Quadratic roots, narrow range

Percentage Accurate: 56.0% → 99.1%

Time: 10.5s

Alternatives: 9

Speedup: 3.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 56.0% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{\frac{-c}{a}}\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (pow
  (/ (* (/ 0.5 a) (+ (sqrt (fma (* -4.0 c) a (* b b))) b)) (/ (- c) a))
  -1.0))

C

double code(double a, double b, double c) {
	return pow((((0.5 / a) * (sqrt(fma((-4.0 * c), a, (b * b))) + b)) / (-c / a)), -1.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)) / Float64(Float64(-c) / a)) ^ -1.0
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[((-c) / a), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 56.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 56.3

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

   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 56.3

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

4. Applied rewrites 56.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. lower-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/r* N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. associate-/r* N/A

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

   18. lift-*.f64 N/A

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

   19. lower-*.f64 N/A

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

6. Applied rewrites 55.6%

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

8. Applied rewrites 56.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{{\left(\frac{0.5}{a}\right)}^{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - {\left(\frac{0.5}{a} \cdot b\right)}^{2}}}} \]

9. Taylor expanded in a around 0

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

    1. associate-*r/ N/A

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

    2. lower-/.f64 N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 99.1

       \[\leadsto \frac{1}{\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{\frac{\color{blue}{-c}}{a}}} \]

11. Applied rewrites 99.1%

    \[\leadsto \frac{1}{\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{\color{blue}{\frac{-c}{a}}}} \]

12. Final simplification 99.1%

    \[\leadsto {\left(\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{\frac{-c}{a}}\right)}^{-1} \]
13. Add Preprocessing

Alternative 2: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00058:\\ \;\;\;\;\frac{0.5 \cdot \left(t\_0 - b \cdot b\right)}{a \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00058)
     (/ (* 0.5 (- t_0 (* b b))) (* a (+ (sqrt t_0) b)))
     (pow (/ (fma a (/ c b) (- b)) c) -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00058) {
		tmp = (0.5 * (t_0 - (b * b))) / (a * (sqrt(t_0) + b));
	} else {
		tmp = pow((fma(a, (c / b), -b) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.00058)
		tmp = Float64(Float64(0.5 * Float64(t_0 - Float64(b * b))) / Float64(a * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -0.00058], N[(N[(0.5 * N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.8e-4

   1. Initial program 79.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 79.9

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

      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 79.9

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

   4. Applied rewrites 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 81.0%

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

   if -5.8e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 42.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. 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 42.1

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

      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 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. associate-/r* N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 41.8%

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

   8. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{{\left(\frac{0.5}{a}\right)}^{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - {\left(\frac{0.5}{a} \cdot b\right)}^{2}}}} \]

   9. Taylor expanded in c around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 89.7

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

   11. Applied rewrites 89.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00058:\\ \;\;\;\;\frac{0.5 \cdot \left(\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b\right)}{a \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00058:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00058)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 c) a)))) (* 2.0 a))
   (pow (/ (fma a (/ c b) (- b)) c) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00058) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * c) * a)))) / (2.0 * a);
	} else {
		tmp = pow((fma(a, (c / b), -b) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.00058)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a)))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.00058], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.8e-4

   1. Initial program 79.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 80.2

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

   4. Applied rewrites 80.2%

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

   if -5.8e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 42.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. 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 42.1

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

      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 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. associate-/r* N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 41.8%

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

   8. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{{\left(\frac{0.5}{a}\right)}^{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - {\left(\frac{0.5}{a} \cdot b\right)}^{2}}}} \]

   9. Taylor expanded in c around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 89.7

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

   11. Applied rewrites 89.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00058:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00058:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00058)
   (/ (- (sqrt (fma (* -4.0 c) a (* b b))) b) (* 2.0 a))
   (pow (/ (fma a (/ c b) (- b)) c) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00058) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = pow((fma(a, (c / b), -b) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.00058)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.00058], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.8e-4

   1. Initial program 79.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 79.9

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 79.9

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

   4. Applied rewrites 79.9%

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

   if -5.8e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 42.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. 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 42.1

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

      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 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. associate-/r* N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 41.8%

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

   8. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{{\left(\frac{0.5}{a}\right)}^{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - {\left(\frac{0.5}{a} \cdot b\right)}^{2}}}} \]

   9. Taylor expanded in c around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 89.7

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

   11. Applied rewrites 89.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00058:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00058:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00058)
   (* (/ 0.5 a) (- (sqrt (fma (* -4.0 c) a (* b b))) b))
   (pow (/ (fma a (/ c b) (- b)) c) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00058) {
		tmp = (0.5 / a) * (sqrt(fma((-4.0 * c), a, (b * b))) - b);
	} else {
		tmp = pow((fma(a, (c / b), -b) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.00058)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.00058], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.8e-4

   1. Initial program 79.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 79.9

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

      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 79.9

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

   4. Applied rewrites 79.9%

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

   if -5.8e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 42.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. 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 42.1

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

      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 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. associate-/r* N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 41.8%

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

   8. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{{\left(\frac{0.5}{a}\right)}^{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - {\left(\frac{0.5}{a} \cdot b\right)}^{2}}}} \]

   9. Taylor expanded in c around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 89.7

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

   11. Applied rewrites 89.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00058:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (pow (/ (fma a (/ c b) (- b)) c) -1.0))

C

double code(double a, double b, double c) {
	return pow((fma(a, (c / b), -b) / c), -1.0);
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(c / b), Float64(-b)) / c) ^ -1.0
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 56.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 56.3

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

   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 56.3

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

4. Applied rewrites 56.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. lower-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/r* N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. associate-/r* N/A

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

   18. lift-*.f64 N/A

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

   19. lower-*.f64 N/A

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

6. Applied rewrites 55.6%

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

8. Applied rewrites 56.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{{\left(\frac{0.5}{a}\right)}^{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - {\left(\frac{0.5}{a} \cdot b\right)}^{2}}}} \]

9. Taylor expanded in c around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. associate-/l* N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. mul-1-neg N/A

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

    7. lower-neg.f64 79.1

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

11. Applied rewrites 79.1%

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

12. Final simplification 79.1%

    \[\leadsto {\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1} \]
13. Add Preprocessing

Alternative 7: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (pow (- (/ a b) (/ b c)) -1.0))

C

double code(double a, double b, double c) {
	return pow(((a / b) - (b / c)), -1.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((a / b) - (b / c)) ** (-1.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return Math.pow(((a / b) - (b / c)), -1.0);
}

Python

def code(a, b, c):
	return math.pow(((a / b) - (b / c)), -1.0)

Julia

function code(a, b, c)
	return Float64(Float64(a / b) - Float64(b / c)) ^ -1.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a / b) - (b / c)) ^ -1.0;
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 56.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 56.3

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

   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 56.3

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

4. Applied rewrites 56.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. lower-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/r* N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. associate-/r* N/A

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

   18. lift-*.f64 N/A

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

   19. lower-*.f64 N/A

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

6. Applied rewrites 55.6%

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

8. Applied rewrites 56.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{{\left(\frac{0.5}{a}\right)}^{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - {\left(\frac{0.5}{a} \cdot b\right)}^{2}}}} \]

9. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. unsub-neg N/A

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

    4. lower--.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 79.1

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

11. Applied rewrites 79.1%

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

12. Final simplification 79.1%

    \[\leadsto {\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1} \]
13. Add Preprocessing

Alternative 8: 63.8% 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(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 56.3%

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

3. Taylor expanded in a 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 63.3

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

5. Applied rewrites 63.3%

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

Alternative 9: 3.2% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 56.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 56.3

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

   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 56.3

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

4. Applied rewrites 56.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. lower-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/r* N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. associate-/r* N/A

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

   18. lift-*.f64 N/A

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

   19. lower-*.f64 N/A

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

6. Applied rewrites 55.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 56.9

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 56.9

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

8. Applied rewrites 56.9%

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

9. Taylor expanded in c around 0

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

    1. distribute-rgt-out N/A

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

    2. metadata-eval N/A

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

    3. mul0-rgt 3.2

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

11. Applied rewrites 3.2%

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

Reproduce

herbie shell --seed 2024322 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))