The quadratic formula (r2)

Percentage Accurate: 52.1% → 85.0%

Time: 13.0s

Alternatives: 8

Speedup: 2.5×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-115)
   (/ c (- b))
   (if (<= b 5e+55)
     (/ (+ b (sqrt (fma b b (* c (* a -4.0))))) (* a -2.0))
     (- (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-115) {
		tmp = c / -b;
	} else if (b <= 5e+55) {
		tmp = (b + sqrt(fma(b, b, (c * (a * -4.0))))) / (a * -2.0);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-115)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5e+55)
		tmp = Float64(Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))) / Float64(a * -2.0));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-115], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5e+55], N[(N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -8.49999999999999953e-115

   1. Initial program 18.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -8.49999999999999953e-115 < b < 5.00000000000000046e55

   1. Initial program 81.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}} \]

   if 5.00000000000000046e55 < b

   1. Initial program 64.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 95.3

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

   5. Applied rewrites 95.3%

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

Alternative 2: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+55}:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-115)
   (/ c (- b))
   (if (<= b 4.8e+55)
     (* (+ b (sqrt (fma b b (* c (* a -4.0))))) (/ -0.5 a))
     (- (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-115) {
		tmp = c / -b;
	} else if (b <= 4.8e+55) {
		tmp = (b + sqrt(fma(b, b, (c * (a * -4.0))))) * (-0.5 / a);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-115)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 4.8e+55)
		tmp = Float64(Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))) * Float64(-0.5 / a));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-115], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 4.8e+55], N[(N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -8.49999999999999953e-115

   1. Initial program 18.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -8.49999999999999953e-115 < b < 4.7999999999999998e55

   1. Initial program 81.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]

   if 4.7999999999999998e55 < b

   1. Initial program 64.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 95.3

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

   5. Applied rewrites 95.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+55}:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.45 \cdot 10^{-31}:\\ \;\;\;\;\frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-115)
   (/ c (- b))
   (if (<= b 3.45e-31)
     (/ (+ b (sqrt (* -4.0 (* c a)))) (* a -2.0))
     (fma b (/ c (* b b)) (- (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-115) {
		tmp = c / -b;
	} else if (b <= 3.45e-31) {
		tmp = (b + sqrt((-4.0 * (c * a)))) / (a * -2.0);
	} else {
		tmp = fma(b, (c / (b * b)), -(b / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-115)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 3.45e-31)
		tmp = Float64(Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))) / Float64(a * -2.0));
	else
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(-Float64(b / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-115], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 3.45e-31], N[(N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.49999999999999953e-115

   1. Initial program 18.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -8.49999999999999953e-115 < b < 3.4500000000000002e-31

   1. Initial program 80.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a \cdot -2} \]

      2. *-commutative N/A

         \[\leadsto \frac{b + \sqrt{-4 \cdot \color{blue}{\left(c \cdot a\right)}}}{a \cdot -2} \]

      3. lower-*.f64 69.4

         \[\leadsto \frac{b + \sqrt{-4 \cdot \color{blue}{\left(c \cdot a\right)}}}{a \cdot -2} \]

   7. Applied rewrites 69.4%

      \[\leadsto \frac{b + \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right)}}}{a \cdot -2} \]

   if 3.4500000000000002e-31 < b

   1. Initial program 69.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + b \cdot \frac{\color{blue}{-1}}{a} \]

      5. associate-/l* N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \color{blue}{\frac{b \cdot -1}{a}} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \frac{\color{blue}{-1 \cdot b}}{a} \]

      7. associate-*r/ N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.45 \cdot 10^{-31}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-115)
   (/ c (- b))
   (if (<= b 3.45e-31)
     (* (/ -0.5 a) (+ b (sqrt (* a (* c -4.0)))))
     (fma b (/ c (* b b)) (- (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-115) {
		tmp = c / -b;
	} else if (b <= 3.45e-31) {
		tmp = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = fma(b, (c / (b * b)), -(b / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-115)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 3.45e-31)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(-Float64(b / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-115], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 3.45e-31], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.49999999999999953e-115

   1. Initial program 18.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -8.49999999999999953e-115 < b < 3.4500000000000002e-31

   1. Initial program 80.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 79.9%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(-4 \cdot c\right)}}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \]

      6. lower-*.f64 69.3

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \]

   7. Applied rewrites 69.3%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \]

   if 3.4500000000000002e-31 < b

   1. Initial program 69.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + b \cdot \frac{\color{blue}{-1}}{a} \]

      5. associate-/l* N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \color{blue}{\frac{b \cdot -1}{a}} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \frac{\color{blue}{-1 \cdot b}}{a} \]

      7. associate-*r/ N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

Alternative 5: 67.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-311) (/ c (- b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-311)) then
        tmp = c / -b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-311:
		tmp = c / -b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-311)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-311)
		tmp = c / -b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-311], N[(c / (-b)), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9999999999999e-311

   1. Initial program 30.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 72.9

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

   5. Applied rewrites 72.9%

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

   if -1.9999999999999e-311 < b

   1. Initial program 74.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b}} - \frac{b}{a} \]

      6. lower-/.f64 67.5

         \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]

   5. Applied rewrites 67.5%

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

Alternative 6: 67.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-311) (/ c (- b)) (- (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		tmp = c / -b;
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-311)) then
        tmp = c / -b
    else
        tmp = -(b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		tmp = c / -b;
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-311:
		tmp = c / -b
	else:
		tmp = -(b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-311)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-311)
		tmp = c / -b;
	else
		tmp = -(b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-311], N[(c / (-b)), $MachinePrecision], (-N[(b / a), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < -1.9999999999999e-311

   1. Initial program 30.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 72.9

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

   5. Applied rewrites 72.9%

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

   if -1.9999999999999e-311 < b

   1. Initial program 74.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 67.0

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

   5. Applied rewrites 67.0%

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

Alternative 7: 42.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-311}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -2e-311) 0.0 (- (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		tmp = 0.0;
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-311)) then
        tmp = 0.0d0
    else
        tmp = -(b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		tmp = 0.0;
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-311:
		tmp = 0.0
	else:
		tmp = -(b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-311)
		tmp = 0.0;
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-311)
		tmp = 0.0;
	else
		tmp = -(b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-311], 0.0, (-N[(b / a), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < -1.9999999999999e-311

   1. Initial program 30.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\frac{\left(b \cdot 0.5\right) \cdot a - a \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot 0.5\right)}{a \cdot a}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot \frac{1}{2}\right)} \cdot a - a \cdot \left(\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \frac{1}{2}\right)}{a \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot a} - a \cdot \left(\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \frac{1}{2}\right)}{a \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\sqrt{b \cdot b + c \cdot \color{blue}{\left(a \cdot -4\right)}} \cdot \frac{1}{2}\right)}{a \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\sqrt{b \cdot b + \color{blue}{c \cdot \left(a \cdot -4\right)}} \cdot \frac{1}{2}\right)}{a \cdot a} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \cdot \frac{1}{2}\right)}{a \cdot a} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\color{blue}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \cdot \frac{1}{2}\right)}{a \cdot a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \frac{1}{2}\right)}}{a \cdot a} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - \color{blue}{a \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \frac{1}{2}\right)}}{a \cdot a} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \frac{1}{2}\right)}{\color{blue}{a \cdot a}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \frac{1}{2}\right)}}{a \cdot a} \]

      11. lift-/.f64 17.1

          \[\leadsto \color{blue}{\frac{\left(b \cdot 0.5\right) \cdot a - a \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot 0.5\right)}{a \cdot a}} \]

   6. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\frac{a \cdot \mathsf{fma}\left(b, 0.5, \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} \cdot -0.5\right)}{a \cdot a}} \]

   7. Taylor expanded in a around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{b \cdot \color{blue}{0}}{a} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0}}{a} \]

      4. lower-/.f64 25.6

         \[\leadsto \color{blue}{\frac{0}{a}} \]

   9. Applied rewrites 25.6%

      \[\leadsto \color{blue}{\frac{0}{a}} \]
   10. Step-by-step derivation

       1. div0 25.6

          \[\leadsto \color{blue}{0} \]

   11. Applied rewrites 25.6%

       \[\leadsto \color{blue}{0} \]

   if -1.9999999999999e-311 < b

   1. Initial program 74.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 67.0

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

   5. Applied rewrites 67.0%

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

Alternative 8: 10.9% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 50.5%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 23.4%

   \[\leadsto \color{blue}{\frac{\left(b \cdot 0.5\right) \cdot a - a \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot 0.5\right)}{a \cdot a}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot \frac{1}{2}\right)} \cdot a - a \cdot \left(\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \frac{1}{2}\right)}{a \cdot a} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot a} - a \cdot \left(\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \frac{1}{2}\right)}{a \cdot a} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\sqrt{b \cdot b + c \cdot \color{blue}{\left(a \cdot -4\right)}} \cdot \frac{1}{2}\right)}{a \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\sqrt{b \cdot b + \color{blue}{c \cdot \left(a \cdot -4\right)}} \cdot \frac{1}{2}\right)}{a \cdot a} \]

   5. lift-fma.f64 N/A

      \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \cdot \frac{1}{2}\right)}{a \cdot a} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\color{blue}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \cdot \frac{1}{2}\right)}{a \cdot a} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \frac{1}{2}\right)}}{a \cdot a} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - \color{blue}{a \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \frac{1}{2}\right)}}{a \cdot a} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \frac{1}{2}\right)}{\color{blue}{a \cdot a}} \]

   10. lift--.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot a - a \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \frac{1}{2}\right)}}{a \cdot a} \]

   11. lift-/.f64 23.4

       \[\leadsto \color{blue}{\frac{\left(b \cdot 0.5\right) \cdot a - a \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot 0.5\right)}{a \cdot a}} \]

6. Applied rewrites 23.5%

   \[\leadsto \color{blue}{\frac{a \cdot \mathsf{fma}\left(b, 0.5, \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} \cdot -0.5\right)}{a \cdot a}} \]

7. Taylor expanded in a around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{b \cdot \color{blue}{0}}{a} \]

   3. mul0-rgt N/A

      \[\leadsto \frac{\color{blue}{0}}{a} \]

   4. lower-/.f64 15.3

      \[\leadsto \color{blue}{\frac{0}{a}} \]

9. Applied rewrites 15.3%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
10. Step-by-step derivation

    1. div0 15.3

       \[\leadsto \color{blue}{0} \]

11. Applied rewrites 15.3%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Developer Target 1: 70.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp = 0.0;
	if (b < 0.0)
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-b - t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024219 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))