Cubic critical, narrow range

Percentage Accurate: 55.3% → 91.0%

Time: 13.5s

Alternatives: 12

Speedup: 2.9×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.3% 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: 91.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot 3, \mathsf{fma}\left(-1.40625, c \cdot c, 0.84375 \cdot \left(c \cdot c\right)\right), \mathsf{fma}\left(-1.125, a \cdot c, \left(b \cdot b\right) \cdot -1.5\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}}, a, 2 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  -1.0
  (fma
   (/
    (fma
     (* (* a a) 3.0)
     (fma -1.40625 (* c c) (* 0.84375 (* c c)))
     (* (fma -1.125 (* a c) (* (* b b) -1.5)) (* b b)))
    (pow b 5.0))
   a
   (* 2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	return -1.0 / fma((fma(((a * a) * 3.0), fma(-1.40625, (c * c), (0.84375 * (c * c))), (fma(-1.125, (a * c), ((b * b) * -1.5)) * (b * b))) / pow(b, 5.0)), a, (2.0 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(-1.0 / fma(Float64(fma(Float64(Float64(a * a) * 3.0), fma(-1.40625, Float64(c * c), Float64(0.84375 * Float64(c * c))), Float64(fma(-1.125, Float64(a * c), Float64(Float64(b * b) * -1.5)) * Float64(b * b))) / (b ^ 5.0)), a, Float64(2.0 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 / N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 3.0), $MachinePrecision] * N[(-1.40625 * N[(c * c), $MachinePrecision] + N[(0.84375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.125 * N[(a * c), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * a + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-times N/A

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

4. Applied rewrites 53.4%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 93.2%

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.75}{b}, \frac{\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot c}{b}, \mathsf{fma}\left(\frac{-0.2222222222222222}{c}, \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{c}, \frac{0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right), a, \frac{c}{{b}^{3}} \cdot -0.375\right), a, \frac{-1.5}{b}\right), a, \frac{b}{c} \cdot 2\right)}} \]

8. Taylor expanded in c around 0

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

10. Applied rewrites 93.2%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(3 \cdot \mathsf{fma}\left(\frac{-0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}, a, \frac{c}{{b}^{3}} \cdot -0.375\right), a, \frac{-1.5}{b}\right), a, \frac{b}{c} \cdot 2\right)} \]

11. Taylor expanded in b around 0

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{3 \cdot \left({a}^{2} \cdot \left(\frac{-45}{32} \cdot {c}^{2} + \left(\frac{9}{32} \cdot {c}^{2} + \frac{9}{16} \cdot {c}^{2}\right)\right)\right) + {b}^{2} \cdot \left(\frac{-3}{2} \cdot {b}^{2} + \frac{-9}{8} \cdot \left(a \cdot c\right)\right)}{{b}^{5}}, a, \frac{b}{c} \cdot 2\right)} \]
12. Step-by-step derivation

13. Applied rewrites 93.2%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.40625, c \cdot c, \left(c \cdot c\right) \cdot 0.84375\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(-1.125, a \cdot c, -1.5 \cdot \left(b \cdot b\right)\right)\right)}{{b}^{5}}, a, \frac{b}{c} \cdot 2\right)} \]

14. Final simplification 93.2%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot 3, \mathsf{fma}\left(-1.40625, c \cdot c, 0.84375 \cdot \left(c \cdot c\right)\right), \mathsf{fma}\left(-1.125, a \cdot c, \left(b \cdot b\right) \cdot -1.5\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}}, a, 2 \cdot \frac{b}{c}\right)} \]
15. Add Preprocessing

Alternative 2: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot -1.5\right), c, 2 \cdot b\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  -1.0
  (/
   (fma
    (fma (* 3.0 c) (* (/ (* a a) (pow b 3.0)) -0.375) (* (/ a b) -1.5))
    c
    (* 2.0 b))
   c)))

C

double code(double a, double b, double c) {
	return -1.0 / (fma(fma((3.0 * c), (((a * a) / pow(b, 3.0)) * -0.375), ((a / b) * -1.5)), c, (2.0 * b)) / c);
}

Julia

function code(a, b, c)
	return Float64(-1.0 / Float64(fma(fma(Float64(3.0 * c), Float64(Float64(Float64(a * a) / (b ^ 3.0)) * -0.375), Float64(Float64(a / b) * -1.5)), c, Float64(2.0 * b)) / c))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 / N[(N[(N[(N[(3.0 * c), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision] * c + N[(2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-times N/A

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

4. Applied rewrites 53.4%

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

5. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 90.3%

   \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot -1.5\right), c, b \cdot 2\right)}{c}}} \]

8. Final simplification 90.3%

   \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, \frac{a}{b} \cdot -1.5\right), c, 2 \cdot b\right)}{c}} \]
9. Add Preprocessing

Alternative 3: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.125, \frac{c}{b \cdot b} \cdot a, -1.5\right)}{b}, a, 2 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ -1.0 (fma (/ (fma -1.125 (* (/ c (* b b)) a) -1.5) b) a (* 2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	return -1.0 / fma((fma(-1.125, ((c / (b * b)) * a), -1.5) / b), a, (2.0 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(-1.0 / fma(Float64(fma(-1.125, Float64(Float64(c / Float64(b * b)) * a), -1.5) / b), a, Float64(2.0 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 / N[(N[(N[(-1.125 * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] + -1.5), $MachinePrecision] / b), $MachinePrecision] * a + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-times N/A

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

4. Applied rewrites 53.4%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 93.2%

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.75}{b}, \frac{\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot c}{b}, \mathsf{fma}\left(\frac{-0.2222222222222222}{c}, \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{c}, \frac{0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right), a, \frac{c}{{b}^{3}} \cdot -0.375\right), a, \frac{-1.5}{b}\right), a, \frac{b}{c} \cdot 2\right)}} \]

8. Taylor expanded in c around 0

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

10. Applied rewrites 93.2%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(3 \cdot \mathsf{fma}\left(\frac{-0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}, a, \frac{c}{{b}^{3}} \cdot -0.375\right), a, \frac{-1.5}{b}\right), a, \frac{b}{c} \cdot 2\right)} \]

11. Taylor expanded in b around inf

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

13. Applied rewrites 90.2%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.125, a \cdot \frac{c}{b \cdot b}, -1.5\right)}{b}, a, \frac{b}{c} \cdot 2\right)} \]

14. Final simplification 90.2%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.125, \frac{c}{b \cdot b} \cdot a, -1.5\right)}{b}, a, 2 \cdot \frac{b}{c}\right)} \]
15. Add Preprocessing

Alternative 4: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, -1.5, 2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 44.0)
   (/ 1.0 (/ a (* 0.3333333333333333 (- (sqrt (fma b b (* (* c -3.0) a))) b))))
   (/ -1.0 (/ (fma (* (/ c b) a) -1.5 (* 2.0 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 44.0) {
		tmp = 1.0 / (a / (0.3333333333333333 * (sqrt(fma(b, b, ((c * -3.0) * a))) - b)));
	} else {
		tmp = -1.0 / (fma(((c / b) * a), -1.5, (2.0 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 44.0)
		tmp = Float64(1.0 / Float64(a / Float64(0.3333333333333333 * Float64(sqrt(fma(b, b, Float64(Float64(c * -3.0) * a))) - b))));
	else
		tmp = Float64(-1.0 / Float64(fma(Float64(Float64(c / b) * a), -1.5, Float64(2.0 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 44.0], N[(1.0 / N[(a / N[(0.3333333333333333 * N[(N[Sqrt[N[(b * b + N[(N[(c * -3.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * -1.5 + N[(2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 44

   1. Initial program 78.8%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 78.8%

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

   6. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 79.1

         \[\leadsto \frac{1}{\frac{a}{\left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3 \cdot c\right) \cdot a}\right)} - b\right) \cdot 0.3333333333333333}} \]

   8. Applied rewrites 79.1%

      \[\leadsto \frac{1}{\frac{a}{\left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}} - b\right) \cdot 0.3333333333333333}} \]

   if 44 < b

   1. Initial program 45.0%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 89.3

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, -1.5, \color{blue}{b \cdot 2}\right)}{c}} \]

   7. Applied rewrites 89.3%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, -1.5, b \cdot 2\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{1}{\frac{a}{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, -1.5, 2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, -1.5, 2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 44.0)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/ -1.0 (/ (fma (* (/ c b) a) -1.5 (* 2.0 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 44.0) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = -1.0 / (fma(((c / b) * a), -1.5, (2.0 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 44.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-1.0 / Float64(fma(Float64(Float64(c / b) * a), -1.5, Float64(2.0 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 44.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * -1.5 + N[(2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 44

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 79.0

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

   4. Applied rewrites 79.0%

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

   if 44 < b

   1. Initial program 45.0%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 89.3

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, -1.5, \color{blue}{b \cdot 2}\right)}{c}} \]

   7. Applied rewrites 89.3%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, -1.5, b \cdot 2\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, -1.5, 2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 44.0)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/ -1.0 (fma (/ a b) -1.5 (* 2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 44.0) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = -1.0 / fma((a / b), -1.5, (2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 44.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-1.0 / fma(Float64(a / b), -1.5, Float64(2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 44.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(a / b), $MachinePrecision] * -1.5 + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 44

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 79.0

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

   4. Applied rewrites 79.0%

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

   if 44 < b

   1. Initial program 45.0%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.2

         \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, \color{blue}{\frac{b}{c}} \cdot 2\right)} \]

   7. Applied rewrites 89.2%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, -1.5, \frac{b}{c} \cdot 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 44.0)
   (/ (* (- (sqrt (fma (* c -3.0) a (* b b))) b) 0.3333333333333333) a)
   (/ -1.0 (fma (/ a b) -1.5 (* 2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 44.0) {
		tmp = ((sqrt(fma((c * -3.0), a, (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = -1.0 / fma((a / b), -1.5, (2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 44.0)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(-1.0 / fma(Float64(a / b), -1.5, Float64(2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 44.0], N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(-1.0 / N[(N[(a / b), $MachinePrecision] * -1.5 + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 44

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 78.9%

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

   if 44 < b

   1. Initial program 45.0%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.2

         \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, \color{blue}{\frac{b}{c}} \cdot 2\right)} \]

   7. Applied rewrites 89.2%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, -1.5, \frac{b}{c} \cdot 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 44.0)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma (* c -3.0) a (* b b))) b))
   (/ -1.0 (fma (/ a b) -1.5 (* 2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 44.0) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((c * -3.0), a, (b * b))) - b);
	} else {
		tmp = -1.0 / fma((a / b), -1.5, (2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 44.0)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b));
	else
		tmp = Float64(-1.0 / fma(Float64(a / b), -1.5, Float64(2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 44.0], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(a / b), $MachinePrecision] * -1.5 + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 44

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 78.8

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

      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 78.8

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

   4. Applied rewrites 78.8%

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

   if 44 < b

   1. Initial program 45.0%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.2

         \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, \color{blue}{\frac{b}{c}} \cdot 2\right)} \]

   7. Applied rewrites 89.2%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, -1.5, \frac{b}{c} \cdot 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, 2 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ -1.0 (fma (/ a b) -1.5 (* 2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	return -1.0 / fma((a / b), -1.5, (2.0 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(-1.0 / fma(Float64(a / b), -1.5, Float64(2.0 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 / N[(N[(a / b), $MachinePrecision] * -1.5 + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-times N/A

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

4. Applied rewrites 53.4%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 83.3

      \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, \color{blue}{\frac{b}{c}} \cdot 2\right)} \]

7. Applied rewrites 83.3%

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, -1.5, \frac{b}{c} \cdot 2\right)}} \]

8. Final simplification 83.3%

   \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{a}{b}, -1.5, 2 \cdot \frac{b}{c}\right)} \]
9. Add Preprocessing

Alternative 10: 81.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. unpow2 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 82.7

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

5. Applied rewrites 82.7%

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

6. Taylor expanded in c around inf

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

8. Applied rewrites 82.6%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 82.6%

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

12. Final simplification 82.6%

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

Alternative 11: 64.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 65.5

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

5. Applied rewrites 65.5%

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

6. Final simplification 65.5%

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

Alternative 12: 64.4% 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 53.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 65.5

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

5. Applied rewrites 65.5%

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

7. Applied rewrites 65.4%

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

8. Final simplification 65.4%

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

Reproduce

herbie shell --seed 2024248 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))