quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.2% → 87.3%

Time: 7.8s

Alternatives: 7

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

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

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

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

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
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]

Initial Program: 51.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

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

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

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

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
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]

Alternative 1: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{+126}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.3 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{elif}\;b\_2 \leq 90000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} + b\_2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e+126)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 2.3e-184)
     (/ (+ (- b_2) (sqrt (fma (- c) a (* b_2 b_2)))) a)
     (if (<= b_2 90000000.0)
       (/ (/ (fma (- c) a 0.0) (+ (sqrt (fma b_2 b_2 (* (- c) a))) b_2)) a)
       (* (/ c b_2) -0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e+126) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 2.3e-184) {
		tmp = (-b_2 + sqrt(fma(-c, a, (b_2 * b_2)))) / a;
	} else if (b_2 <= 90000000.0) {
		tmp = (fma(-c, a, 0.0) / (sqrt(fma(b_2, b_2, (-c * a))) + b_2)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e+126)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 2.3e-184)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(fma(Float64(-c), a, Float64(b_2 * b_2)))) / a);
	elseif (b_2 <= 90000000.0)
		tmp = Float64(Float64(fma(Float64(-c), a, 0.0) / Float64(sqrt(fma(b_2, b_2, Float64(Float64(-c) * a))) + b_2)) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.2e+126], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.3e-184], N[(N[((-b$95$2) + N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 90000000.0], N[(N[(N[((-c) * a + 0.0), $MachinePrecision] / N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[((-c) * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b$95$2), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.19999999999999999e126

   1. Initial program 47.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. lower-/.f64 98.2

         \[\leadsto -2 \cdot \color{blue}{\frac{b\_2}{a}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

   if -2.19999999999999999e126 < b_2 < 2.2999999999999999e-184

   1. Initial program 78.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(\mathsf{neg}\left(a\right)\right) \cdot c}}}{a} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot c + b\_2 \cdot b\_2}}}{a} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(a \cdot c\right)\right)} + b\_2 \cdot b\_2}}{a} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{a \cdot \left(\mathsf{neg}\left(c\right)\right)} + b\_2 \cdot b\_2}}{a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot a} + b\_2 \cdot b\_2}}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(c\right), a, b\_2 \cdot b\_2\right)}}}{a} \]

      9. lower-neg.f64 78.9

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(\color{blue}{-c}, a, b\_2 \cdot b\_2\right)}}{a} \]

   4. Applied rewrites 78.9%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}}{a} \]

   if 2.2999999999999999e-184 < b_2 < 9e7

   1. Initial program 54.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(\mathsf{neg}\left(a\right)\right) \cdot c}}}{a} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot c + b\_2 \cdot b\_2}}}{a} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(a \cdot c\right)\right)} + b\_2 \cdot b\_2}}{a} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{a \cdot \left(\mathsf{neg}\left(c\right)\right)} + b\_2 \cdot b\_2}}{a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot a} + b\_2 \cdot b\_2}}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(c\right), a, b\_2 \cdot b\_2\right)}}}{a} \]

      9. lower-neg.f64 54.2

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(\color{blue}{-c}, a, b\_2 \cdot b\_2\right)}}{a} \]

   4. Applied rewrites 54.2%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}}{a} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}}{a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} + \left(-b\_2\right)}}{a} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}} + \left(-b\_2\right)}{a} \]

      4. rem-square-sqrt N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} \cdot \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}} + \left(-b\_2\right)}{a} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}} \cdot \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}} + \left(-b\_2\right)}{a} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} \cdot \color{blue}{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}} + \left(-b\_2\right)}{a} \]

      7. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}} + \left(-b\_2\right)}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}, -b\_2\right)}}{a} \]

   6. Applied rewrites 53.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}, -b\_2\right)}}{a} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}} + \left(-b\_2\right)}}{a} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}} + \left(-b\_2\right)}{a} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}} \cdot \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}} + \left(-b\_2\right)}{a} \]

      4. rem-square-sqrt N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}} + \left(-b\_2\right)}{a} \]

      5. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} \cdot \sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - \left(-b\_2\right) \cdot \left(-b\_2\right)}{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - \left(-b\_2\right)}}}{a} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} \cdot \sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - \left(-b\_2\right) \cdot \left(-b\_2\right)}{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - \left(-b\_2\right)}}}{a} \]

   8. Applied rewrites 54.3%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right) - b\_2 \cdot b\_2}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} - \left(-b\_2\right)}}}{a} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right) - b\_2 \cdot b\_2}}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} - \left(-b\_2\right)}}{a} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(b\_2 \cdot b\_2 + \left(-c\right) \cdot a\right)} - b\_2 \cdot b\_2}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} - \left(-b\_2\right)}}{a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{b\_2 \cdot b\_2} + \left(-c\right) \cdot a\right) - b\_2 \cdot b\_2}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} - \left(-b\_2\right)}}{a} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(-c\right) \cdot a + b\_2 \cdot b\_2\right)} - b\_2 \cdot b\_2}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} - \left(-b\_2\right)}}{a} \]

      5. associate--l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(-c\right) \cdot a + \left(b\_2 \cdot b\_2 - b\_2 \cdot b\_2\right)}}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} - \left(-b\_2\right)}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(-c\right) \cdot a} + \left(b\_2 \cdot b\_2 - b\_2 \cdot b\_2\right)}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} - \left(-b\_2\right)}}{a} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2 - b\_2 \cdot b\_2\right)}}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} - \left(-b\_2\right)}}{a} \]

      8. lower--.f64 81.1

         \[\leadsto \frac{\frac{\mathsf{fma}\left(-c, a, \color{blue}{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}\right)}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} - \left(-b\_2\right)}}{a} \]

   10. Applied rewrites 81.1%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2 - b\_2 \cdot b\_2\right)}}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} - \left(-b\_2\right)}}{a} \]

   if 9e7 < b_2

   1. Initial program 8.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 93.4

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{+126}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.3 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{elif}\;b\_2 \leq 90000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-c\right) \cdot a\right)} + b\_2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{+126}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e+126)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 7.8e-84)
     (/ (+ (- b_2) (sqrt (fma (- c) a (* b_2 b_2)))) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e+126) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 7.8e-84) {
		tmp = (-b_2 + sqrt(fma(-c, a, (b_2 * b_2)))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e+126)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 7.8e-84)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(fma(Float64(-c), a, Float64(b_2 * b_2)))) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.2e+126], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7.8e-84], N[(N[((-b$95$2) + N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.19999999999999999e126

   1. Initial program 47.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. lower-/.f64 98.2

         \[\leadsto -2 \cdot \color{blue}{\frac{b\_2}{a}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

   if -2.19999999999999999e126 < b_2 < 7.80000000000000045e-84

   1. Initial program 76.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(\mathsf{neg}\left(a\right)\right) \cdot c}}}{a} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot c + b\_2 \cdot b\_2}}}{a} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(a \cdot c\right)\right)} + b\_2 \cdot b\_2}}{a} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{a \cdot \left(\mathsf{neg}\left(c\right)\right)} + b\_2 \cdot b\_2}}{a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot a} + b\_2 \cdot b\_2}}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(c\right), a, b\_2 \cdot b\_2\right)}}}{a} \]

      9. lower-neg.f64 76.8

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(\color{blue}{-c}, a, b\_2 \cdot b\_2\right)}}{a} \]

   4. Applied rewrites 76.8%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}}{a} \]

   if 7.80000000000000045e-84 < b_2

   1. Initial program 13.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 88.9

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{+126}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{b\_2 - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e+126)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 7.8e-84)
     (/ (- b_2 (sqrt (- (* b_2 b_2) (* a c)))) (- a))
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e+126) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 7.8e-84) {
		tmp = (b_2 - sqrt(((b_2 * b_2) - (a * c)))) / -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
    real(8) :: tmp
    if (b_2 <= (-2.2d+126)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 7.8d-84) then
        tmp = (b_2 - sqrt(((b_2 * b_2) - (a * c)))) / -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) {
	double tmp;
	if (b_2 <= -2.2e+126) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 7.8e-84) {
		tmp = (b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.2e+126:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 7.8e-84:
		tmp = (b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / -a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e+126)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 7.8e-84)
		tmp = Float64(Float64(b_2 - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / Float64(-a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.2e+126)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 7.8e-84)
		tmp = (b_2 - sqrt(((b_2 * b_2) - (a * c)))) / -a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.2e+126], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7.8e-84], 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], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.19999999999999999e126

   1. Initial program 47.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. lower-/.f64 98.2

         \[\leadsto -2 \cdot \color{blue}{\frac{b\_2}{a}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

   if -2.19999999999999999e126 < b_2 < 7.80000000000000045e-84

   1. Initial program 76.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 7.80000000000000045e-84 < b_2

   1. Initial program 13.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 88.9

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{+126}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{b\_2 - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.5 \cdot 10^{-116}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{b\_2 - \sqrt{\left(-c\right) \cdot a}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.5e-116)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 7.5e-84)
     (/ (- b_2 (sqrt (* (- c) a))) (- a))
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.5e-116) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 7.5e-84) {
		tmp = (b_2 - sqrt((-c * a))) / -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
    real(8) :: tmp
    if (b_2 <= (-2.5d-116)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 7.5d-84) then
        tmp = (b_2 - sqrt((-c * a))) / -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) {
	double tmp;
	if (b_2 <= -2.5e-116) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 7.5e-84) {
		tmp = (b_2 - Math.sqrt((-c * a))) / -a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.5e-116:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 7.5e-84:
		tmp = (b_2 - math.sqrt((-c * a))) / -a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.5e-116)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 7.5e-84)
		tmp = Float64(Float64(b_2 - sqrt(Float64(Float64(-c) * a))) / Float64(-a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.5e-116)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 7.5e-84)
		tmp = (b_2 - sqrt((-c * a))) / -a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.5e-116], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7.5e-84], N[(N[(b$95$2 - N[Sqrt[N[((-c) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.5000000000000001e-116

   1. Initial program 67.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. lower-/.f64 84.1

         \[\leadsto -2 \cdot \color{blue}{\frac{b\_2}{a}} \]

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

   if -2.5000000000000001e-116 < b_2 < 7.50000000000000026e-84

   1. Initial program 68.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\mathsf{neg}\left(a \cdot c\right)}}}{a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\mathsf{neg}\left(\color{blue}{c \cdot a}\right)}}{a} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot a}}}{a} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot a}}}{a} \]

      5. lower-neg.f64 67.2

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-c\right)} \cdot a}}{a} \]

   5. Applied rewrites 67.2%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-c\right) \cdot a}}}{a} \]

   if 7.50000000000000026e-84 < b_2

   1. Initial program 13.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 88.9

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.5 \cdot 10^{-116}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{b\_2 - \sqrt{\left(-c\right) \cdot a}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (* -2.0 (/ b_2 a)) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = -2.0 * (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
    real(8) :: tmp
    if (b_2 <= (-1d-309)) then
        tmp = (-2.0d0) * (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) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 66.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. lower-/.f64 70.6

         \[\leadsto -2 \cdot \color{blue}{\frac{b\_2}{a}} \]

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

   if -1.000000000000002e-309 < b_2

   1. Initial program 30.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 69.9

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b\_2} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (* -2.0 (/ b_2 a)) (* (/ -0.5 b_2) c)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (-0.5 / b_2) * c;
	}
	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
    real(8) :: tmp
    if (b_2 <= (-1d-309)) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = ((-0.5d0) / b_2) * c
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (-0.5 / b_2) * c;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (-0.5 / b_2) * c
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(-0.5 / b_2) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (-0.5 / b_2) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / b$95$2), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 66.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. lower-/.f64 70.6

         \[\leadsto -2 \cdot \color{blue}{\frac{b\_2}{a}} \]

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

   if -1.000000000000002e-309 < b_2

   1. Initial program 30.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right) \cdot c} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right)} \cdot c \]

      4. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left(a \cdot c\right)}{{b\_2}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right) \cdot c \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{\color{blue}{\left(\frac{-1}{8} \cdot a\right) \cdot c}}{{b\_2}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right) \cdot c \]

      6. associate-*l/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{8} \cdot a}{{b\_2}^{3}} \cdot c} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right) \cdot c \]

      7. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right)} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \left(\color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right) \cdot c \]

      9. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(c \cdot \frac{-1}{8}\right) \cdot \frac{a}{{b\_2}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right) \cdot c \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{{b\_2}^{3}}, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right)} \cdot c \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{c \cdot \frac{-1}{8}}, \frac{a}{{b\_2}^{3}}, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right) \cdot c \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \color{blue}{\frac{a}{{b\_2}^{3}}}, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right) \cdot c \]

      13. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{\color{blue}{{b\_2}^{3}}}, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{b\_2}\right) \cdot c \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{{b\_2}^{3}}, \color{blue}{\frac{-1}{2}} \cdot \frac{1}{b\_2}\right) \cdot c \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{{b\_2}^{3}}, \color{blue}{\frac{\frac{-1}{2} \cdot 1}{b\_2}}\right) \cdot c \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{{b\_2}^{3}}, \frac{\color{blue}{\frac{-1}{2}}}{b\_2}\right) \cdot c \]

      17. lower-/.f64 63.3

          \[\leadsto \mathsf{fma}\left(c \cdot -0.125, \frac{a}{{b\_2}^{3}}, \color{blue}{\frac{-0.5}{b\_2}}\right) \cdot c \]

   5. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot -0.125, \frac{a}{{b\_2}^{3}}, \frac{-0.5}{b\_2}\right) \cdot c} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{-1}{2}}{b\_2} \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto \frac{-0.5}{b\_2} \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 35.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{b\_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -2.0 (/ b_2 a)))

C

double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

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 = (-2.0d0) * (b_2 / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

Python

def code(a, b_2, c):
	return -2.0 * (b_2 / a)

Julia

function code(a, b_2, c)
	return Float64(-2.0 * Float64(b_2 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -2.0 * (b_2 / a);
end

Wolfram

code[a_, b$95$2_, c_] := N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.5%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in b_2 around -inf

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

   2. lower-/.f64 36.6

      \[\leadsto -2 \cdot \color{blue}{\frac{b\_2}{a}} \]

5. Applied rewrites 36.6%

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
6. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b\_2\right| - t\_0} \cdot \sqrt{\left|b\_2\right| + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b\_2, t\_0\right)\\ \end{array}\\ \mathbf{if}\;b\_2 < 0:\\ \;\;\;\;\frac{t\_1 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b\_2 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024326 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))