Cubic critical, wide range

Percentage Accurate: 17.9% → 99.3%

Time: 10.2s

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: 17.9% 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: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}{a}}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}}{3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (/ (fma (* -3.0 c) a 0.0) a) (+ (sqrt (fma (* -3.0 c) a (* b b))) b))
  3.0))

C

double code(double a, double b, double c) {
	return ((fma((-3.0 * c), a, 0.0) / a) / (sqrt(fma((-3.0 * c), a, (b * b))) + b)) / 3.0;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(fma(Float64(-3.0 * c), a, 0.0) / a) / Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) + b)) / 3.0)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(-3.0 * c), $MachinePrecision] * a + 0.0), $MachinePrecision] / a), $MachinePrecision] / N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 19.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-+.f64 20.1

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

6. Applied rewrites 20.1%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 20.1

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

   7. lift--.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. associate--l+ N/A

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

   10. +-inverses N/A

       \[\leadsto \frac{\frac{\frac{\left(-3 \cdot c\right) \cdot a + \color{blue}{0}}{a}}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}}{3} \]

   11. lower-fma.f64 99.3

       \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}}{a}}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}}{3} \]

8. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}{a}}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}}}{3} \]
9. Add Preprocessing

Alternative 2: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 19.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-+.f64 20.1

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

6. Applied rewrites 20.1%

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

7. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 99.3

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

9. Applied rewrites 99.3%

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

Alternative 3: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}{3 \cdot \left(\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma (* -3.0 c) a 0.0)
  (* 3.0 (* (+ (sqrt (fma (* -3.0 c) a (* b b))) b) a))))

C

double code(double a, double b, double c) {
	return fma((-3.0 * c), a, 0.0) / (3.0 * ((sqrt(fma((-3.0 * c), a, (b * b))) + b) * a));
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(-3.0 * c), a, 0.0) / Float64(3.0 * Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) + b) * a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-3.0 * c), $MachinePrecision] * a + 0.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 19.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-+.f64 20.1

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

6. Applied rewrites 20.1%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. associate--l+ N/A

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

   10. +-inverses N/A

       \[\leadsto \frac{\left(-3 \cdot c\right) \cdot a + \color{blue}{0}}{3 \cdot \left(a \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}}{3 \cdot \left(a \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}{\color{blue}{3 \cdot \left(a \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)\right)}} \]

8. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}{3 \cdot \left(\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a\right)}} \]
9. Add Preprocessing

Alternative 4: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right) \cdot 0.3333333333333333}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (fma (* -3.0 c) a 0.0) 0.3333333333333333)
  (* (+ (sqrt (fma (* -3.0 c) a (* b b))) b) a)))

C

double code(double a, double b, double c) {
	return (fma((-3.0 * c), a, 0.0) * 0.3333333333333333) / ((sqrt(fma((-3.0 * c), a, (b * b))) + b) * a);
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(-3.0 * c), a, 0.0) * 0.3333333333333333) / Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) + b) * a))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-3.0 * c), $MachinePrecision] * a + 0.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 19.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-+.f64 20.1

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

6. Applied rewrites 20.1%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

8. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right) \cdot 0.3333333333333333}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a}} \]
9. Add Preprocessing

Alternative 5: 95.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := 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. Initial program 19.5%

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

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

   12. lower-/.f64 95.2

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

5. Applied rewrites 95.2%

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

7. Applied rewrites 95.2%

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

Alternative 6: 95.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

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

   5. unpow2 N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 95.2

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

5. Applied rewrites 95.2%

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

Alternative 7: 95.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, \mathsf{fma}\left(\frac{\left({c}^{3} \cdot a\right) \cdot a}{{b}^{4}}, -0.5625, -0.5 \cdot c\right)\right)}{b}} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 95.2%

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

Alternative 8: 90.4% 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 19.5%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 89.2

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

5. Applied rewrites 89.2%

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

Reproduce

herbie shell --seed 2024321 
(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)))