Cubic critical, wide range

Percentage Accurate: 17.5% → 97.3%

Time: 12.2s

Alternatives: 8

Speedup: 2.9×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.5% 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: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}}, \frac{\left(c \cdot -0.16666666666666666\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot 6.328125\right)\right)}{\left(a \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot t\_0\right)\right)}\right), \frac{a \cdot -0.375}{t\_0}\right), \frac{-0.5}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (*
    c
    (fma
     c
     (fma
      c
      (fma
       -0.5625
       (/ (* a a) (pow b 5.0))
       (/
        (* (* c -0.16666666666666666) (* (* a a) (* (* a a) 6.328125)))
        (* (* a b) (* (* b b) (* b t_0)))))
      (/ (* a -0.375) t_0))
     (/ -0.5 b)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return c * fma(c, fma(c, fma(-0.5625, ((a * a) / pow(b, 5.0)), (((c * -0.16666666666666666) * ((a * a) * ((a * a) * 6.328125))) / ((a * b) * ((b * b) * (b * t_0))))), ((a * -0.375) / t_0)), (-0.5 / b));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(c * fma(c, fma(c, fma(-0.5625, Float64(Float64(a * a) / (b ^ 5.0)), Float64(Float64(Float64(c * -0.16666666666666666) * Float64(Float64(a * a) * Float64(Float64(a * a) * 6.328125))) / Float64(Float64(a * b) * Float64(Float64(b * b) * Float64(b * t_0))))), Float64(Float64(a * -0.375) / t_0)), Float64(-0.5 / b)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(c * N[(c * N[(c * N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * -0.16666666666666666), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around 0

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

5. Simplified 97.9%

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

7. Applied egg-rr 97.9%

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

8. Final simplification 97.9%

   \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}}, \frac{\left(c \cdot -0.16666666666666666\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot 6.328125\right)\right)}{\left(a \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right) \]
9. Add Preprocessing

Alternative 2: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in b around inf

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

5. Simplified 97.6%

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

Alternative 3: 97.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 18.1

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

4. Applied egg-rr 18.1%

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

5. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

7. Simplified 97.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-/.f64 97.6

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

   7. lift-pow.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-prod-up N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 97.6

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

9. Applied egg-rr 97.6%

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

Alternative 4: 95.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. cube-mult N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. associate-*r/ N/A

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

   19. lower-/.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 95.8

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

5. Simplified 95.8%

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

6. Final simplification 95.8%

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

Alternative 5: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

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

5. Simplified 95.8%

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

Alternative 6: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. distribute-rgt-in N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r* N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

5. Simplified 95.5%

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

6. Final simplification 95.5%

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

Alternative 7: 90.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 90.3

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

5. Simplified 90.3%

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

Alternative 8: 90.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 90.3

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

5. Simplified 90.3%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 90.0

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

7. Applied egg-rr 90.0%

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

8. Final simplification 90.0%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))