Cubic critical, medium range

Percentage Accurate: 31.8% → 95.4%

Time: 13.5s

Alternatives: 15

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% 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 29.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 98.0%

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

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 98.0%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 98.0%

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

13. Applied rewrites 98.0%

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

14. Final simplification 98.0%

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

Alternative 2: 95.2% 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 29.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 98.0%

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

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 98.0%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 98.0%

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

13. Applied rewrites 97.7%

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

14. Final simplification 97.7%

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

Alternative 3: 93.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return fma(Float64(Float64(Float64(Float64(Float64(c * a) * -0.5625) / (b ^ 5.0)) - 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[(N[(c * a), $MachinePrecision] * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - 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 29.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 98.0%

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

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 96.9%

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

9. Final simplification 96.9%

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

Alternative 4: 93.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.9%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.6%

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

6. Final simplification 96.6%

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

Alternative 5: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -9 \cdot \left(c \cdot a\right)\\ \frac{\left(\left(\frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {t\_0}^{2} \cdot -0.25\right)}{\left(b \cdot b\right) \cdot a} + \frac{t\_0}{a}\right) \cdot 0.16666666666666666\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(-1.5, \frac{c}{b} \cdot a, b\right), b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -9.0 (* c a))))
   (/
    (*
     (*
      (+
       (/ (fma (* (* c c) (* a a)) 27.0 (* (pow t_0 2.0) -0.25)) (* (* b b) a))
       (/ t_0 a))
      0.16666666666666666)
     b)
    (fma b b (fma (fma -1.5 (* (/ c b) a) b) b (fma (* -3.0 c) a (* b b)))))))

C

double code(double a, double b, double c) {
	double t_0 = -9.0 * (c * a);
	return ((((fma(((c * c) * (a * a)), 27.0, (pow(t_0, 2.0) * -0.25)) / ((b * b) * a)) + (t_0 / a)) * 0.16666666666666666) * b) / fma(b, b, fma(fma(-1.5, ((c / b) * a), b), b, fma((-3.0 * c), a, (b * b))));
}

Julia

function code(a, b, c)
	t_0 = Float64(-9.0 * Float64(c * a))
	return Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(c * c) * Float64(a * a)), 27.0, Float64((t_0 ^ 2.0) * -0.25)) / Float64(Float64(b * b) * a)) + Float64(t_0 / a)) * 0.16666666666666666) * b) / fma(b, b, fma(fma(-1.5, Float64(Float64(c / b) * a), b), b, fma(Float64(-3.0 * c), a, Float64(b * b)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-9.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * 27.0 + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * b), $MachinePrecision] / N[(b * b + N[(N[(-1.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision] * b + N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 29.9%

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

6. Applied rewrites 31.0%

   \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot \frac{0.3333333333333333}{a}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)}} \]

7. Taylor expanded in b around inf

   \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{1}{6} \cdot \frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{1}{6} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{1}{6} \cdot \frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{1}{6} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{b \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{b \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{b \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

9. Applied rewrites 93.6%

   \[\leadsto \frac{\color{blue}{b \cdot \left(0.16666666666666666 \cdot \left(\frac{\left(a \cdot c\right) \cdot -9}{a} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

10. Taylor expanded in c around 0

    \[\leadsto \frac{b \cdot \left(\frac{1}{6} \cdot \left(\frac{\left(a \cdot c\right) \cdot -9}{a} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\color{blue}{b + \frac{-3}{2} \cdot \frac{a \cdot c}{b}}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{b \cdot \left(\frac{1}{6} \cdot \left(\frac{\left(a \cdot c\right) \cdot -9}{a} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b} + b}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \frac{b \cdot \left(\frac{1}{6} \cdot \left(\frac{\left(a \cdot c\right) \cdot -9}{a} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-3}{2}, \frac{a \cdot c}{b}, b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

    3. associate-/l* N/A

       \[\leadsto \frac{b \cdot \left(\frac{1}{6} \cdot \left(\frac{\left(a \cdot c\right) \cdot -9}{a} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{2}, \color{blue}{a \cdot \frac{c}{b}}, b\right), b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{b \cdot \left(\frac{1}{6} \cdot \left(\frac{\left(a \cdot c\right) \cdot -9}{a} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{2}, \color{blue}{a \cdot \frac{c}{b}}, b\right), b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

    5. lower-/.f64 96.5

       \[\leadsto \frac{b \cdot \left(0.16666666666666666 \cdot \left(\frac{\left(a \cdot c\right) \cdot -9}{a} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(-1.5, a \cdot \color{blue}{\frac{c}{b}}, b\right), b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

12. Applied rewrites 96.5%

    \[\leadsto \frac{b \cdot \left(0.16666666666666666 \cdot \left(\frac{\left(a \cdot c\right) \cdot -9}{a} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.5, a \cdot \frac{c}{b}, b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

13. Final simplification 96.5%

    \[\leadsto \frac{\left(\left(\frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right), 27, {\left(-9 \cdot \left(c \cdot a\right)\right)}^{2} \cdot -0.25\right)}{\left(b \cdot b\right) \cdot a} + \frac{-9 \cdot \left(c \cdot a\right)}{a}\right) \cdot 0.16666666666666666\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(-1.5, \frac{c}{b} \cdot a, b\right), b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]
14. Add Preprocessing

Alternative 6: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \frac{\mathsf{fma}\left(\frac{0.16666666666666666}{b}, \frac{6.75 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b}, -1.5 \cdot c\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{t\_0}, b, t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (/
    (*
     (fma (/ 0.16666666666666666 b) (/ (* 6.75 (* (* c c) a)) b) (* -1.5 c))
     b)
    (fma b b (fma (sqrt t_0) b t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	return (fma((0.16666666666666666 / b), ((6.75 * ((c * c) * a)) / b), (-1.5 * c)) * b) / fma(b, b, fma(sqrt(t_0), b, t_0));
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	return Float64(Float64(fma(Float64(0.16666666666666666 / b), Float64(Float64(6.75 * Float64(Float64(c * c) * a)) / b), Float64(-1.5 * c)) * b) / fma(b, b, fma(sqrt(t_0), b, t_0)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(0.16666666666666666 / b), $MachinePrecision] * N[(N[(6.75 * N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(-1.5 * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(b * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 29.9%

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

6. Applied rewrites 31.0%

   \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot \frac{0.3333333333333333}{a}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)}} \]

7. Taylor expanded in b around inf

   \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{1}{6} \cdot \frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{1}{6} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{1}{6} \cdot \frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{1}{6} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{b \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{b \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{b \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

9. Applied rewrites 93.6%

   \[\leadsto \frac{\color{blue}{b \cdot \left(0.16666666666666666 \cdot \left(\frac{\left(a \cdot c\right) \cdot -9}{a} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

10. Taylor expanded in b around inf

    \[\leadsto \frac{b \cdot \left(\frac{-3}{2} \cdot c + \color{blue}{\frac{1}{6} \cdot \frac{\frac{-81}{4} \cdot \left(a \cdot {c}^{2}\right) + 27 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]
11. Step-by-step derivation

12. Applied rewrites 93.8%

    \[\leadsto \frac{b \cdot \mathsf{fma}\left(\frac{0.16666666666666666}{b}, \color{blue}{\frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot 6.75}{b}}, -1.5 \cdot c\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

13. Final simplification 93.8%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{0.16666666666666666}{b}, \frac{6.75 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b}, -1.5 \cdot c\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]
14. Add Preprocessing

Alternative 7: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \frac{\left(\mathsf{fma}\left(0.16666666666666666, \left(\frac{a}{b \cdot b} \cdot 6.75\right) \cdot c, -1.5\right) \cdot c\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{t\_0}, b, t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (/
    (* (* (fma 0.16666666666666666 (* (* (/ a (* b b)) 6.75) c) -1.5) c) b)
    (fma b b (fma (sqrt t_0) b t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	return ((fma(0.16666666666666666, (((a / (b * b)) * 6.75) * c), -1.5) * c) * b) / fma(b, b, fma(sqrt(t_0), b, t_0));
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	return Float64(Float64(Float64(fma(0.16666666666666666, Float64(Float64(Float64(a / Float64(b * b)) * 6.75) * c), -1.5) * c) * b) / fma(b, b, fma(sqrt(t_0), b, t_0)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(0.16666666666666666 * N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * 6.75), $MachinePrecision] * c), $MachinePrecision] + -1.5), $MachinePrecision] * c), $MachinePrecision] * b), $MachinePrecision] / N[(b * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 29.9%

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

6. Applied rewrites 31.0%

   \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}\right) \cdot \frac{0.3333333333333333}{a}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)}} \]

7. Taylor expanded in b around inf

   \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{1}{6} \cdot \frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{1}{6} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{1}{6} \cdot \frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{1}{6} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{b \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{b \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{b \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

9. Applied rewrites 93.6%

   \[\leadsto \frac{\color{blue}{b \cdot \left(0.16666666666666666 \cdot \left(\frac{\left(a \cdot c\right) \cdot -9}{a} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

10. Taylor expanded in c around 0

    \[\leadsto \frac{b \cdot \left(c \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(c \cdot \left(\frac{-81}{4} \cdot \frac{a}{{b}^{2}} + 27 \cdot \frac{a}{{b}^{2}}\right)\right) - \frac{3}{2}\right)}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]
11. Step-by-step derivation

12. Applied rewrites 93.7%

    \[\leadsto \frac{b \cdot \left(c \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, c \cdot \left(\frac{a}{b \cdot b} \cdot 6.75\right), -1.5\right)}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]

13. Final simplification 93.7%

    \[\leadsto \frac{\left(\mathsf{fma}\left(0.16666666666666666, \left(\frac{a}{b \cdot b} \cdot 6.75\right) \cdot c, -1.5\right) \cdot c\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)\right)} \]
14. Add Preprocessing

Alternative 8: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -160:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -160.0)
   (/ (- (sqrt (fma b b (* (* -3.0 a) c))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -160.0) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -160.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -160.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -160

   1. Initial program 69.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 70.3

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

   4. Applied rewrites 70.3%

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

   if -160 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 25.4%

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

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

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -160:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -160:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -160.0)
   (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -160.0) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -160.0)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -160.0], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -160

   1. Initial program 69.8%

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

   4. Applied rewrites 69.8%

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

   if -160 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 25.4%

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

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

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -160:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -160:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -160.0)
   (/ (* (- (sqrt (fma (* -3.0 c) a (* b b))) b) 0.3333333333333333) a)
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -160.0) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -160.0)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -160.0], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -160

   1. Initial program 69.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 69.8%

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

   if -160 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 25.4%

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

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

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -160:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -160:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -160.0)
   (* (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a) 0.3333333333333333)
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -160.0) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -160.0)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -160.0], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -160

   1. Initial program 69.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 69.8%

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

   if -160 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 25.4%

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

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

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -160:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -160:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -160.0)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma (* -3.0 c) a (* b b))) b))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -160.0) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((-3.0 * c), a, (b * b))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -160.0)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -160.0], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -160

   1. Initial program 69.8%

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

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 69.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 69.8

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

   4. Applied rewrites 69.8%

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

   if -160 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 25.4%

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

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

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -160:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 29.9%

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

5. Taylor expanded in b around inf

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

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 93.7

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

7. Applied rewrites 93.7%

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

8. Final simplification 93.7%

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

Alternative 14: 81.0% 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 29.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 83.1

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

5. Applied rewrites 83.1%

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

6. Final simplification 83.1%

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

Alternative 15: 80.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 83.1

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

5. Applied rewrites 83.1%

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

7. Applied rewrites 82.9%

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

8. Final simplification 82.9%

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

Reproduce

herbie shell --seed 2024249 
(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)))