Cubic critical, wide range

Percentage Accurate: 17.1% → 97.9%

Time: 12.8s

Alternatives: 11

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in a around 0

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

5. Simplified 97.9%

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

6. Final simplification 97.9%

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

Alternative 2: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}}, \frac{-1.0546875 \cdot \left(a \cdot \left(a \cdot \left(a \cdot c\right)\right)\right)}{{b}^{7}}\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (fma
   c
   (fma
    c
    (fma
     -0.5625
     (/ (* a a) (pow b 5.0))
     (/ (* -1.0546875 (* a (* a (* a c)))) (pow b 7.0)))
    (/ (* a -0.375) (* b (* b b))))
   (/ -0.5 b))))

C

double code(double a, double b, double c) {
	return c * fma(c, fma(c, fma(-0.5625, ((a * a) / pow(b, 5.0)), ((-1.0546875 * (a * (a * (a * c)))) / pow(b, 7.0))), ((a * -0.375) / (b * (b * b)))), (-0.5 / b));
}

Julia

function code(a, b, c)
	return Float64(c * fma(c, fma(c, fma(-0.5625, Float64(Float64(a * a) / (b ^ 5.0)), Float64(Float64(-1.0546875 * Float64(a * Float64(a * Float64(a * c)))) / (b ^ 7.0))), Float64(Float64(a * -0.375) / Float64(b * Float64(b * b)))), Float64(-0.5 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(c * N[(c * N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[(a * N[(a * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in c around 0

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

5. Simplified 97.5%

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

6. Taylor expanded in c around 0

   \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \color{blue}{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \color{blue}{\frac{\frac{-135}{128} \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   2. lower-/.f64 N/A

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \color{blue}{\frac{\frac{-135}{128} \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \frac{\color{blue}{\frac{-135}{128} \cdot \left({a}^{3} \cdot c\right)}}{{b}^{7}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   4. cube-mult N/A

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \frac{\frac{-135}{128} \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \cdot c\right)}{{b}^{7}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   5. unpow2 N/A

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \frac{\frac{-135}{128} \cdot \left(\left(a \cdot \color{blue}{{a}^{2}}\right) \cdot c\right)}{{b}^{7}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   6. associate-*l* N/A

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \frac{\frac{-135}{128} \cdot \color{blue}{\left(a \cdot \left({a}^{2} \cdot c\right)\right)}}{{b}^{7}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \frac{\frac{-135}{128} \cdot \color{blue}{\left(a \cdot \left({a}^{2} \cdot c\right)\right)}}{{b}^{7}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   8. unpow2 N/A

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot c\right)\right)}{{b}^{7}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   9. associate-*l* N/A

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(a \cdot c\right)\right)}\right)}{{b}^{7}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(a \cdot c\right)\right)}\right)}{{b}^{7}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot a}{{b}^{5}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(a \cdot c\right)}\right)\right)}{{b}^{7}}\right), \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{\frac{-1}{2}}{b}\right) \]

   12. lower-pow.f64 97.5

       \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}}, \frac{-1.0546875 \cdot \left(a \cdot \left(a \cdot \left(a \cdot c\right)\right)\right)}{\color{blue}{{b}^{7}}}\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right) \]

8. Simplified 97.5%

   \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}}, \color{blue}{\frac{-1.0546875 \cdot \left(a \cdot \left(a \cdot \left(a \cdot c\right)\right)\right)}{{b}^{7}}}\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right) \]
9. Add Preprocessing

Alternative 3: 97.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

5. Simplified 97.0%

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

Alternative 4: 97.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 20.0

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

5. Simplified 20.0%

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

6. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

8. Simplified 96.6%

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

9. Taylor expanded in a around 0

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

11. Simplified 96.9%

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

Alternative 5: 97.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 20.0

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

5. Simplified 20.0%

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

6. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

8. Simplified 96.6%

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

9. Taylor expanded in b around inf

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

11. Simplified 96.9%

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

Alternative 6: 96.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in c around 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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. associate-*r/ N/A

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

   5. associate-*r* N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

   8. lower-fma.f64 N/A

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

5. Simplified 96.6%

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

6. Final simplification 96.6%

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

Alternative 7: 96.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 20.0

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

5. Simplified 20.0%

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

6. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

8. Simplified 96.6%

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

9. Taylor expanded in b around 0

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

    1. lower-/.f64 N/A

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

11. Simplified 96.1%

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

12. Final simplification 96.1%

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

Alternative 8: 95.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. cube-mult N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. associate-*r/ N/A

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

   19. lower-/.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 94.8

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

5. Simplified 94.8%

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

Alternative 9: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Simplified 94.8%

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

Alternative 10: 95.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. distribute-rgt-in N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r* N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

5. Simplified 94.4%

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

6. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. cube-mult N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 94.4

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

8. Simplified 94.4%

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

9. Taylor expanded in b around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. unpow2 N/A

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

    4. associate-*r* N/A

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

    5. associate-*r* N/A

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

    6. associate-*r* N/A

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

    7. distribute-rgt-out N/A

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

    8. lower-*.f64 N/A

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

    9. +-commutative N/A

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

    10. lower-fma.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. *-commutative N/A

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

    14. associate-*l* N/A

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

    15. *-commutative N/A

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

    16. lower-*.f64 N/A

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

    17. *-commutative N/A

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

    18. lower-*.f64 N/A

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

    19. cube-mult N/A

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

    20. unpow2 N/A

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

    21. lower-*.f64 N/A

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

    22. unpow2 N/A

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

    23. lower-*.f64 94.4

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

11. Simplified 94.4%

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

12. Taylor expanded in c around 0

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

    1. sub-neg N/A

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

    2. distribute-rgt-in N/A

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

    3. associate-*r/ N/A

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

    4. associate-*l/ N/A

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

    5. unpow3 N/A

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

    6. unpow2 N/A

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

    7. times-frac N/A

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

    8. associate-*r/ N/A

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

    9. metadata-eval N/A

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

    10. distribute-neg-frac N/A

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

    11. metadata-eval N/A

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

    12. associate-*l/ N/A

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

    13. associate-*r/ N/A

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

    14. distribute-rgt-out N/A

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

    15. lower-*.f64 N/A

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

    16. lower-/.f64 N/A

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

14. Simplified 94.8%

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

Alternative 11: 90.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 88.7

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

5. Simplified 88.7%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))