Cubic critical, wide range

Percentage Accurate: 17.8% → 97.5%

Time: 9.0s

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: 17.8% 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: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (* (* a a) -0.5625)
   (* (/ (* c c) (* b b)) (/ c (* b b)))
   (fma
    (/ -0.16666666666666666 (pow b 6.0))
    (* (* (pow a 4.0) (pow c 4.0)) (/ 6.328125 a))
    (fma (/ -0.375 b) (/ (* (* c c) a) b) (* -0.5 c))))
  b))

C

double code(double a, double b, double c) {
	return fma(((a * a) * -0.5625), (((c * c) / (b * b)) * (c / (b * b))), fma((-0.16666666666666666 / pow(b, 6.0)), ((pow(a, 4.0) * pow(c, 4.0)) * (6.328125 / a)), fma((-0.375 / b), (((c * c) * a) / b), (-0.5 * c)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(a * a) * -0.5625), Float64(Float64(Float64(c * c) / Float64(b * b)) * Float64(c / Float64(b * b))), fma(Float64(-0.16666666666666666 / (b ^ 6.0)), Float64(Float64((a ^ 4.0) * (c ^ 4.0)) * Float64(6.328125 / a)), fma(Float64(-0.375 / b), Float64(Float64(Float64(c * c) * a) / b), Float64(-0.5 * c)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 / b), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 95.7%

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

8. Final simplification 95.7%

   \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b} \]
9. Add Preprocessing

Alternative 2: 97.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 95.7%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 95.7%

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

11. Taylor expanded in c around 0

    \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{-9}{16}, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left({c}^{2} \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{6}} - \frac{3}{8} \cdot \frac{1}{{b}^{2}}\right), a, \frac{-1}{2} \cdot c\right)\right)}{b} \]
12. Step-by-step derivation

13. Applied rewrites 95.7%

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

14. Final simplification 95.7%

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

Alternative 3: 96.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 95.7%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 95.7%

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

11. Taylor expanded in a around 0

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

13. Applied rewrites 94.7%

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

14. Final simplification 94.7%

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

Alternative 4: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 95.7%

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

8. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
9. 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 92.5

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

10. Applied rewrites 92.5%

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

12. Applied rewrites 92.5%

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

13. Final simplification 92.5%

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

Alternative 5: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((((a * (c / (b * b))) * -0.375) - 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 = ((((a * (c / (b * b))) * (-0.375d0)) - 0.5d0) * c) / b
end function

Java

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

Python

def code(a, b, c):
	return ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b;
end

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 95.7%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 95.7%

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

11. Taylor expanded in c around 0

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

13. Applied rewrites 92.5%

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

14. Final simplification 92.5%

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

Alternative 6: 90.4% 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 23.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. *-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 86.6

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

5. Applied rewrites 86.6%

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

6. Final simplification 86.6%

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

Alternative 7: 90.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. *-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 86.6

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

5. Applied rewrites 86.6%

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

7. Applied rewrites 86.4%

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

8. Final simplification 86.4%

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

Alternative 8: 3.3% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in c around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 23.0

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

5. Applied rewrites 23.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 22.9

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 22.9

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

7. Applied rewrites 22.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. div-add N/A

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

   5. div-add N/A

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

   6. associate-/r* N/A

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

   7. *-commutative N/A

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

   8. lower-+.f64 N/A

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

9. Applied rewrites 23.0%

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

10. Taylor expanded in a around 0

    \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot b + \frac{1}{3} \cdot b}{a}} \]
11. Step-by-step derivation

    1. div-add N/A

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

    2. associate-*r/ N/A

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

    3. associate-*r/ N/A

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

    4. fp-cancel-sign-sub-inv N/A

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

    5. metadata-eval N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{b}{a} - \color{blue}{\frac{-1}{3}} \cdot \frac{b}{a} \]

    6. +-inverses 3.3

       \[\leadsto \color{blue}{0} \]

12. Applied rewrites 3.3%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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)))