Cubic critical, medium range

Percentage Accurate: 31.6% → 95.4%

Time: 10.0s

Alternatives: 13

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Initial Program: 31.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Alternative 1: 95.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot -0.16666666666666666, \frac{6.328125}{b} \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{-0.5625 \cdot {c}^{3}}{{b}^{5}}\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{{b}^{3}}\right), a, -0.5 \cdot \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (fma
   (fma
    (* a -0.16666666666666666)
    (* (/ 6.328125 b) (/ (pow c 4.0) (pow b 6.0)))
    (/ (* -0.5625 (pow c 3.0)) (pow b 5.0)))
   a
   (/ (* -0.375 (* c c)) (pow b 3.0)))
  a
  (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	return fma(fma(fma((a * -0.16666666666666666), ((6.328125 / b) * (pow(c, 4.0) / pow(b, 6.0))), ((-0.5625 * pow(c, 3.0)) / pow(b, 5.0))), a, ((-0.375 * (c * c)) / pow(b, 3.0))), a, (-0.5 * (c / b)));
}

Julia

function code(a, b, c)
	return fma(fma(fma(Float64(a * -0.16666666666666666), Float64(Float64(6.328125 / b) * Float64((c ^ 4.0) / (b ^ 6.0))), Float64(Float64(-0.5625 * (c ^ 3.0)) / (b ^ 5.0))), a, Float64(Float64(-0.375 * Float64(c * c)) / (b ^ 3.0))), a, Float64(-0.5 * Float64(c / b)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(a * -0.16666666666666666), $MachinePrecision] * N[(N[(6.328125 / b), $MachinePrecision] * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5625 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Final simplification 96.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot -0.16666666666666666, \frac{6.328125}{b} \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{-0.5625 \cdot {c}^{3}}{{b}^{5}}\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{{b}^{3}}\right), a, -0.5 \cdot \frac{c}{b}\right) \]
7. Add Preprocessing

Alternative 2: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{c}{b}, -0.5, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(c \cdot a, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot c\right) \cdot c, b \cdot b, \left(a \cdot a\right) \cdot \left(-1.0546875 \cdot {c}^{4}\right)\right) \cdot {b}^{-7}\right) \cdot a\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/ c b)
  -0.5
  (*
   (*
    (fma
     (* (* (fma (* c a) -0.5625 (* (* b b) -0.375)) c) c)
     (* b b)
     (* (* a a) (* -1.0546875 (pow c 4.0))))
    (pow b -7.0))
   a)))

C

double code(double a, double b, double c) {
	return fma((c / b), -0.5, ((fma(((fma((c * a), -0.5625, ((b * b) * -0.375)) * c) * c), (b * b), ((a * a) * (-1.0546875 * pow(c, 4.0)))) * pow(b, -7.0)) * a));
}

Julia

function code(a, b, c)
	return fma(Float64(c / b), -0.5, Float64(Float64(fma(Float64(Float64(fma(Float64(c * a), -0.5625, Float64(Float64(b * b) * -0.375)) * c) * c), Float64(b * b), Float64(Float64(a * a) * Float64(-1.0546875 * (c ^ 4.0)))) * (b ^ -7.0)) * a))
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * -0.5 + N[(N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * -0.5625 + N[(N[(b * b), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(-1.0546875 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[b, -7.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
7. Step-by-step derivation

8. Applied rewrites 96.6%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128} \cdot \left(a \cdot a\right), {c}^{4}, \left({c}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
10. Step-by-step derivation

11. Applied rewrites 96.6%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \left(\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot c, -0.375 \cdot \left(b \cdot b\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

13. Applied rewrites 96.6%

    \[\leadsto \mathsf{fma}\left(\frac{c}{b}, \color{blue}{-0.5}, \left({b}^{-7} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(c \cdot a, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot c\right) \cdot c, b \cdot b, \left({c}^{4} \cdot -1.0546875\right) \cdot \left(a \cdot a\right)\right)\right) \cdot a\right) \]

14. Final simplification 96.6%

    \[\leadsto \mathsf{fma}\left(\frac{c}{b}, -0.5, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(c \cdot a, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot c\right) \cdot c, b \cdot b, \left(a \cdot a\right) \cdot \left(-1.0546875 \cdot {c}^{4}\right)\right) \cdot {b}^{-7}\right) \cdot a\right) \]
15. Add Preprocessing

Alternative 3: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-0.5}{b}, c, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(c \cdot a, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot c\right) \cdot c, b \cdot b, \left(a \cdot a\right) \cdot \left(-1.0546875 \cdot {c}^{4}\right)\right) \cdot {b}^{-7}\right) \cdot a\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/ -0.5 b)
  c
  (*
   (*
    (fma
     (* (* (fma (* c a) -0.5625 (* (* b b) -0.375)) c) c)
     (* b b)
     (* (* a a) (* -1.0546875 (pow c 4.0))))
    (pow b -7.0))
   a)))

C

double code(double a, double b, double c) {
	return fma((-0.5 / b), c, ((fma(((fma((c * a), -0.5625, ((b * b) * -0.375)) * c) * c), (b * b), ((a * a) * (-1.0546875 * pow(c, 4.0)))) * pow(b, -7.0)) * a));
}

Julia

function code(a, b, c)
	return fma(Float64(-0.5 / b), c, Float64(Float64(fma(Float64(Float64(fma(Float64(c * a), -0.5625, Float64(Float64(b * b) * -0.375)) * c) * c), Float64(b * b), Float64(Float64(a * a) * Float64(-1.0546875 * (c ^ 4.0)))) * (b ^ -7.0)) * a))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 / b), $MachinePrecision] * c + N[(N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * -0.5625 + N[(N[(b * b), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(-1.0546875 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[b, -7.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
7. Step-by-step derivation

8. Applied rewrites 96.6%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128} \cdot \left(a \cdot a\right), {c}^{4}, \left({c}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
10. Step-by-step derivation

11. Applied rewrites 96.6%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \left(\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot c, -0.375 \cdot \left(b \cdot b\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

13. Applied rewrites 96.4%

    \[\leadsto \mathsf{fma}\left(\frac{-0.5}{b}, \color{blue}{c}, \left({b}^{-7} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(c \cdot a, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot c\right) \cdot c, b \cdot b, \left({c}^{4} \cdot -1.0546875\right) \cdot \left(a \cdot a\right)\right)\right) \cdot a\right) \]

14. Final simplification 96.4%

    \[\leadsto \mathsf{fma}\left(\frac{-0.5}{b}, c, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(c \cdot a, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot c\right) \cdot c, b \cdot b, \left(a \cdot a\right) \cdot \left(-1.0546875 \cdot {c}^{4}\right)\right) \cdot {b}^{-7}\right) \cdot a\right) \]
15. Add Preprocessing

Alternative 4: 93.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{5}} \cdot a, -0.5625, \frac{-0.375}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, -0.5 \cdot \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (* (fma (* (/ c (pow b 5.0)) a) -0.5625 (/ -0.375 (pow b 3.0))) (* c c))
  a
  (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	return fma((fma(((c / pow(b, 5.0)) * a), -0.5625, (-0.375 / pow(b, 3.0))) * (c * c)), a, (-0.5 * (c / b)));
}

Julia

function code(a, b, c)
	return fma(Float64(fma(Float64(Float64(c / (b ^ 5.0)) * a), -0.5625, Float64(-0.375 / (b ^ 3.0))) * Float64(c * c)), a, Float64(-0.5 * Float64(c / b)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -0.5625 + N[(-0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{5}} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right), a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
7. Step-by-step derivation

8. Applied rewrites 94.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{5}}, -0.5625, \frac{-0.375}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right) \]

9. Final simplification 94.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{5}} \cdot a, -0.5625, \frac{-0.375}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, -0.5 \cdot \frac{c}{b}\right) \]
10. Add Preprocessing

Alternative 5: 93.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 94.9%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 94.9%

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

9. Final simplification 94.9%

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

Alternative 6: 93.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.375 \cdot \left(a \cdot a\right)}{{b}^{3}}, c, 0.5 \cdot \frac{a}{b}\right), c, -0.6666666666666666 \cdot b\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.3333333333333333
  (/
   (fma
    (fma (/ (* 0.375 (* a a)) (pow b 3.0)) c (* 0.5 (/ a b)))
    c
    (* -0.6666666666666666 b))
   c)))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / (fma(fma(((0.375 * (a * a)) / pow(b, 3.0)), c, (0.5 * (a / b))), c, (-0.6666666666666666 * b)) / c);
}

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / Float64(fma(fma(Float64(Float64(0.375 * Float64(a * a)) / (b ^ 3.0)), c, Float64(0.5 * Float64(a / b))), c, Float64(-0.6666666666666666 * b)) / c))
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(N[(N[(N[(N[(0.375 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * c + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.5%

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

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 28.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 28.5

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

4. Applied rewrites 28.5%

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

5. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 94.8%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.375 \cdot \left(a \cdot a\right)}{{b}^{3}}, c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}}} \]

8. Final simplification 94.8%

   \[\leadsto \frac{0.3333333333333333}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.375 \cdot \left(a \cdot a\right)}{{b}^{3}}, c, 0.5 \cdot \frac{a}{b}\right), c, -0.6666666666666666 \cdot b\right)}{c}} \]
9. Add Preprocessing

Alternative 7: 93.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{3}} \cdot 0.375, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.3333333333333333
  (fma
   (fma (* (/ c (pow b 3.0)) 0.375) a (/ 0.5 b))
   a
   (* (/ b c) -0.6666666666666666))))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / fma(fma(((c / pow(b, 3.0)) * 0.375), a, (0.5 / b)), a, ((b / c) * -0.6666666666666666));
}

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / fma(fma(Float64(Float64(c / (b ^ 3.0)) * 0.375), a, Float64(0.5 / b)), a, Float64(Float64(b / c) * -0.6666666666666666)))
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(N[(N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision] * a + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.5%

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

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 28.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 28.5

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

4. Applied rewrites 28.5%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 94.8%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{3}} \cdot 0.375, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}} \]
8. Add Preprocessing

Alternative 8: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.375 \cdot a, \frac{c}{{b}^{3}} \cdot c, -0.5 \cdot \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma (* -0.375 a) (* (/ c (pow b 3.0)) c) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	return fma((-0.375 * a), ((c / pow(b, 3.0)) * c), (-0.5 * (c / b)));
}

Julia

function code(a, b, c)
	return fma(Float64(-0.375 * a), Float64(Float64(c / (b ^ 3.0)) * c), Float64(-0.5 * Float64(c / b)))
end

Wolfram

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

Derivation

1. Initial program 28.5%

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

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 92.0

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

5. Applied rewrites 92.0%

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

6. Final simplification 92.0%

   \[\leadsto \mathsf{fma}\left(-0.375 \cdot a, \frac{c}{{b}^{3}} \cdot c, -0.5 \cdot \frac{c}{b}\right) \]
7. Add Preprocessing

Alternative 9: 90.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. unpow2 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 92.0

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

5. Applied rewrites 92.0%

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

6. Final simplification 92.0%

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

Alternative 10: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
7. Step-by-step derivation

8. Applied rewrites 96.6%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128} \cdot \left(a \cdot a\right), {c}^{4}, \left({c}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
10. Step-by-step derivation

11. Applied rewrites 96.6%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \left(\left(c \cdot c\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot c, -0.375 \cdot \left(b \cdot b\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

12. Taylor expanded in b around inf

    \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
13. 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. lower-fma.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 92.0

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

14. Applied rewrites 92.0%

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

15. Final simplification 92.0%

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

Alternative 11: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* 0.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / fma(-0.6666666666666666, (b / c), (0.5 * (a / b)));
}

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(0.5 * Float64(a / b))))
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.5%

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

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 28.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 28.5

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

4. Applied rewrites 28.5%

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

5. Taylor expanded in c around 0

   \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\frac{\frac{-2}{3} \cdot b + \frac{1}{2} \cdot \frac{a \cdot c}{b}}{c}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 91.9

      \[\leadsto \frac{0.3333333333333333}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 0.5, \color{blue}{-0.6666666666666666 \cdot b}\right)}{c}} \]

7. Applied rewrites 91.9%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 0.5, -0.6666666666666666 \cdot b\right)}{c}}} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\frac{-2}{3} \cdot \frac{b}{c} + \frac{1}{2} \cdot \frac{a}{b}}} \]
9. Step-by-step derivation

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 91.9

      \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \color{blue}{\frac{a}{b}}\right)} \]

10. Applied rewrites 91.9%

    \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right)}} \]
11. Add Preprocessing

Alternative 12: 81.1% 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 28.5%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 83.6

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

5. Applied rewrites 83.6%

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

6. Final simplification 83.6%

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

Alternative 13: 80.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.5%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 83.6

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

5. Applied rewrites 83.6%

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

7. Applied rewrites 83.4%

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

8. Final simplification 83.4%

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

Reproduce

herbie shell --seed 2024288 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))