Cubic critical, narrow range

Percentage Accurate: 55.7% → 99.1%

Time: 13.9s

Alternatives: 9

Speedup: 2.9×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{3 \cdot \left(c \cdot a\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ (* 3.0 (* c a)) (* a (+ b (sqrt (fma c (* a -3.0) (* b b)))))) -3.0))

C

double code(double a, double b, double c) {
	return ((3.0 * (c * a)) / (a * (b + sqrt(fma(c, (a * -3.0), (b * b)))))) / -3.0;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(3.0 * Float64(c * a)) / Float64(a * Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b)))))) / -3.0)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

4. Applied rewrites 55.3%

   \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]

6. Applied rewrites 57.2%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}}{-3} \]

7. Taylor expanded in c around 0

   \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3} \]

   2. lower-*.f64 99.1

      \[\leadsto \frac{\frac{3 \cdot \color{blue}{\left(a \cdot c\right)}}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3} \]

9. Applied rewrites 99.1%

   \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3} \]

10. Final simplification 99.1%

    \[\leadsto \frac{\frac{3 \cdot \left(c \cdot a\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3} \]
11. Add Preprocessing

Alternative 2: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.2)
   (/ (- b (sqrt (fma b b (* a (* c -3.0))))) (* a -3.0))
   (fma a (/ (* -0.375 (* c c)) (* b (* b b))) (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2) {
		tmp = (b - sqrt(fma(b, b, (a * (c * -3.0))))) / (a * -3.0);
	} else {
		tmp = fma(a, ((-0.375 * (c * c)) / (b * (b * b))), ((c * -0.5) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / Float64(a * -3.0));
	else
		tmp = fma(a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * Float64(b * b))), Float64(Float64(c * -0.5) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2], N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -3.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000018

   1. Initial program 79.5%

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

   4. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}}{-3} \]

      3. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{a \cdot \left(-3 \cdot c\right) + b \cdot b}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      7. *-commutative N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(-3 \cdot c\right) \cdot a} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(-3 \cdot c\right)} \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      9. *-commutative N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(c \cdot -3\right)} \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      10. associate-*l* N/A

          \[\leadsto \left(b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      11. metadata-eval N/A

          \[\leadsto \left(b - \sqrt{c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

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

          \[\leadsto \left(b - \sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(3 \cdot a\right), b \cdot b\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      14. *-commutative N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \mathsf{neg}\left(\color{blue}{a \cdot 3}\right), b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

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

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      16. metadata-eval N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-3}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot -3}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      18. lower-/.f64 N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{\color{blue}{\frac{1}{a}}}{-3} \]

      4. associate-/r* N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{1}{a \cdot -3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{1}{\color{blue}{a \cdot -3}} \]

      6. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a \cdot -3}} \]

      7. lower-/.f64 79.5

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a \cdot -3}} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right) + b \cdot b}}}{a \cdot -3} \]

      9. +-commutative N/A

         \[\leadsto \frac{b - \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -3\right)}}}{a \cdot -3} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{b - \sqrt{\color{blue}{b \cdot b} + c \cdot \left(a \cdot -3\right)}}{a \cdot -3} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{a \cdot -3} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot -3\right)}\right)}}{a \cdot -3} \]

      13. *-commutative N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(-3 \cdot a\right)}\right)}}{a \cdot -3} \]

      14. associate-*r* N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -3\right) \cdot a}\right)}}{a \cdot -3} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -3\right) \cdot a}\right)}}{a \cdot -3} \]

      16. lower-*.f64 79.7

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -3\right)} \cdot a\right)}}{a \cdot -3} \]

   8. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)}}{a \cdot -3}} \]

   if 6.20000000000000018 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot {c}^{2}}{{b}^{3}} \cdot \frac{-3}{8}} + \frac{-1}{2} \cdot \frac{c}{b} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \cdot \frac{-3}{8} + \frac{-1}{2} \cdot \frac{c}{b} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot \frac{-3}{8}\right)} + \frac{-1}{2} \cdot \frac{c}{b} \]

      5. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)} + \frac{-1}{2} \cdot \frac{c}{b} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}}\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}}\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b}\right) \]

      20. lower-*.f64 87.1

          \[\leadsto \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \frac{\color{blue}{c \cdot -0.5}}{b}\right) \]

   5. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \frac{c \cdot -0.5}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.2)
   (/ (- b (sqrt (fma b b (* a (* c -3.0))))) (* a -3.0))
   (/ (fma (* a -0.375) (* c (/ c (* b b))) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2) {
		tmp = (b - sqrt(fma(b, b, (a * (c * -3.0))))) / (a * -3.0);
	} else {
		tmp = fma((a * -0.375), (c * (c / (b * b))), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / Float64(a * -3.0));
	else
		tmp = Float64(fma(Float64(a * -0.375), Float64(c * Float64(c / Float64(b * b))), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2], N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * -0.375), $MachinePrecision] * N[(c * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000018

   1. Initial program 79.5%

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

   4. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}}{-3} \]

      3. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{a \cdot \left(-3 \cdot c\right) + b \cdot b}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      7. *-commutative N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(-3 \cdot c\right) \cdot a} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(-3 \cdot c\right)} \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      9. *-commutative N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(c \cdot -3\right)} \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      10. associate-*l* N/A

          \[\leadsto \left(b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      11. metadata-eval N/A

          \[\leadsto \left(b - \sqrt{c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

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

          \[\leadsto \left(b - \sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(3 \cdot a\right), b \cdot b\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      14. *-commutative N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \mathsf{neg}\left(\color{blue}{a \cdot 3}\right), b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

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

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      16. metadata-eval N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-3}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot -3}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      18. lower-/.f64 N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{\color{blue}{\frac{1}{a}}}{-3} \]

      4. associate-/r* N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{1}{a \cdot -3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{1}{\color{blue}{a \cdot -3}} \]

      6. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a \cdot -3}} \]

      7. lower-/.f64 79.5

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a \cdot -3}} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right) + b \cdot b}}}{a \cdot -3} \]

      9. +-commutative N/A

         \[\leadsto \frac{b - \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -3\right)}}}{a \cdot -3} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{b - \sqrt{\color{blue}{b \cdot b} + c \cdot \left(a \cdot -3\right)}}{a \cdot -3} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{a \cdot -3} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot -3\right)}\right)}}{a \cdot -3} \]

      13. *-commutative N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(-3 \cdot a\right)}\right)}}{a \cdot -3} \]

      14. associate-*r* N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -3\right) \cdot a}\right)}}{a \cdot -3} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -3\right) \cdot a}\right)}}{a \cdot -3} \]

      16. lower-*.f64 79.7

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -3\right)} \cdot a\right)}}{a \cdot -3} \]

   8. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)}}{a \cdot -3}} \]

   if 6.20000000000000018 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{-3}{8} \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)} + \frac{-1}{2} \cdot c}{b} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot \frac{{c}^{2}}{{b}^{2}}} + \frac{-1}{2} \cdot c}{b} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}}{b} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{a \cdot \frac{-3}{8}}, \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{a \cdot \frac{-3}{8}}, \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a \cdot \frac{-3}{8}, \frac{\color{blue}{c \cdot c}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(a \cdot \frac{-3}{8}, \color{blue}{c \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a \cdot \frac{-3}{8}, \color{blue}{c \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a \cdot \frac{-3}{8}, c \cdot \color{blue}{\frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a \cdot \frac{-3}{8}, c \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a \cdot \frac{-3}{8}, c \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(a \cdot \frac{-3}{8}, c \cdot \frac{c}{b \cdot b}, \color{blue}{c \cdot \frac{-1}{2}}\right)}{b} \]

      15. lower-*.f64 87.0

          \[\leadsto \frac{\mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{b \cdot b}, \color{blue}{c \cdot -0.5}\right)}{b} \]

   5. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.2)
   (/ (- b (sqrt (fma b b (* a (* c -3.0))))) (* a -3.0))
   (/ (* c (fma -0.375 (* a (/ c (* b b))) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2) {
		tmp = (b - sqrt(fma(b, b, (a * (c * -3.0))))) / (a * -3.0);
	} else {
		tmp = (c * fma(-0.375, (a * (c / (b * b))), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / Float64(a * -3.0));
	else
		tmp = Float64(Float64(c * fma(-0.375, Float64(a * Float64(c / Float64(b * b))), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2], N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(-0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000018

   1. Initial program 79.5%

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

   4. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}}{-3} \]

      3. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{a \cdot \left(-3 \cdot c\right) + b \cdot b}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      7. *-commutative N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(-3 \cdot c\right) \cdot a} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(-3 \cdot c\right)} \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      9. *-commutative N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(c \cdot -3\right)} \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      10. associate-*l* N/A

          \[\leadsto \left(b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      11. metadata-eval N/A

          \[\leadsto \left(b - \sqrt{c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

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

          \[\leadsto \left(b - \sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(3 \cdot a\right), b \cdot b\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      14. *-commutative N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \mathsf{neg}\left(\color{blue}{a \cdot 3}\right), b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

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

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      16. metadata-eval N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-3}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot -3}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      18. lower-/.f64 N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{\color{blue}{\frac{1}{a}}}{-3} \]

      4. associate-/r* N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{1}{a \cdot -3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{1}{\color{blue}{a \cdot -3}} \]

      6. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a \cdot -3}} \]

      7. lower-/.f64 79.5

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a \cdot -3}} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right) + b \cdot b}}}{a \cdot -3} \]

      9. +-commutative N/A

         \[\leadsto \frac{b - \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -3\right)}}}{a \cdot -3} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{b - \sqrt{\color{blue}{b \cdot b} + c \cdot \left(a \cdot -3\right)}}{a \cdot -3} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{a \cdot -3} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot -3\right)}\right)}}{a \cdot -3} \]

      13. *-commutative N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(-3 \cdot a\right)}\right)}}{a \cdot -3} \]

      14. associate-*r* N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -3\right) \cdot a}\right)}}{a \cdot -3} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -3\right) \cdot a}\right)}}{a \cdot -3} \]

      16. lower-*.f64 79.7

          \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -3\right)} \cdot a\right)}}{a \cdot -3} \]

   8. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)}}{a \cdot -3}} \]

   if 6.20000000000000018 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}}, c \cdot -0.5\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.9%

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.2)
   (* (/ (- b (sqrt (fma b b (* a (* c -3.0))))) a) -0.3333333333333333)
   (/ (* c (fma -0.375 (* a (/ c (* b b))) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2) {
		tmp = ((b - sqrt(fma(b, b, (a * (c * -3.0))))) / a) * -0.3333333333333333;
	} else {
		tmp = (c * fma(-0.375, (a * (c / (b * b))), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2)
		tmp = Float64(Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / a) * -0.3333333333333333);
	else
		tmp = Float64(Float64(c * fma(-0.375, Float64(a * Float64(c / Float64(b * b))), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2], N[(N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(c * N[(-0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000018

   1. Initial program 79.5%

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

   4. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}}{-3} \]

      3. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{a \cdot \left(-3 \cdot c\right) + b \cdot b}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      7. *-commutative N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(-3 \cdot c\right) \cdot a} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(-3 \cdot c\right)} \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      9. *-commutative N/A

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(c \cdot -3\right)} \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      10. associate-*l* N/A

          \[\leadsto \left(b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      11. metadata-eval N/A

          \[\leadsto \left(b - \sqrt{c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

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

          \[\leadsto \left(b - \sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} + b \cdot b}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(3 \cdot a\right), b \cdot b\right)}}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      14. *-commutative N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \mathsf{neg}\left(\color{blue}{a \cdot 3}\right), b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

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

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      16. metadata-eval N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-3}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot -3}, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3} \]

      18. lower-/.f64 N/A

          \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   6. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

   8. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)}}{a} \cdot -0.3333333333333333} \]

   if 6.20000000000000018 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}}, c \cdot -0.5\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.9%

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.2)
   (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* c -3.0) (* b b)))))
   (/ (* c (fma -0.375 (* a (/ c (* b b))) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (c * -3.0), (b * b))));
	} else {
		tmp = (c * fma(-0.375, (a * (c / (b * b))), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))));
	else
		tmp = Float64(Float64(c * fma(-0.375, Float64(a * Float64(c / Float64(b * b))), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(-0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000018

   1. Initial program 79.5%

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

   4. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

   if 6.20000000000000018 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}}, c \cdot -0.5\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.9%

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (fma -0.375 (* a (/ c (* b b))) -0.5)) b))

C

double code(double a, double b, double c) {
	return (c * fma(-0.375, (a * (c / (b * b))), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(-0.375, Float64(a * Float64(c / Float64(b * b))), -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

5. Applied rewrites 88.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}}, c \cdot -0.5\right)\right)}{b}} \]

6. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
7. Step-by-step derivation

8. Applied rewrites 82.0%

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b} \]
9. Add Preprocessing

Alternative 8: 64.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (* c -0.5) b))

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 * (-0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * -0.5) / b;
}

Python

def code(a, b, c):
	return (c * -0.5) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * -0.5) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * -0.5) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 64.7

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

5. Applied rewrites 64.7%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
6. Add Preprocessing

Alternative 9: 64.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* c (/ -0.5 b)))

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 * ((-0.5d0) / b)
end function

Java

public static double code(double a, double b, double c) {
	return c * (-0.5 / b);
}

Python

def code(a, b, c):
	return c * (-0.5 / b)

Julia

function code(a, b, c)
	return Float64(c * Float64(-0.5 / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (-0.5 / b);
end

Wolfram

code[a_, b_, c_] := N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 64.7

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

5. Applied rewrites 64.7%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 64.7%

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

8. Final simplification 64.7%

   \[\leadsto c \cdot \frac{-0.5}{b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))