Quadratic roots, full range

Percentage Accurate: 52.8% → 90.9%

Time: 8.4s

Alternatives: 9

Speedup: 2.5×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.8% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -5.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+157)
   (/ (- b) a)
   (if (<= b -5.8e-300)
     (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* 2.0 a))
     (if (<= b 2e+129)
       (/ (* 2.0 c) (- (- b) (sqrt (fma (* -4.0 c) a (* b b)))))
       (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+157) {
		tmp = -b / a;
	} else if (b <= -5.8e-300) {
		tmp = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (2.0 * a);
	} else if (b <= 2e+129) {
		tmp = (2.0 * c) / (-b - sqrt(fma((-4.0 * c), a, (b * b))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e+157)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -5.8e-300)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(2.0 * a));
	elseif (b <= 2e+129)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+157], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -5.8e-300], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+129], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.6e157

   1. Initial program 37.4%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.6e157 < b < -5.79999999999999985e-300

   1. Initial program 94.8%

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

   if -5.79999999999999985e-300 < b < 2e129

   1. Initial program 48.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 48.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 48.3

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

   4. Applied rewrites 48.3%

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

   6. Applied rewrites 81.2%

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

   7. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 88.1

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

   9. Applied rewrites 88.1%

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

   if 2e129 < b

   1. Initial program 6.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -5.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-245}:\\ \;\;\;\;\left(t\_0 - b\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 c) a (* b b)))))
   (if (<= b -1.5e+131)
     (/ (- b) a)
     (if (<= b -4.5e-245)
       (* (- t_0 b) (/ 0.5 a))
       (if (<= b 2e+129) (/ (* 2.0 c) (- (- b) t_0)) (/ (- c) b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * c), a, (b * b)));
	double tmp;
	if (b <= -1.5e+131) {
		tmp = -b / a;
	} else if (b <= -4.5e-245) {
		tmp = (t_0 - b) * (0.5 / a);
	} else if (b <= 2e+129) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))
	tmp = 0.0
	if (b <= -1.5e+131)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -4.5e-245)
		tmp = Float64(Float64(t_0 - b) * Float64(0.5 / a));
	elseif (b <= 2e+129)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.5e+131], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -4.5e-245], N[(N[(t$95$0 - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+129], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.5000000000000001e131

   1. Initial program 49.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.5000000000000001e131 < b < -4.49999999999999969e-245

   1. Initial program 96.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 96.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 96.5

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

   4. Applied rewrites 96.5%

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

   if -4.49999999999999969e-245 < b < 2e129

   1. Initial program 49.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 49.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 49.0

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

   4. Applied rewrites 49.0%

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

   6. Applied rewrites 80.1%

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

   7. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.7

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

   9. Applied rewrites 86.7%

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

   if 2e129 < b

   1. Initial program 6.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-245}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-69}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+131)
   (/ (- b) a)
   (if (<= b 2.15e-69)
     (* (- (sqrt (fma (* -4.0 c) a (* b b))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+131) {
		tmp = -b / a;
	} else if (b <= 2.15e-69) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+131)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.15e-69)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+131], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.15e-69], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5000000000000001e131

   1. Initial program 49.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.5000000000000001e131 < b < 2.15e-69

   1. Initial program 88.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 87.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 87.9

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

   4. Applied rewrites 87.9%

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

   if 2.15e-69 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 85.4

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

   5. Applied rewrites 85.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-69}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-92}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-69}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\left(c \cdot a\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.65e-92)
   (- (/ c b) (/ b a))
   (if (<= b 2.15e-69)
     (/ (- (* 2.0 c)) (+ (sqrt (* (* c a) -4.0)) b))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e-92) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.15e-69) {
		tmp = -(2.0 * c) / (sqrt(((c * a) * -4.0)) + b);
	} else {
		tmp = -c / b;
	}
	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 <= (-2.65d-92)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.15d-69) then
        tmp = -(2.0d0 * c) / (sqrt(((c * a) * (-4.0d0))) + b)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e-92) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.15e-69) {
		tmp = -(2.0 * c) / (Math.sqrt(((c * a) * -4.0)) + b);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.65e-92:
		tmp = (c / b) - (b / a)
	elif b <= 2.15e-69:
		tmp = -(2.0 * c) / (math.sqrt(((c * a) * -4.0)) + b)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.65e-92)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.15e-69)
		tmp = Float64(Float64(-Float64(2.0 * c)) / Float64(sqrt(Float64(Float64(c * a) * -4.0)) + b));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.65e-92)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.15e-69)
		tmp = -(2.0 * c) / (sqrt(((c * a) * -4.0)) + b);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.65e-92], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.15e-69], N[((-N[(2.0 * c), $MachinePrecision]) / N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.65000000000000015e-92

   1. Initial program 74.7%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 87.9

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

   8. Applied rewrites 87.9%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 88.3%

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

   if -2.65000000000000015e-92 < b < 2.15e-69

   1. Initial program 79.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 79.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 79.4

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

   4. Applied rewrites 79.4%

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.2

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

   9. Applied rewrites 82.2%

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

   10. Taylor expanded in c around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 77.5

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

   12. Applied rewrites 77.5%

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

   if 2.15e-69 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 85.4

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

   5. Applied rewrites 85.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-92}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-69}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\left(c \cdot a\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-70}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot a\right) \cdot -4} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e-85)
   (- (/ c b) (/ b a))
   (if (<= b 9.5e-70)
     (* (- (sqrt (* (* c a) -4.0)) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-85) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9.5e-70) {
		tmp = (sqrt(((c * a) * -4.0)) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	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 <= (-6.5d-85)) then
        tmp = (c / b) - (b / a)
    else if (b <= 9.5d-70) then
        tmp = (sqrt(((c * a) * (-4.0d0))) - b) * (0.5d0 / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-85) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9.5e-70) {
		tmp = (Math.sqrt(((c * a) * -4.0)) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-85:
		tmp = (c / b) - (b / a)
	elif b <= 9.5e-70:
		tmp = (math.sqrt(((c * a) * -4.0)) - b) * (0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-85)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 9.5e-70)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.5e-85)
		tmp = (c / b) - (b / a);
	elseif (b <= 9.5e-70)
		tmp = (sqrt(((c * a) * -4.0)) - b) * (0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-85], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.5e-70], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e-85

   1. Initial program 74.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 88.7

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

   8. Applied rewrites 88.7%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 89.1%

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

   if -6.5e-85 < b < 9.4999999999999994e-70

   1. Initial program 80.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 80.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if 9.4999999999999994e-70 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 85.4

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

   5. Applied rewrites 85.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-70}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot a\right) \cdot -4} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.3%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 69.2

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

   5. Applied rewrites 69.2%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 68.9

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

   8. Applied rewrites 68.9%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 69.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.3%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 69.1

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

   5. Applied rewrites 69.1%

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

Alternative 7: 67.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-272}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.8e-272) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-272) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	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 <= 2.8d-272) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-272) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.8e-272:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.8e-272)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.8e-272)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.8e-272], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.79999999999999994e-272

   1. Initial program 77.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 2.79999999999999994e-272 < b

   1. Initial program 30.1%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.0

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

   5. Applied rewrites 71.0%

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

Alternative 8: 43.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5500000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5500000.0) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5500000.0) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	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 <= 5500000.0d0) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5500000.0) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5500000.0:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5500000.0)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5500000.0)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5500000.0], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5e6

   1. Initial program 72.1%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 48.9

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

   5. Applied rewrites 48.9%

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

   if 5.5e6 < b

   1. Initial program 15.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 2.2

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

   8. Applied rewrites 2.2%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 24.9%

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

Alternative 9: 11.0% 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 51.8%

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

3. Taylor expanded in b around -inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 32.3

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

5. Applied rewrites 32.3%

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

6. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 31.8

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

8. Applied rewrites 31.8%

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

9. Taylor expanded in c around inf

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

11. Applied rewrites 10.8%

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

Reproduce

herbie shell --seed 2024278 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))