quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.6% → 83.5%

Time: 20.6s

Precision: binary64

Cost: 7496

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{+91}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 10^{-198}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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 -1.55e+91)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1e-198)
     (- (/ (sqrt (- (* b_2 b_2) (* a c))) 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 <= -1.55e+91) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1e-198) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) / 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 <= (-1.55d+91)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 1d-198) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) / 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 <= -1.55e+91) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1e-198) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) / 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 <= -1.55e+91:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 1e-198:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) / 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 <= -1.55e+91)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1e-198)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) / 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 <= -1.55e+91)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 1e-198)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) / 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, -1.55e+91], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1e-198], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $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 < -1.54999999999999999e91

   1. Initial program 53.0%

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

   2. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 53.0%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 53.0%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 53.0%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 93.0%

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

   4. Simplified 93.0%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
      Step-by-step derivation

      [Start] 93.0%	\[ \frac{-2 \cdot b_2}{a} \]
      *-commutative [=>] 93.0%	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if -1.54999999999999999e91 < b_2 < 9.9999999999999991e-199

   1. Initial program 85.2%

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

   2. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 85.2%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 85.2%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 85.2%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 84.8%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]
      Step-by-step derivation

      [Start] 85.2%	\[ \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a} \]
      add-sqr-sqrt [=>] 84.8%	\[ \frac{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]
      pow2 [=>] 84.8%	\[ \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]
      pow1/2 [=>] 84.8%	\[ \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]
      sqrt-pow1 [=>] 84.8%	\[ \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]
      fma-neg [=>] 84.8%	\[ \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]
      *-commutative [=>] 84.8%	\[ \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]
      distribute-rgt-neg-in [=>] 84.8%	\[ \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]
      metadata-eval [=>] 84.8%	\[ \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

   4. Applied egg-rr 85.2%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.5}}{a} - \frac{b_2}{a}} \]
      Step-by-step derivation

      [Start] 84.8%	\[ \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2} - b_2}{a} \]
      div-sub [=>] 84.8%	\[ \color{blue}{\frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}}{a} - \frac{b_2}{a}} \]
      pow-pow [=>] 85.2%	\[ \frac{\color{blue}{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\left(0.25 \cdot 2\right)}}}{a} - \frac{b_2}{a} \]
      metadata-eval [=>] 85.2%	\[ \frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.5}}}{a} - \frac{b_2}{a} \]

   5. Applied egg-rr 84.8%

      \[\leadsto \frac{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25}}}{a} - \frac{b_2}{a} \]
      Step-by-step derivation

      [Start] 85.2%	\[ \frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.5}}{a} - \frac{b_2}{a} \]
      sqr-pow [=>] 84.8%	\[ \frac{\color{blue}{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}}{a} - \frac{b_2}{a} \]
      distribute-rgt-neg-out [=>] 84.8%	\[ \frac{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}{a} - \frac{b_2}{a} \]
      fma-neg [<=] 84.8%	\[ \frac{{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right)}}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}{a} - \frac{b_2}{a} \]
      metadata-eval [=>] 84.8%	\[ \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.25}} \cdot {\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}{a} - \frac{b_2}{a} \]
      distribute-rgt-neg-out [=>] 84.8%	\[ \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}}{a} - \frac{b_2}{a} \]
      fma-neg [<=] 84.8%	\[ \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25} \cdot {\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right)}}^{\left(\frac{0.5}{2}\right)}}{a} - \frac{b_2}{a} \]
      metadata-eval [=>] 84.8%	\[ \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.25}}}{a} - \frac{b_2}{a} \]

   6. Simplified 85.2%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} - \frac{b_2}{a} \]
      Step-by-step derivation

      [Start] 84.8%	\[ \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.25}}{a} - \frac{b_2}{a} \]
      pow-sqr [=>] 85.2%	\[ \frac{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(2 \cdot 0.25\right)}}}{a} - \frac{b_2}{a} \]
      metadata-eval [=>] 85.2%	\[ \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.5}}}{a} - \frac{b_2}{a} \]
      unpow1/2 [=>] 85.2%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} - \frac{b_2}{a} \]

   if 9.9999999999999991e-199 < b_2

   1. Initial program 20.5%

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

   2. Simplified 20.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 20.5%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 20.5%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 20.5%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 78.2%

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

   4. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      Step-by-step derivation

      [Start] 78.2%	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 78.2%	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{+91}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 10^{-198}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.5%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{+91}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 10^{-198}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2
Accuracy	83.5%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 10^{-198}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Accuracy	78.7%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{-23}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{-198}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4
Accuracy	78.7%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.2 \cdot 10^{-24}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{-198}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 5
Accuracy	68.6%
Cost	836

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

Alternative 6
Accuracy	47.9%
Cost	452

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

Alternative 7
Accuracy	68.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.5 \cdot 10^{-309}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 8
Accuracy	68.3%
Cost	452

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

Alternative 9
Accuracy	68.4%
Cost	452

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

Alternative 10
Accuracy	23.6%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 5.6 \cdot 10^{-305}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 11
Accuracy	11.1%
Cost	192

\[\frac{0}{a} \]

Reproduce

herbie shell --seed 2023263 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))