quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.1% → 86.1%

Time: 10.0s

Alternatives: 8

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: 52.1% 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: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.25 \cdot 10^{-74}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(0.125, \frac{a \cdot \frac{c}{b\_2}}{b\_2}, 0.5\right)}{-b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -1, \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.25e-74)
   (/ (* c (fma 0.125 (/ (* a (/ c b_2)) b_2) 0.5)) (- b_2))
   (if (<= b_2 1.35e+79)
     (fma (/ b_2 a) -1.0 (/ (sqrt (- (* b_2 b_2) (* a c))) (- a)))
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.25e-74) {
		tmp = (c * fma(0.125, ((a * (c / b_2)) / b_2), 0.5)) / -b_2;
	} else if (b_2 <= 1.35e+79) {
		tmp = fma((b_2 / a), -1.0, (sqrt(((b_2 * b_2) - (a * c))) / -a));
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.25e-74)
		tmp = Float64(Float64(c * fma(0.125, Float64(Float64(a * Float64(c / b_2)) / b_2), 0.5)) / Float64(-b_2));
	elseif (b_2 <= 1.35e+79)
		tmp = fma(Float64(b_2 / a), -1.0, Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) / Float64(-a)));
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.25e-74], N[(N[(c * N[(0.125 * N[(N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] / b$95$2), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / (-b$95$2)), $MachinePrecision], If[LessEqual[b$95$2, 1.35e+79], N[(N[(b$95$2 / a), $MachinePrecision] * -1.0 + N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.2500000000000001e-74

   1. Initial program 15.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 87.1%

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(0.125, a \cdot \frac{c}{b\_2 \cdot b\_2}, 0.5\right)}{-\color{blue}{b\_2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 87.4%

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

   if -3.2500000000000001e-74 < b_2 < 1.35e79

   1. Initial program 83.8%

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

   4. Applied rewrites 83.8%

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

   if 1.35e79 < b_2

   1. Initial program 61.4%

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

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

      2. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.25 \cdot 10^{-74}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(0.125, \frac{a \cdot \frac{c}{b\_2}}{b\_2}, 0.5\right)}{-b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -1, \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.25 \cdot 10^{-74}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(0.125, \frac{a \cdot \frac{c}{b\_2}}{b\_2}, 0.5\right)}{-b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.25e-74)
   (/ (* c (fma 0.125 (/ (* a (/ c b_2)) b_2) 0.5)) (- b_2))
   (if (<= b_2 1.35e+79)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.25e-74) {
		tmp = (c * fma(0.125, ((a * (c / b_2)) / b_2), 0.5)) / -b_2;
	} else if (b_2 <= 1.35e+79) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.25e-74)
		tmp = Float64(Float64(c * fma(0.125, Float64(Float64(a * Float64(c / b_2)) / b_2), 0.5)) / Float64(-b_2));
	elseif (b_2 <= 1.35e+79)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.25e-74], N[(N[(c * N[(0.125 * N[(N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] / b$95$2), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / (-b$95$2)), $MachinePrecision], If[LessEqual[b$95$2, 1.35e+79], 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[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.2500000000000001e-74

   1. Initial program 15.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 87.1%

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(0.125, a \cdot \frac{c}{b\_2 \cdot b\_2}, 0.5\right)}{-\color{blue}{b\_2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 87.4%

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

   if -3.2500000000000001e-74 < b_2 < 1.35e79

   1. Initial program 83.8%

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

   if 1.35e79 < b_2

   1. Initial program 61.4%

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

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

      2. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.25 \cdot 10^{-74}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(0.125, \frac{a \cdot \frac{c}{b\_2}}{b\_2}, 0.5\right)}{-b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-74)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.35e+79)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-74) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.35e+79) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.2d-74)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 1.35d+79) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-74) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.35e+79) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.2e-74:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 1.35e+79:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-74)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.35e+79)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.2e-74)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 1.35e+79)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.2e-74], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.35e+79], 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[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.1999999999999999e-74

   1. Initial program 15.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -3.1999999999999999e-74 < b_2 < 1.35e79

   1. Initial program 83.8%

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

   if 1.35e79 < b_2

   1. Initial program 61.4%

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

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

      2. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

Alternative 4: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.32 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, \frac{c \cdot 0.5}{b\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-74)
   (/ (* c -0.5) b_2)
   (if (<= b_2 2.32e-66)
     (/ (- (- b_2) (sqrt (- (* a c)))) a)
     (fma -2.0 (/ b_2 a) (/ (* c 0.5) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-74) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.32e-66) {
		tmp = (-b_2 - sqrt(-(a * c))) / a;
	} else {
		tmp = fma(-2.0, (b_2 / a), ((c * 0.5) / b_2));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-74)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 2.32e-66)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(-Float64(a * c)))) / a);
	else
		tmp = fma(-2.0, Float64(b_2 / a), Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.2e-74], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.32e-66], N[(N[((-b$95$2) - N[Sqrt[(-N[(a * c), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.1999999999999999e-74

   1. Initial program 15.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -3.1999999999999999e-74 < b_2 < 2.32000000000000004e-66

   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 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if 2.32000000000000004e-66 < b_2

   1. Initial program 74.4%

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 89.1

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

   8. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{b\_2}{a}, \frac{0.5 \cdot c}{b\_2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.32 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, \frac{c \cdot 0.5}{b\_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, \frac{c \cdot 0.5}{b\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310)
   (/ (* c -0.5) b_2)
   (fma -2.0 (/ b_2 a) (/ (* c 0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = fma(-2.0, (b_2 / a), ((c * 0.5) / b_2));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = fma(-2.0, Float64(b_2 / a), Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 28.5%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 78.9%

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

   3. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 65.0

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

   8. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{b\_2}{a}, \frac{0.5 \cdot c}{b\_2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b\_2}{a}, \frac{c \cdot 0.5}{b\_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (/ (* c -0.5) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 28.5%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 78.9%

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

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

      2. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

Alternative 7: 67.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (/ (* c -0.5) b_2) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = (c * -0.5) / b_2
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = (c * -0.5) / b_2;
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 28.5%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 78.9%

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

   3. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 64.8

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

   5. Applied rewrites 64.8%

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

Alternative 8: 35.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / 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 * ((-2.0d0) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 36.0

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

5. Applied rewrites 36.0%

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

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

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

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) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

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 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	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 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	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 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	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(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	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 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	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[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024222 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :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) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

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