Cubic critical, medium range

Percentage Accurate: 31.4% → 99.0%

Time: 12.1s

Alternatives: 9

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.4% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(0.3333333333333333 \cdot \frac{\left(c \cdot a\right) \cdot 3}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}\right) \cdot \frac{-1}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (*
   0.3333333333333333
   (/ (* (* c a) 3.0) (+ (sqrt (fma (* -3.0 a) c (* b b))) b)))
  (/ -1.0 a)))

C

double code(double a, double b, double c) {
	return (0.3333333333333333 * (((c * a) * 3.0) / (sqrt(fma((-3.0 * a), c, (b * b))) + b))) * (-1.0 / a);
}

Julia

function code(a, b, c)
	return Float64(Float64(0.3333333333333333 * Float64(Float64(Float64(c * a) * 3.0) / Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + b))) * Float64(-1.0 / a))
end

Wolfram

code[a_, b_, c_] := N[(N[(0.3333333333333333 * N[(N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

4. Applied rewrites 31.1%

   \[\leadsto \color{blue}{\frac{-1}{a} \cdot \left(\left(b - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right) \cdot 0.3333333333333333\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. lower-*.f64 31.1

       \[\leadsto \frac{-1}{a} \cdot \left(\left(b - \sqrt{\mathsf{fma}\left(\color{blue}{-3 \cdot a}, c, b \cdot b\right)}\right) \cdot 0.3333333333333333\right) \]

6. Applied rewrites 31.1%

   \[\leadsto \frac{-1}{a} \cdot \left(\left(b - \sqrt{\color{blue}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}\right) \cdot 0.3333333333333333\right) \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. flip-- N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. rem-square-sqrt N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 N/A

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

8. Applied rewrites 33.7%

   \[\leadsto \frac{-1}{a} \cdot \left(\color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b, b \cdot b, \left(-\left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b\right)\right) \cdot \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}} \cdot 0.3333333333333333\right) \]

9. Taylor expanded in c around inf

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 99.0

       \[\leadsto \frac{-1}{a} \cdot \left(\frac{3 \cdot \color{blue}{\left(a \cdot c\right)}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b} \cdot 0.3333333333333333\right) \]

11. Applied rewrites 99.0%

    \[\leadsto \frac{-1}{a} \cdot \left(\frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b} \cdot 0.3333333333333333\right) \]

12. Final simplification 99.0%

    \[\leadsto \left(0.3333333333333333 \cdot \frac{\left(c \cdot a\right) \cdot 3}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}\right) \cdot \frac{-1}{a} \]
13. Add Preprocessing

Alternative 2: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.084:\\ \;\;\;\;\mathsf{fma}\left(b, 0.3333333333333333, \left(-0.3333333333333333\right) \cdot \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}\right) \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot a, \frac{\frac{c \cdot c}{b \cdot b}}{b}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.084)
   (*
    (fma
     b
     0.3333333333333333
     (* (- 0.3333333333333333) (sqrt (fma (* c a) -3.0 (* b b)))))
    (/ -1.0 a))
   (fma (* -0.375 a) (/ (/ (* c c) (* b b)) b) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.084) {
		tmp = fma(b, 0.3333333333333333, (-0.3333333333333333 * sqrt(fma((c * a), -3.0, (b * b))))) * (-1.0 / a);
	} else {
		tmp = fma((-0.375 * a), (((c * c) / (b * b)) / b), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.084)
		tmp = Float64(fma(b, 0.3333333333333333, Float64(Float64(-0.3333333333333333) * sqrt(fma(Float64(c * a), -3.0, Float64(b * b))))) * Float64(-1.0 / a));
	else
		tmp = fma(Float64(-0.375 * a), Float64(Float64(Float64(c * c) / Float64(b * b)) / b), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.084], N[(N[(b * 0.3333333333333333 + N[((-0.3333333333333333) * N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * a), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0840000000000000052

   1. Initial program 75.3%

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

   4. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{-1}{a} \cdot \left(\left(b - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right) \cdot 0.3333333333333333\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 76.0%

      \[\leadsto \frac{-1}{a} \cdot \color{blue}{\mathsf{fma}\left(b, 0.3333333333333333, \left(-\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}\right) \cdot 0.3333333333333333\right)} \]

   if 0.0840000000000000052 < b

   1. Initial program 24.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 93.5

          \[\leadsto \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{{b}^{3}}, \color{blue}{\frac{c}{b}} \cdot -0.5\right) \]

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{{b}^{3}}, \frac{c}{b} \cdot -0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.084:\\ \;\;\;\;\mathsf{fma}\left(b, 0.3333333333333333, \left(-0.3333333333333333\right) \cdot \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}\right) \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot a, \frac{\frac{c \cdot c}{b \cdot b}}{b}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.084:\\ \;\;\;\;\mathsf{fma}\left(b, 0.3333333333333333, \left(-0.3333333333333333\right) \cdot \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}\right) \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot -0.375, \frac{a}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.084)
   (*
    (fma
     b
     0.3333333333333333
     (* (- 0.3333333333333333) (sqrt (fma (* c a) -3.0 (* b b)))))
    (/ -1.0 a))
   (/ (fma (* (* c c) -0.375) (/ a (* b b)) (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.084) {
		tmp = fma(b, 0.3333333333333333, (-0.3333333333333333 * sqrt(fma((c * a), -3.0, (b * b))))) * (-1.0 / a);
	} else {
		tmp = fma(((c * c) * -0.375), (a / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.084)
		tmp = Float64(fma(b, 0.3333333333333333, Float64(Float64(-0.3333333333333333) * sqrt(fma(Float64(c * a), -3.0, Float64(b * b))))) * Float64(-1.0 / a));
	else
		tmp = Float64(fma(Float64(Float64(c * c) * -0.375), Float64(a / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.084], N[(N[(b * 0.3333333333333333 + N[((-0.3333333333333333) * N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0840000000000000052

   1. Initial program 75.3%

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

   4. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{-1}{a} \cdot \left(\left(b - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right) \cdot 0.3333333333333333\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 76.0%

      \[\leadsto \frac{-1}{a} \cdot \color{blue}{\mathsf{fma}\left(b, 0.3333333333333333, \left(-\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}\right) \cdot 0.3333333333333333\right)} \]

   if 0.0840000000000000052 < b

   1. Initial program 24.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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-*r/ N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b}, \frac{a}{b}, -0.5 \cdot c\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.084:\\ \;\;\;\;\mathsf{fma}\left(b, 0.3333333333333333, \left(-0.3333333333333333\right) \cdot \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}\right) \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot -0.375, \frac{a}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.084:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot -0.375, \frac{a}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.084)
   (/ (- (sqrt (fma b b (* (* -3.0 a) c))) b) (* 3.0 a))
   (/ (fma (* (* c c) -0.375) (/ a (* b b)) (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.084) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = fma(((c * c) * -0.375), (a / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.084)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(c * c) * -0.375), Float64(a / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.084], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0840000000000000052

   1. Initial program 75.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 75.4

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

   4. Applied rewrites 75.4%

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

   if 0.0840000000000000052 < b

   1. Initial program 24.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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-*r/ N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b}, \frac{a}{b}, -0.5 \cdot c\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.084:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot -0.375, \frac{a}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.084:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{c \cdot a}{b \cdot b}, -0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.084)
   (/ (- (sqrt (fma b b (* (* -3.0 a) c))) b) (* 3.0 a))
   (/ (* (fma -0.375 (/ (* c a) (* b b)) -0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.084) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (fma(-0.375, ((c * a) / (b * b)), -0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.084)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(fma(-0.375, Float64(Float64(c * a) / Float64(b * b)), -0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.084], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0840000000000000052

   1. Initial program 75.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 75.4

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

   4. Applied rewrites 75.4%

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

   if 0.0840000000000000052 < b

   1. Initial program 24.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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-*r/ N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   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 93.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.084:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{c \cdot a}{b \cdot b}, -0.5\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.084:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{c \cdot a}{b \cdot b}, -0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.084)
   (* (- (sqrt (fma (* c -3.0) a (* b b))) b) (/ 0.3333333333333333 a))
   (/ (* (fma -0.375 (/ (* c a) (* b b)) -0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.084) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (fma(-0.375, ((c * a) / (b * b)), -0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.084)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(fma(-0.375, Float64(Float64(c * a) / Float64(b * b)), -0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.084], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0840000000000000052

   1. Initial program 75.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 75.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 75.3

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

   4. Applied rewrites 75.3%

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

   if 0.0840000000000000052 < b

   1. Initial program 24.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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-*r/ N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   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 93.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.084:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{c \cdot a}{b \cdot b}, -0.5\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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-*r/ N/A

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

   4. unpow2 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 89.3

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

5. Applied rewrites 89.3%

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

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 89.2%

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

9. Final simplification 89.2%

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

Alternative 8: 81.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 80.9

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

5. Applied rewrites 80.9%

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

6. Final simplification 80.9%

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

Alternative 9: 81.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) / b) * c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 80.9

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

5. Applied rewrites 80.9%

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

7. Applied rewrites 80.7%

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

8. Final simplification 80.7%

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

Reproduce

herbie shell --seed 2024247 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))