quad2p (problem 3.2.1, positive)

Average Accuracy: 45.4% → 83.9%

Time: 18.4s

Precision: binary64

Cost: 7880

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.2 \cdot 10^{+149}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{a \cdot c + \left(\left(b_2 \cdot b_2 - a \cdot c\right) - a \cdot c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

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
 (if (<= b_2 -4.2e+149)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 9e-41)
     (/ (- (sqrt (+ (* a c) (- (- (* b_2 b_2) (* a c)) (* a c)))) b_2) a)
     (* (/ 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 tmp;
	if (b_2 <= -4.2e+149) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 9e-41) {
		tmp = (sqrt(((a * c) + (((b_2 * b_2) - (a * c)) - (a * c)))) - b_2) / a;
	} else {
		tmp = (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) :: tmp
    if (b_2 <= (-4.2d+149)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 9d-41) then
        tmp = (sqrt(((a * c) + (((b_2 * b_2) - (a * c)) - (a * c)))) - b_2) / a
    else
        tmp = (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 tmp;
	if (b_2 <= -4.2e+149) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 9e-41) {
		tmp = (Math.sqrt(((a * c) + (((b_2 * b_2) - (a * c)) - (a * c)))) - b_2) / a;
	} else {
		tmp = (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):
	tmp = 0
	if b_2 <= -4.2e+149:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 9e-41:
		tmp = (math.sqrt(((a * c) + (((b_2 * b_2) - (a * c)) - (a * c)))) - b_2) / a
	else:
		tmp = (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)
	tmp = 0.0
	if (b_2 <= -4.2e+149)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 9e-41)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) + Float64(Float64(Float64(b_2 * b_2) - Float64(a * c)) - Float64(a * c)))) - b_2) / a);
	else
		tmp = 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)
	tmp = 0.0;
	if (b_2 <= -4.2e+149)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 9e-41)
		tmp = (sqrt(((a * c) + (((b_2 * b_2) - (a * c)) - (a * c)))) - b_2) / a;
	else
		tmp = (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_] := If[LessEqual[b$95$2, -4.2e+149], 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, 9e-41], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] + N[(N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.2000000000000003e149

   1. Initial program 3.4%

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

   2. Simplified 3.4%

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

      [Start] 3.4	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 3.4	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 3.4	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 96.6%

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

   if -4.2000000000000003e149 < b_2 < 9e-41

   1. Initial program 77.6%

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

   2. Simplified 77.6%

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

      [Start] 77.6	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 77.6	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 77.6	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 77.6%

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

      [Start] 77.6	\[ \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a} \]
      prod-diff [=>] 77.6	\[ \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}} - b_2}{a} \]
      *-commutative [<=] 77.6	\[ \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)} - b_2}{a} \]
      fma-neg [<=] 77.6	\[ \frac{\sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)} - b_2}{a} \]
      *-commutative [<=] 77.6	\[ \frac{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)} - b_2}{a} \]
      fma-udef [=>] 77.6	\[ \frac{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}} - b_2}{a} \]
      distribute-lft-neg-in [<=] 77.6	\[ \frac{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)} - b_2}{a} \]
      *-commutative [<=] 77.6	\[ \frac{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)} - b_2}{a} \]
      associate-+r+ [=>] 77.6	\[ \frac{\sqrt{\color{blue}{\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(-a \cdot c\right)\right) + a \cdot c}} - b_2}{a} \]
      distribute-rgt-neg-in [=>] 77.6	\[ \frac{\sqrt{\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \color{blue}{a \cdot \left(-c\right)}\right) + a \cdot c} - b_2}{a} \]

   if 9e-41 < b_2

   1. Initial program 14.7%

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

   2. Simplified 14.7%

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

      [Start] 14.7	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 14.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 14.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 88.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.2 \cdot 10^{+149}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{a \cdot c + \left(\left(b_2 \cdot b_2 - a \cdot c\right) - a \cdot c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.9%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8.5 \cdot 10^{+148}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 7.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 2
Accuracy	78.6%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.4 \cdot 10^{-48}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 3
Accuracy	65.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 7.4 \cdot 10^{-241}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 4
Accuracy	28.7%
Cost	320

\[-2 \cdot \frac{b_2}{a} \]

Reproduce

herbie shell --seed 2023138 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))