quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.4% → 87.4%

Time: 8.4s

Alternatives: 9

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: 52.4% 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: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 8 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}{a} - \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.18 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+152)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 8e-292)
     (- (/ (sqrt (fma a (- c) (* b_2 b_2))) a) (/ b_2 a))
     (if (<= b_2 1.18e+34)
       (/ (/ (* c a) a) (- (- b_2) (sqrt (fma (- a) c (* b_2 b_2)))))
       (* -0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+152) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 8e-292) {
		tmp = (sqrt(fma(a, -c, (b_2 * b_2))) / a) - (b_2 / a);
	} else if (b_2 <= 1.18e+34) {
		tmp = ((c * a) / a) / (-b_2 - sqrt(fma(-a, c, (b_2 * b_2))));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+152)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 8e-292)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-c), Float64(b_2 * b_2))) / a) - Float64(b_2 / a));
	elseif (b_2 <= 1.18e+34)
		tmp = Float64(Float64(Float64(c * a) / a) / Float64(Float64(-b_2) - sqrt(fma(Float64(-a), c, Float64(b_2 * b_2)))));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+152], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 8e-292], N[(N[(N[Sqrt[N[(a * (-c) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.18e+34], N[(N[(N[(c * a), $MachinePrecision] / a), $MachinePrecision] / N[((-b$95$2) - N[Sqrt[N[((-a) * c + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1e152

   1. Initial program 48.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1e152 < b_2 < 8.0000000000000004e-292

   1. Initial program 86.9%

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

   4. Applied rewrites 36.9%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\frac{{b\_2}^{6} - {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right), c \cdot a, {b\_2}^{4}\right)}}}}{a} \]

   6. Applied rewrites 86.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 86.9

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

   8. Applied rewrites 86.9%

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

   if 8.0000000000000004e-292 < b_2 < 1.18000000000000002e34

   1. Initial program 63.0%

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

   4. Applied rewrites 32.4%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\frac{{b\_2}^{6} - {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right), c \cdot a, {b\_2}^{4}\right)}}}}{a} \]

   6. Applied rewrites 56.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 75.3

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

   9. Applied rewrites 75.3%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 86.9

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

       6. lift-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(c \cdot a\right)\right)}{a}}{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-c\right) \cdot a + b\_2 \cdot b\_2}}} \]

   11. Applied rewrites 86.9%

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

   if 1.18000000000000002e34 < b_2

   1. Initial program 11.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 8 \cdot 10^{-292}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}{a} - \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.18 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+152)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 3.7e-96)
     (- (/ (sqrt (fma a (- c) (* b_2 b_2))) a) (/ b_2 a))
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+152) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3.7e-96) {
		tmp = (sqrt(fma(a, -c, (b_2 * b_2))) / a) - (b_2 / a);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+152)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 3.7e-96)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-c), Float64(b_2 * b_2))) / a) - Float64(b_2 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+152], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.7e-96], N[(N[(N[Sqrt[N[(a * (-c) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1e152

   1. Initial program 48.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1e152 < b_2 < 3.69999999999999986e-96

   1. Initial program 85.8%

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

   4. Applied rewrites 37.8%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\frac{{b\_2}^{6} - {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right), c \cdot a, {b\_2}^{4}\right)}}}}{a} \]

   6. Applied rewrites 85.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 85.8

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

   8. Applied rewrites 85.8%

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

   if 3.69999999999999986e-96 < b_2

   1. Initial program 16.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 89.6

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

   5. Applied rewrites 89.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+152)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 3.7e-96)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+152) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3.7e-96) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / 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
    real(8) :: tmp
    if (b_2 <= (-1d+152)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 3.7d-96) then
        tmp = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+152) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3.7e-96) {
		tmp = (Math.sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e+152:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 3.7e-96:
		tmp = (math.sqrt(((b_2 * b_2) - (c * a))) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+152)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 3.7e-96)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e+152)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 3.7e-96)
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+152], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.7e-96], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1e152

   1. Initial program 48.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1e152 < b_2 < 3.69999999999999986e-96

   1. Initial program 85.8%

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

   if 3.69999999999999986e-96 < b_2

   1. Initial program 16.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 89.6

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

   5. Applied rewrites 89.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+152)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 3.7e-96)
     (/ (- (sqrt (fma (- c) a (* b_2 b_2))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+152) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3.7e-96) {
		tmp = (sqrt(fma(-c, a, (b_2 * b_2))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+152)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 3.7e-96)
		tmp = Float64(Float64(sqrt(fma(Float64(-c), a, Float64(b_2 * b_2))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+152], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.7e-96], N[(N[(N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1e152

   1. Initial program 48.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1e152 < b_2 < 3.69999999999999986e-96

   1. Initial program 85.8%

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

   4. Applied rewrites 37.8%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\frac{{b\_2}^{6} - {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right), c \cdot a, {b\_2}^{4}\right)}}}}{a} \]

   6. Applied rewrites 85.8%

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

   if 3.69999999999999986e-96 < b_2

   1. Initial program 16.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 89.6

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

   5. Applied rewrites 89.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-10}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.95 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3e-10)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.95e-96) (/ (- (sqrt (* (- a) c)) b_2) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-10) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.95e-96) {
		tmp = (sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / 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
    real(8) :: tmp
    if (b_2 <= (-3d-10)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.95d-96) then
        tmp = (sqrt((-a * c)) - b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-10) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.95e-96) {
		tmp = (Math.sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3e-10:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.95e-96:
		tmp = (math.sqrt((-a * c)) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3e-10)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.95e-96)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3e-10)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.95e-96)
		tmp = (sqrt((-a * c)) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3e-10], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.95e-96], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3e-10

   1. Initial program 73.7%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if -3e-10 < b_2 < 1.9499999999999999e-96

   1. Initial program 82.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. associate-*r* N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if 1.9499999999999999e-96 < b_2

   1. Initial program 16.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 89.6

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

   5. Applied rewrites 89.6%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-10}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.95 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 1.7× speedup

Math

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

FPCore

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

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

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	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 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 77.9%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 34.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

4. Final simplification 66.2%

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

Alternative 7: 34.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -0.5 (/ c b_2)))

C

double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

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 = (-0.5d0) * (c / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

Python

def code(a, b_2, c):
	return -0.5 * (c / b_2)

Julia

function code(a, b_2, c)
	return Float64(-0.5 * Float64(c / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -0.5 * (c / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
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 35.4

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

5. Applied rewrites 35.4%

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

6. Final simplification 35.4%

   \[\leadsto -0.5 \cdot \frac{c}{b\_2} \]
7. Add Preprocessing

Alternative 8: 34.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{b\_2} \cdot c \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* (/ -0.5 b_2) c))

C

double code(double a, double b_2, double c) {
	return (-0.5 / b_2) * c;
}

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 = ((-0.5d0) / b_2) * c
end function

Java

public static double code(double a, double b_2, double c) {
	return (-0.5 / b_2) * c;
}

Python

def code(a, b_2, c):
	return (-0.5 / b_2) * c

Julia

function code(a, b_2, c)
	return Float64(Float64(-0.5 / b_2) * c)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-0.5 / b_2) * c;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(-0.5 / b$95$2), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 55.6%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
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 35.4

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

5. Applied rewrites 35.4%

   \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 35.3%

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

8. Final simplification 35.3%

   \[\leadsto \frac{-0.5}{b\_2} \cdot c \]
9. Add Preprocessing

Alternative 9: 10.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* 0.5 (/ c b_2)))

C

double code(double a, double b_2, double c) {
	return 0.5 * (c / b_2);
}

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 = 0.5d0 * (c / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.5 * (c / b_2);
}

Python

def code(a, b_2, c):
	return 0.5 * (c / b_2)

Julia

function code(a, b_2, c)
	return Float64(0.5 * Float64(c / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.5 * (c / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in b_2 around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 33.0

       \[\leadsto \left(-b\_2\right) \cdot \mathsf{fma}\left(\frac{\frac{c}{b\_2}}{b\_2}, -0.5, \color{blue}{\frac{2}{a}}\right) \]

5. Applied rewrites 33.0%

   \[\leadsto \color{blue}{\left(-b\_2\right) \cdot \mathsf{fma}\left(\frac{\frac{c}{b\_2}}{b\_2}, -0.5, \frac{2}{a}\right)} \]

6. Taylor expanded in a around inf

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

8. Applied rewrites 7.6%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{c}{b\_2}} \]
9. 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 2024255 
(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))