Cubic critical, medium range

Percentage Accurate: 31.8% → 95.3%

Time: 12.8s

Alternatives: 10

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: 31.8% 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: 95.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.375, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \mathsf{fma}\left(0.5, c, \mathsf{fma}\left(1.0546875, \frac{{c}^{4}}{{b}^{6}} \cdot \left(\left(a \cdot a\right) \cdot a\right), \frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{4}} \cdot 0.5625\right)\right)\right)}{-b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   0.375
   (/ (* (* c c) a) (* b b))
   (fma
    0.5
    c
    (fma
     1.0546875
     (* (/ (pow c 4.0) (pow b 6.0)) (* (* a a) a))
     (* (/ (* (* (* c c) c) (* a a)) (pow b 4.0)) 0.5625))))
  (- b)))

C

double code(double a, double b, double c) {
	return fma(0.375, (((c * c) * a) / (b * b)), fma(0.5, c, fma(1.0546875, ((pow(c, 4.0) / pow(b, 6.0)) * ((a * a) * a)), (((((c * c) * c) * (a * a)) / pow(b, 4.0)) * 0.5625)))) / -b;
}

Julia

function code(a, b, c)
	return Float64(fma(0.375, Float64(Float64(Float64(c * c) * a) / Float64(b * b)), fma(0.5, c, fma(1.0546875, Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * Float64(Float64(a * a) * a)), Float64(Float64(Float64(Float64(Float64(c * c) * c) * Float64(a * a)) / (b ^ 4.0)) * 0.5625)))) / Float64(-b))
end

Wolfram

code[a_, b_, c_] := N[(N[(0.375 * N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.5 * c + N[(1.0546875 * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * 0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]

Derivation

1. Initial program 30.3%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 29.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. sub-div N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 29.9%

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

7. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{c \cdot \left(\frac{81}{64} \cdot \frac{{a}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

9. Applied rewrites 95.1%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.16666666666666666, c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}, \frac{-0.5625 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), -0.375 \cdot \frac{a}{\left(b \cdot b\right) \cdot b}\right), \frac{-0.5}{b}\right)} \]

10. Taylor expanded in b around -inf

    \[\leadsto -1 \cdot \color{blue}{\frac{\frac{3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(\frac{1}{2} \cdot c + \left(\frac{9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}}\right)\right)}{b}} \]
11. Step-by-step derivation

12. Applied rewrites 95.3%

    \[\leadsto -\frac{\mathsf{fma}\left(0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(0.5, c, \mathsf{fma}\left(1.0546875, \left(\left(a \cdot a\right) \cdot a\right) \cdot \frac{{c}^{4}}{{b}^{6}}, 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}\right)\right)\right)}{b} \]

13. Final simplification 95.3%

    \[\leadsto \frac{\mathsf{fma}\left(0.375, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \mathsf{fma}\left(0.5, c, \mathsf{fma}\left(1.0546875, \frac{{c}^{4}}{{b}^{6}} \cdot \left(\left(a \cdot a\right) \cdot a\right), \frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{4}} \cdot 0.5625\right)\right)\right)}{-b} \]
14. Add Preprocessing

Alternative 2: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{t\_0 \cdot \left(b \cdot b\right)}, \left(-0.5625 \cdot c\right) \cdot c, \left(\frac{6.328125}{\left(t\_0 \cdot b\right) \cdot t\_0} \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot \left(-0.16666666666666666 \cdot a\right)\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{t\_0}\right), a, \frac{c}{b} \cdot -0.5\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)))
   (fma
    (fma
     (fma
      (/ c (* t_0 (* b b)))
      (* (* -0.5625 c) c)
      (*
       (* (/ 6.328125 (* (* t_0 b) t_0)) (* (* (* c c) c) c))
       (* -0.16666666666666666 a)))
     a
     (/ (* (* c c) -0.375) t_0))
    a
    (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	return fma(fma(fma((c / (t_0 * (b * b))), ((-0.5625 * c) * c), (((6.328125 / ((t_0 * b) * t_0)) * (((c * c) * c) * c)) * (-0.16666666666666666 * a))), a, (((c * c) * -0.375) / t_0)), a, ((c / b) * -0.5));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	return fma(fma(fma(Float64(c / Float64(t_0 * Float64(b * b))), Float64(Float64(-0.5625 * c) * c), Float64(Float64(Float64(6.328125 / Float64(Float64(t_0 * b) * t_0)) * Float64(Float64(Float64(c * c) * c) * c)) * Float64(-0.16666666666666666 * a))), a, Float64(Float64(Float64(c * c) * -0.375) / t_0)), a, Float64(Float64(c / b) * -0.5))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(N[(N[(c / N[(t$95$0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5625 * c), $MachinePrecision] * c), $MachinePrecision] + N[(N[(N[(6.328125 / N[(N[(t$95$0 * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, -0.5625 \cdot \left(\left(c \cdot c\right) \cdot \frac{c}{{b}^{5}}\right)\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

7. Applied rewrites 95.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)}, \left(-0.5625 \cdot c\right) \cdot c, \left(a \cdot -0.16666666666666666\right) \cdot \left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot c\right) \cdot \frac{6.328125}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right)}\right)\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right), a, \frac{c}{b} \cdot -0.5\right) \]

8. Final simplification 95.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)}, \left(-0.5625 \cdot c\right) \cdot c, \left(\frac{6.328125}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot \left(-0.16666666666666666 \cdot a\right)\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{\left(b \cdot b\right) \cdot b}\right), a, \frac{c}{b} \cdot -0.5\right) \]
9. Add Preprocessing

Alternative 3: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot 6.328125\right) \cdot \left(a \cdot a\right)}{\left(\left(t\_0 \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot a\right)}, -0.16666666666666666 \cdot c, \frac{\left(a \cdot a\right) \cdot -0.5625}{t\_0 \cdot \left(b \cdot b\right)}\right), \frac{a}{t\_0} \cdot -0.375\right), \frac{-0.5}{b}\right) \cdot c \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)))
   (*
    (fma
     c
     (fma
      c
      (fma
       (/ (* (* (* a a) 6.328125) (* a a)) (* (* (* t_0 b) (* b b)) (* b a)))
       (* -0.16666666666666666 c)
       (/ (* (* a a) -0.5625) (* t_0 (* b b))))
      (* (/ a t_0) -0.375))
     (/ -0.5 b))
    c)))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	return fma(c, fma(c, fma(((((a * a) * 6.328125) * (a * a)) / (((t_0 * b) * (b * b)) * (b * a))), (-0.16666666666666666 * c), (((a * a) * -0.5625) / (t_0 * (b * b)))), ((a / t_0) * -0.375)), (-0.5 / b)) * c;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	return Float64(fma(c, fma(c, fma(Float64(Float64(Float64(Float64(a * a) * 6.328125) * Float64(a * a)) / Float64(Float64(Float64(t_0 * b) * Float64(b * b)) * Float64(b * a))), Float64(-0.16666666666666666 * c), Float64(Float64(Float64(a * a) * -0.5625) / Float64(t_0 * Float64(b * b)))), Float64(Float64(a / t_0) * -0.375)), Float64(-0.5 / b)) * c)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(c * N[(c * N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 6.328125), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$0 * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * c), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] / N[(t$95$0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a / t$95$0), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Initial program 30.3%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 29.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. sub-div N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 29.9%

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

7. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{c \cdot \left(\frac{81}{64} \cdot \frac{{a}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

9. Applied rewrites 95.1%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.16666666666666666, c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}, \frac{-0.5625 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), -0.375 \cdot \frac{a}{\left(b \cdot b\right) \cdot b}\right), \frac{-0.5}{b}\right)} \]
10. Step-by-step derivation

11. Applied rewrites 95.1%

    \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot 6.328125\right)}{\left(b \cdot a\right) \cdot \left(\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)\right)}, \color{blue}{c \cdot -0.16666666666666666}, \frac{\left(a \cdot a\right) \cdot -0.5625}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)}\right), -0.375 \cdot \frac{a}{\left(b \cdot b\right) \cdot b}\right), \frac{-0.5}{b}\right) \]

12. Final simplification 95.1%

    \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot 6.328125\right) \cdot \left(a \cdot a\right)}{\left(\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot a\right)}, -0.16666666666666666 \cdot c, \frac{\left(a \cdot a\right) \cdot -0.5625}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)}\right), \frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375\right), \frac{-0.5}{b}\right) \cdot c \]
13. Add Preprocessing

Alternative 4: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.375 \cdot c, c, \frac{\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot a\right) \cdot -0.5625}{b \cdot b}\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{c}{b} \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/
   (fma (* -0.375 c) c (/ (* (* (* (* c c) c) a) -0.5625) (* b b)))
   (* (* b b) b))
  a
  (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	return fma((fma((-0.375 * c), c, (((((c * c) * c) * a) * -0.5625) / (b * b))) / ((b * b) * b)), a, ((c / b) * -0.5));
}

Julia

function code(a, b, c)
	return fma(Float64(fma(Float64(-0.375 * c), c, Float64(Float64(Float64(Float64(Float64(c * c) * c) * a) * -0.5625) / Float64(b * b))) / Float64(Float64(b * b) * b)), a, Float64(Float64(c / b) * -0.5))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-0.375 * c), $MachinePrecision] * c + N[(N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * -0.5625), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, -0.5625 \cdot \left(\left(c \cdot c\right) \cdot \frac{c}{{b}^{5}}\right)\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Taylor expanded in b around inf

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

8. Applied rewrites 93.8%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.375 \cdot c, c, \frac{\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot a\right) \cdot -0.5625}{b \cdot b}\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{c}{b} \cdot -0.5\right) \]
9. Add Preprocessing

Alternative 5: 93.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot a\right) \cdot a, -0.5625, -0.375 \cdot a\right)}{\left(b \cdot b\right) \cdot b}, c, \frac{-0.5}{b}\right) \cdot c \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (fma
   (/ (fma (* (* (/ c (* b b)) a) a) -0.5625 (* -0.375 a)) (* (* b b) b))
   c
   (/ -0.5 b))
  c))

C

double code(double a, double b, double c) {
	return fma((fma((((c / (b * b)) * a) * a), -0.5625, (-0.375 * a)) / ((b * b) * b)), c, (-0.5 / b)) * c;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(fma(Float64(Float64(Float64(c / Float64(b * b)) * a) * a), -0.5625, Float64(-0.375 * a)) / Float64(Float64(b * b) * b)), c, Float64(-0.5 / b)) * c)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -0.5625 + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * c + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

5. Applied rewrites 93.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]

6. Taylor expanded in b around inf

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

8. Applied rewrites 93.5%

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

9. Final simplification 93.5%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot a\right) \cdot a, -0.5625, -0.375 \cdot a\right)}{\left(b \cdot b\right) \cdot b}, c, \frac{-0.5}{b}\right) \cdot c \]
10. Add Preprocessing

Alternative 6: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 91.0

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

5. Applied rewrites 91.0%

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

6. Final simplification 91.0%

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

Alternative 7: 90.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 91.0

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

5. Applied rewrites 91.0%

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

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 91.0%

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

9. Final simplification 91.0%

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

Alternative 8: 90.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

5. Applied rewrites 93.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]

6. Taylor expanded in b around inf

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

8. Applied rewrites 90.7%

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

9. Final simplification 90.7%

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

Alternative 9: 81.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
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 82.2

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

5. Applied rewrites 82.2%

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

Alternative 10: 80.8% 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 30.3%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

5. Applied rewrites 93.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 82.0%

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

Reproduce

herbie shell --seed 2024235 
(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)))