The quadratic formula (r2)

Percentage Accurate: 52.4% → 85.9%

Time: 9.6s

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: 52.4% 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: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{+43}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-119}:\\ \;\;\;\;\frac{\left(a \cdot 4\right) \cdot c}{\left(-2 \cdot a\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+67}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.65e+43)
   (/ c (- b))
   (if (<= b -1.16e-119)
     (/ (* (* a 4.0) c) (* (* -2.0 a) (- b (sqrt (fma -4.0 (* c a) (* b b))))))
     (if (<= b 4.7e+67)
       (/ (* (+ (sqrt (fma (* c a) -4.0 (* b b))) b) -0.5) a)
       (- (/ c b) (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.65e+43) {
		tmp = c / -b;
	} else if (b <= -1.16e-119) {
		tmp = ((a * 4.0) * c) / ((-2.0 * a) * (b - sqrt(fma(-4.0, (c * a), (b * b)))));
	} else if (b <= 4.7e+67) {
		tmp = ((sqrt(fma((c * a), -4.0, (b * b))) + b) * -0.5) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.65e+43)
		tmp = Float64(c / Float64(-b));
	elseif (b <= -1.16e-119)
		tmp = Float64(Float64(Float64(a * 4.0) * c) / Float64(Float64(-2.0 * a) * Float64(b - sqrt(fma(-4.0, Float64(c * a), Float64(b * b))))));
	elseif (b <= 4.7e+67)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b) * -0.5) / a);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.65e+43], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, -1.16e-119], N[(N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision] / N[(N[(-2.0 * a), $MachinePrecision] * N[(b - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.7e+67], N[(N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.6500000000000001e43

   1. Initial program 13.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 89.2

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

   5. Applied rewrites 89.2%

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

   if -1.6500000000000001e43 < b < -1.16e-119

   1. Initial program 37.9%

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

   4. Applied rewrites 32.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

   if -1.16e-119 < b < 4.70000000000000017e67

   1. Initial program 87.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 87.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 87.9

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 87.9

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

   6. Applied rewrites 87.9%

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

   if 4.70000000000000017e67 < b

   1. Initial program 61.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{+43}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-119}:\\ \;\;\;\;\frac{\left(a \cdot 4\right) \cdot c}{\left(-2 \cdot a\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+67}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-111}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+67}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-111)
   (/ c (- b))
   (if (<= b 4.7e+67)
     (/ (* (+ (sqrt (fma (* c a) -4.0 (* b b))) b) -0.5) a)
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-111) {
		tmp = c / -b;
	} else if (b <= 4.7e+67) {
		tmp = ((sqrt(fma((c * a), -4.0, (b * b))) + b) * -0.5) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-111)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 4.7e+67)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b) * -0.5) / a);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-111], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 4.7e+67], N[(N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000008e-111

   1. Initial program 21.5%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 78.9

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

   5. Applied rewrites 78.9%

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

   if -3.00000000000000008e-111 < b < 4.70000000000000017e67

   1. Initial program 87.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 87.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 87.9

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 87.9

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

   6. Applied rewrites 87.9%

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

   if 4.70000000000000017e67 < b

   1. Initial program 61.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-111}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+67}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-111}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-96}:\\ \;\;\;\;\left(\sqrt{\left(-4 \cdot a\right) \cdot c} + b\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-111)
   (/ c (- b))
   (if (<= b 1.15e-96)
     (* (+ (sqrt (* (* -4.0 a) c)) b) (/ -0.5 a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-111) {
		tmp = c / -b;
	} else if (b <= 1.15e-96) {
		tmp = (sqrt(((-4.0 * a) * c)) + b) * (-0.5 / a);
	} 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 <= (-3d-111)) then
        tmp = c / -b
    else if (b <= 1.15d-96) then
        tmp = (sqrt((((-4.0d0) * a) * c)) + b) * ((-0.5d0) / a)
    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 <= -3e-111) {
		tmp = c / -b;
	} else if (b <= 1.15e-96) {
		tmp = (Math.sqrt(((-4.0 * a) * c)) + b) * (-0.5 / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-111:
		tmp = c / -b
	elif b <= 1.15e-96:
		tmp = (math.sqrt(((-4.0 * a) * c)) + b) * (-0.5 / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-111)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1.15e-96)
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) + b) * Float64(-0.5 / a));
	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 <= -3e-111)
		tmp = c / -b;
	elseif (b <= 1.15e-96)
		tmp = (sqrt(((-4.0 * a) * c)) + b) * (-0.5 / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-111], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1.15e-96], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000008e-111

   1. Initial program 21.5%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 78.9

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

   5. Applied rewrites 78.9%

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

   if -3.00000000000000008e-111 < b < 1.15e-96

   1. Initial program 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 82.4%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 76.0

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

   9. Applied rewrites 76.0%

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

   if 1.15e-96 < b

   1. Initial program 73.8%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 87.3

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

   5. Applied rewrites 87.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-111}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-96}:\\ \;\;\;\;\left(\sqrt{\left(-4 \cdot a\right) \cdot c} + b\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ c (- b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = c / -b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		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 <= -5e-310)
		tmp = c / -b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(c / (-b)), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 32.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 63.7

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

   5. Applied rewrites 63.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 77.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 69.5

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

   5. Applied rewrites 69.5%

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

Alternative 5: 67.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		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 <= -5e-310)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 32.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 63.7

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

   5. Applied rewrites 63.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 77.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 69.2

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

   5. Applied rewrites 69.2%

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

Alternative 6: 35.2% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (a b c) :precision binary64 (/ c (- b)))

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / -b
end function

Java

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

Python

def code(a, b, c):
	return c / -b

Julia

function code(a, b, c)
	return Float64(c / Float64(-b))
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c / (-b)), $MachinePrecision]

Derivation

1. Initial program 57.1%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
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 30.4

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

5. Applied rewrites 30.4%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Alternative 7: 10.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 57.1%

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

4. Applied rewrites 32.2%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 11.4

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

7. Applied rewrites 11.4%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
8. Add Preprocessing

Alternative 8: 2.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return b / 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 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(b / a), $MachinePrecision]

Derivation

1. Initial program 57.1%

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

4. Applied rewrites 32.2%

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

5. Taylor expanded in b around -inf

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

   1. lower-/.f64 2.5

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

7. Applied rewrites 2.5%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
8. Add Preprocessing

Developer Target 1: 70.7% 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 2024295 
(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)))