Cubic critical, medium range

Percentage Accurate: 31.3% → 95.7%

Time: 18.4s

Alternatives: 10

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.3% 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: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}, -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(a \cdot 1.5\right) \cdot \left(c \cdot \left(2.25 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (fma
   -0.16666666666666666
   (/ (pow (* c a) 4.0) (/ (* a (pow b 7.0)) 6.328125))
   (*
    -0.16666666666666666
    (+
     (/ (* (pow (* c a) 2.0) 2.25) (* a (pow b 3.0)))
     (/
      (fma
       -3.0
       (* a (* c 0.0))
       (* (* a 1.5) (* c (* 2.25 (* (* c a) (* c a))))))
      (* a (pow b 5.0))))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), fma(-0.16666666666666666, (pow((c * a), 4.0) / ((a * pow(b, 7.0)) / 6.328125)), (-0.16666666666666666 * (((pow((c * a), 2.0) * 2.25) / (a * pow(b, 3.0))) + (fma(-3.0, (a * (c * 0.0)), ((a * 1.5) * (c * (2.25 * ((c * a) * (c * a)))))) / (a * pow(b, 5.0)))))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), fma(-0.16666666666666666, Float64((Float64(c * a) ^ 4.0) / Float64(Float64(a * (b ^ 7.0)) / 6.328125)), Float64(-0.16666666666666666 * Float64(Float64(Float64((Float64(c * a) ^ 2.0) * 2.25) / Float64(a * (b ^ 3.0))) + Float64(fma(-3.0, Float64(a * Float64(c * 0.0)), Float64(Float64(a * 1.5) * Float64(c * Float64(2.25 * Float64(Float64(c * a) * Float64(c * a)))))) / Float64(a * (b ^ 5.0)))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[(N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] / 6.328125), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision] * 2.25), $MachinePrecision] / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 * N[(a * N[(c * 0.0), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] * N[(c * N[(2.25 * N[(N[(c * a), $MachinePrecision] * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.7%

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

   1. flip3-- 29.7%

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

   2. div-inv 29.8%

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

   3. pow2 29.8%

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

   4. pow-pow 29.8%

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

   5. metadata-eval 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   6. associate-*l* 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   7. unpow-prod-down 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{{3}^{3} \cdot {\left(a \cdot c\right)}^{3}}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   8. metadata-eval 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   9. pow2 29.8%

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

   10. pow2 29.8%

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

   11. pow-prod-up 29.9%

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

   12. metadata-eval 29.9%

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

   13. distribute-rgt-out 29.9%

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

4. Applied egg-rr 29.9%

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

5. Taylor expanded in b around inf 95.7%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(1.5 \cdot \left(a \cdot \left(c \cdot \left(-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

7. Simplified 95.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)} \]

8. Taylor expanded in c around 0 95.7%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]
9. Step-by-step derivation

   1. distribute-rgt-out 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{7}}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

   2. associate-*r* 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

   3. *-commutative 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

   4. associate-/l* 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{a}^{4} \cdot {c}^{4}}{\frac{a \cdot {b}^{7}}{1.265625 + 5.0625}}}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

10. Simplified 95.7%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]
11. Step-by-step derivation

    1. unpow2 95.7%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + \color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

12. Applied egg-rr 95.7%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + \color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]

13. Final simplification 95.7%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}, -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(a \cdot 1.5\right) \cdot \left(c \cdot \left(2.25 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right) \]
14. Add Preprocessing

Alternative 2: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+
   (* -0.5 (/ c b))
   (+
    (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
    (*
     -0.16666666666666666
     (* (/ (pow (* c a) 4.0) a) (/ 6.328125 (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((c * a), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.16666666666666666d0) * ((((c * a) ** 4.0d0) / a) * (6.328125d0 / (b ** 7.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.16666666666666666 * ((Math.pow((c * a), 4.0) / a) * (6.328125 / Math.pow(b, 7.0))))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.16666666666666666 * ((math.pow((c * a), 4.0) / a) * (6.328125 / math.pow(b, 7.0))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(c * a) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.16666666666666666 * ((((c * a) ^ 4.0) / a) * (6.328125 / (b ^ 7.0))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf 95.6%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

4. Taylor expanded in c around 0 95.6%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
5. Step-by-step derivation

   1. distribute-rgt-in 95.6%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left(1.265625 \cdot {a}^{4}\right) \cdot {c}^{4} + \left(5.0625 \cdot {a}^{4}\right) \cdot {c}^{4}}}{a \cdot {b}^{7}}\right)\right) \]

   2. associate-*r* 95.6%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right)} + \left(5.0625 \cdot {a}^{4}\right) \cdot {c}^{4}}{a \cdot {b}^{7}}\right)\right) \]

   3. associate-*r* 95.6%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}}{a \cdot {b}^{7}}\right)\right) \]

   4. distribute-rgt-out 95.6%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

   5. times-frac 95.6%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{1.265625 + 5.0625}{{b}^{7}}\right)}\right)\right) \]

6. Simplified 95.6%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)}\right)\right) \]

7. Final simplification 95.6%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right) \]
8. Add Preprocessing

Alternative 3: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(c \cdot a\right)}^{2}\\ \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{t_0}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left(t_0 \cdot 2.25\right) \cdot \left(a \cdot \left(c \cdot 1.5\right)\right)}{a \cdot {b}^{5}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* c a) 2.0)))
   (fma
    -0.5
    (/ c b)
    (*
     -0.16666666666666666
     (+
      (* (/ t_0 a) (/ 2.25 (pow b 3.0)))
      (/ (* (* t_0 2.25) (* a (* c 1.5))) (* a (pow b 5.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((c * a), 2.0);
	return fma(-0.5, (c / b), (-0.16666666666666666 * (((t_0 / a) * (2.25 / pow(b, 3.0))) + (((t_0 * 2.25) * (a * (c * 1.5))) / (a * pow(b, 5.0))))));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * a) ^ 2.0
	return fma(-0.5, Float64(c / b), Float64(-0.16666666666666666 * Float64(Float64(Float64(t_0 / a) * Float64(2.25 / (b ^ 3.0))) + Float64(Float64(Float64(t_0 * 2.25) * Float64(a * Float64(c * 1.5))) / Float64(a * (b ^ 5.0))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision]}, N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(t$95$0 / a), $MachinePrecision] * N[(2.25 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 * 2.25), $MachinePrecision] * N[(a * N[(c * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.7%

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

   1. flip3-- 29.7%

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

   2. div-inv 29.8%

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

   3. pow2 29.8%

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

   4. pow-pow 29.8%

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

   5. metadata-eval 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   6. associate-*l* 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   7. unpow-prod-down 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{{3}^{3} \cdot {\left(a \cdot c\right)}^{3}}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   8. metadata-eval 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   9. pow2 29.8%

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

   10. pow2 29.8%

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

   11. pow-prod-up 29.9%

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

   12. metadata-eval 29.9%

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

   13. distribute-rgt-out 29.9%

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

4. Applied egg-rr 29.9%

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

5. Taylor expanded in b around inf 94.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)} \]
6. Step-by-step derivation

   1. fma-def 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)} \]

7. Simplified 94.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]
8. Step-by-step derivation

   1. distribute-lft-in 94.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}}\right) \]

   2. +-lft-identity 94.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{2} \cdot 2.25}}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(-3, a \cdot \left(c \cdot 0\right), \left(1.5 \cdot a\right) \cdot \left(c \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)\right)}{a \cdot {b}^{5}}\right) \]

9. Applied egg-rr 94.2%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}}\right) \]
10. Step-by-step derivation

    1. distribute-lft-in 94.2%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)}\right) \]

    2. times-frac 94.2%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\color{blue}{\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}}} + \frac{0 + \left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{5}}\right)\right) \]

    3. +-lft-identity 94.2%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\color{blue}{\left(\left(a \cdot 1.5\right) \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}}{a \cdot {b}^{5}}\right)\right) \]

    4. *-commutative 94.2%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot 2.25\right) \cdot \left(\left(a \cdot 1.5\right) \cdot c\right)}}{a \cdot {b}^{5}}\right)\right) \]

    5. *-commutative 94.2%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\color{blue}{\left(2.25 \cdot {\left(a \cdot c\right)}^{2}\right)} \cdot \left(\left(a \cdot 1.5\right) \cdot c\right)}{a \cdot {b}^{5}}\right)\right) \]

    6. associate-*l* 94.2%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left(2.25 \cdot {\left(a \cdot c\right)}^{2}\right) \cdot \color{blue}{\left(a \cdot \left(1.5 \cdot c\right)\right)}}{a \cdot {b}^{5}}\right)\right) \]

11. Simplified 94.2%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left(2.25 \cdot {\left(a \cdot c\right)}^{2}\right) \cdot \left(a \cdot \left(1.5 \cdot c\right)\right)}{a \cdot {b}^{5}}\right)}\right) \]

12. Final simplification 94.2%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{2}}{a} \cdot \frac{2.25}{{b}^{3}} + \frac{\left({\left(c \cdot a\right)}^{2} \cdot 2.25\right) \cdot \left(a \cdot \left(c \cdot 1.5\right)\right)}{a \cdot {b}^{5}}\right)\right) \]
13. Add Preprocessing

Alternative 4: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf 94.2%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

4. Final simplification 94.2%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -2e-5)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma b b (* (* c a) -3.0))) b))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -2e-5) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma(b, b, ((c * a) * -3.0))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -2e-5)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))) - b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -2e-5], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.00000000000000016e-5

   1. Initial program 69.4%

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

   3. Taylor expanded in a around 0 69.4%

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

      1. *-commutative 69.4%

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

   5. Simplified 69.4%

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

      1. add-log-exp 49.4%

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

      2. neg-mul-1 49.4%

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

      3. fma-def 49.4%

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

      4. pow2 49.4%

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

      5. associate-*l* 49.4%

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

      6. *-commutative 49.4%

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

   7. Applied egg-rr 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. log-prod 49.4%

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

      3. metadata-eval 49.4%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a \cdot 3}}\right) \]

      4. rem-log-exp 69.4%

         \[\leadsto 0 + \color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a \cdot 3}} \]

   9. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{0 + \frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}{a \cdot 3}} \]
   10. Step-by-step derivation

       1. +-lft-identity 69.4%

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

       2. *-lft-identity 69.4%

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

       3. associate-*l/ 69.4%

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

       4. fma-def 69.4%

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

       5. +-commutative 69.4%

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

       6. *-commutative 69.4%

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

       7. associate-/r* 69.4%

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

       8. metadata-eval 69.4%

          \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \left(\sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)} + -1 \cdot b\right) \]

       9. mul-1-neg 69.4%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)} + \color{blue}{\left(-b\right)}\right) \]

       10. unsub-neg 69.4%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(\sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)} - b\right)} \]

       11. unpow2 69.4%

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

       12. fma-neg 69.5%

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

       13. associate-*r* 69.5%

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

       14. distribute-rgt-neg-in 69.5%

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

       15. metadata-eval 69.5%

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

   11. Simplified 69.5%

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

   if -2.00000000000000016e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 17.2%

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

   3. Taylor expanded in b around inf 91.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -2e-5) t_0 (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -2e-5) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-2d-5)) then
        tmp = t_0
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -2e-5) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -2e-5:
		tmp = t_0
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -2e-5)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -2e-5)
		tmp = t_0;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-5], t$95$0, N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.00000000000000016e-5

   1. Initial program 69.4%

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

   if -2.00000000000000016e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 17.2%

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

   3. Taylor expanded in b around inf 91.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -2e-5)
   (/ (- (sqrt (- (* b b) (* (* c a) 3.0))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -2e-5) {
		tmp = (sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-2d-5)) then
        tmp = (sqrt(((b * b) - ((c * a) * 3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -2e-5) {
		tmp = (Math.sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -2e-5:
		tmp = (math.sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -2e-5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(c * a) * 3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -2e-5)
		tmp = (sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -2e-5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.00000000000000016e-5

   1. Initial program 69.4%

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

   3. Taylor expanded in a around 0 69.4%

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

      1. *-commutative 69.4%

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

   5. Simplified 69.4%

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

   if -2.00000000000000016e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 17.2%

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

   3. Taylor expanded in b around inf 91.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf 91.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Final simplification 91.4%

   \[\leadsto -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in b around inf 82.7%

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

4. Final simplification 82.7%

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

Alternative 10: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 29.7%

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

3. Taylor expanded in a around 0 29.7%

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

   1. *-commutative 29.7%

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

5. Simplified 29.7%

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

   1. add-log-exp 22.2%

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

   2. neg-mul-1 22.2%

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

   3. fma-def 22.2%

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

   4. pow2 22.2%

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

   5. associate-*l* 22.2%

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

   6. *-commutative 22.2%

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

7. Applied egg-rr 22.2%

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

8. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(b + -1 \cdot b\right)}{a}} \]

   2. distribute-rgt1-in 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

10. Simplified 3.2%

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

11. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))