: 33.5 → 6.4

Time: 15.0s

Precision: binary64

Cost: 7564

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - a \cdot c}\\ \mathbf{if}\;b_2 \leq -3 \cdot 10^{+146}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq -1.2 \cdot 10^{-173}:\\ \;\;\;\;\frac{t_0}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{-c}{b_2 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}\\ \end{array} \]

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 -3e+146)
     (/ (* b_2 -2.0) a)
     (if (<= b_2 -1.2e-173)
       (- (/ t_0 a) (/ b_2 a))
       (if (<= b_2 1.05e+71)
         (/ (- c) (+ b_2 t_0))
         (/ (- c) (+ b_2 (+ b_2 (* -0.5 (/ c (/ b_2 a)))))))))))

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 <= -3e+146) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= -1.2e-173) {
		tmp = (t_0 / a) - (b_2 / a);
	} else if (b_2 <= 1.05e+71) {
		tmp = -c / (b_2 + t_0);
	} else {
		tmp = -c / (b_2 + (b_2 + (-0.5 * (c / (b_2 / a)))));
	}
	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 <= (-3d+146)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= (-1.2d-173)) then
        tmp = (t_0 / a) - (b_2 / a)
    else if (b_2 <= 1.05d+71) then
        tmp = -c / (b_2 + t_0)
    else
        tmp = -c / (b_2 + (b_2 + ((-0.5d0) * (c / (b_2 / a)))))
    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 <= -3e+146) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= -1.2e-173) {
		tmp = (t_0 / a) - (b_2 / a);
	} else if (b_2 <= 1.05e+71) {
		tmp = -c / (b_2 + t_0);
	} else {
		tmp = -c / (b_2 + (b_2 + (-0.5 * (c / (b_2 / a)))));
	}
	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 <= -3e+146:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= -1.2e-173:
		tmp = (t_0 / a) - (b_2 / a)
	elif b_2 <= 1.05e+71:
		tmp = -c / (b_2 + t_0)
	else:
		tmp = -c / (b_2 + (b_2 + (-0.5 * (c / (b_2 / a)))))
	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 <= -3e+146)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= -1.2e-173)
		tmp = Float64(Float64(t_0 / a) - Float64(b_2 / a));
	elseif (b_2 <= 1.05e+71)
		tmp = Float64(Float64(-c) / Float64(b_2 + t_0));
	else
		tmp = Float64(Float64(-c) / Float64(b_2 + Float64(b_2 + Float64(-0.5 * Float64(c / Float64(b_2 / a))))));
	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 <= -3e+146)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= -1.2e-173)
		tmp = (t_0 / a) - (b_2 / a);
	elseif (b_2 <= 1.05e+71)
		tmp = -c / (b_2 + t_0);
	else
		tmp = -c / (b_2 + (b_2 + (-0.5 * (c / (b_2 / a)))));
	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, -3e+146], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -1.2e-173], N[(N[(t$95$0 / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.05e+71], N[((-c) / N[(b$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision], N[((-c) / N[(b$95$2 + N[(b$95$2 + N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.00000000000000002e146

   1. Initial program 61.1

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

   2. Simplified 61.1

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 3 points increase in error, 0 points decrease in error

   3. Taylor expanded in b_2 around -inf 2.9

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

   4. Simplified 2.9

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
      Proof
      (/.f64 (*.f64 b_2 -2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 b_2)) a): 0 points increase in error, 2 points decrease in error

   if -3.00000000000000002e146 < b_2 < -1.20000000000000008e-173

   1. Initial program 6.0

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

   2. Simplified 6.0

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 3 points increase in error, 0 points decrease in error

   3. Applied egg-rr 6.0

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

   if -1.20000000000000008e-173 < b_2 < 1.04999999999999995e71

   1. Initial program 27.3

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

   2. Simplified 27.3

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 3 points increase in error, 0 points decrease in error

   3. Applied egg-rr 27.3

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

   4. Applied egg-rr 27.5

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

   5. Taylor expanded in a around 0 10.1

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

   6. Simplified 10.1

      \[\leadsto \frac{\color{blue}{-c}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)} \]
      Proof
      (/.f64 (neg.f64 c) (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 c)) (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))): 2 points increase in error, 0 points decrease in error

   7. Applied egg-rr 10.1

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

   if 1.04999999999999995e71 < b_2

   1. Initial program 58.1

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

   2. Simplified 58.1

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof
      (/.f64 (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) b_2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))) a): 0 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))))) a): 3 points increase in error, 0 points decrease in error

   3. Applied egg-rr 58.1

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

   4. Applied egg-rr 58.1

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

   5. Taylor expanded in a around 0 27.3

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

   6. Simplified 27.3

      \[\leadsto \frac{\color{blue}{-c}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)} \]
      Proof
      (/.f64 (neg.f64 c) (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 c)) (-.f64 (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c))) (neg.f64 b_2))): 2 points increase in error, 0 points decrease in error

   7. Taylor expanded in b_2 around inf 6.5

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

   8. Simplified 3.1

      \[\leadsto \frac{-c}{\color{blue}{\left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)} - \left(-b_2\right)} \]
      Proof
      (/.f64 (*.f64 b_2 -2) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 b_2)) a): 0 points increase in error, 2 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3 \cdot 10^{+146}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq -1.2 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.4
Cost	7564

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - a \cdot c}\\ \mathbf{if}\;b_2 \leq -5 \cdot 10^{+147}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq -1.2 \cdot 10^{-173}:\\ \;\;\;\;\frac{t_0 - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{-c}{b_2 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}\\ \end{array} \]

Alternative 2
Error	10.0
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.35 \cdot 10^{+146}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 7 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}\\ \end{array} \]

Alternative 3
Error	13.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.6 \cdot 10^{-43}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 7 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}\\ \end{array} \]

Alternative 4
Error	13.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.1 \cdot 10^{-43}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 7 \cdot 10^{-117}:\\ \;\;\;\;\frac{-c}{\sqrt{a \cdot \left(-c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}\\ \end{array} \]

Alternative 5
Error	22.4
Cost	1028

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}\\ \end{array} \]

Alternative 6
Error	22.6
Cost	964

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \end{array} \]

Alternative 7
Error	36.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.2 \cdot 10^{-201}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 8
Error	22.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 10^{-201}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 9
Error	22.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.05 \cdot 10^{-199}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 10
Error	59.2
Cost	256

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

Reproduce

herbie shell --seed 2022343 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))