Cubic critical, wide range

Percentage Accurate: 18.1% → 97.6%

Time: 21.5s

Alternatives: 11

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot {\left(c \cdot \frac{\sqrt{a}}{b}\right)}^{2} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
   (+
    (* c -0.5)
    (+
     (* -0.375 (pow (* c (/ (sqrt a) b)) 2.0))
     (/ (* (pow (* a c) 4.0) -1.0546875) (* a (pow b 6.0))))))
  b))

C

double code(double a, double b, double c) {
	return ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * pow((c * (sqrt(a) / b)), 2.0)) + ((pow((a * c), 4.0) * -1.0546875) / (a * pow(b, 6.0)))))) / 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 = (((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 4.0d0))) + ((c * (-0.5d0)) + (((-0.375d0) * ((c * (sqrt(a) / b)) ** 2.0d0)) + ((((a * c) ** 4.0d0) * (-1.0546875d0)) / (a * (b ** 6.0d0)))))) / b
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, 4.0))) + ((c * -0.5) + ((-0.375 * Math.pow((c * (Math.sqrt(a) / b)), 2.0)) + ((Math.pow((a * c), 4.0) * -1.0546875) / (a * Math.pow(b, 6.0)))))) / b;
}

Python

def code(a, b, c):
	return ((-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * math.pow((c * (math.sqrt(a) / b)), 2.0)) + ((math.pow((a * c), 4.0) * -1.0546875) / (a * math.pow(b, 6.0)))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(Float64(-0.375 * (Float64(c * Float64(sqrt(a) / b)) ^ 2.0)) + Float64(Float64((Float64(a * c) ^ 4.0) * -1.0546875) / Float64(a * (b ^ 6.0)))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + ((c * -0.5) + ((-0.375 * ((c * (sqrt(a) / b)) ^ 2.0)) + ((((a * c) ^ 4.0) * -1.0546875) / (a * (b ^ 6.0)))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(N[(-0.375 * N[Power[N[(c * N[(N[Sqrt[a], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * -1.0546875), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in b around inf 97.7%

   \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left(1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]

   2. distribute-rgt-out 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \color{blue}{\left(\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{6}}\right)\right)}{b} \]

   3. pow-prod-down 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \left(1.265625 + 5.0625\right)\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

   4. metadata-eval 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \color{blue}{6.328125}\right)}{a \cdot {b}^{6}}\right)\right)}{b} \]

5. Applied egg-rr 97.7%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}}\right)\right)}{b} \]
6. Step-by-step derivation

   1. *-commutative 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{\color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right) \cdot -0.16666666666666666}}{a \cdot {b}^{6}}\right)\right)}{b} \]

   2. associate-*l* 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{\color{blue}{{\left(a \cdot c\right)}^{4} \cdot \left(6.328125 \cdot -0.16666666666666666\right)}}{a \cdot {b}^{6}}\right)\right)}{b} \]

   3. metadata-eval 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{{\left(a \cdot c\right)}^{4} \cdot \color{blue}{-1.0546875}}{a \cdot {b}^{6}}\right)\right)}{b} \]

7. Simplified 97.7%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}}\right)\right)}{b} \]
8. Step-by-step derivation

   1. associate-*r/ 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(\color{blue}{\frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

9. Applied egg-rr 97.7%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(\color{blue}{\frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]
10. Step-by-step derivation

    1. associate-*r/ 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(\color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    2. associate-/l* 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    3. rem-square-sqrt 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)} \cdot \frac{{c}^{2}}{{b}^{2}}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    4. unpow2 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\left(\sqrt{a} \cdot \sqrt{a}\right) \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    5. unpow2 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\left(\sqrt{a} \cdot \sqrt{a}\right) \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    6. times-frac 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\left(\sqrt{a} \cdot \sqrt{a}\right) \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    7. swap-sqr 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \color{blue}{\left(\left(\sqrt{a} \cdot \frac{c}{b}\right) \cdot \left(\sqrt{a} \cdot \frac{c}{b}\right)\right)} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    8. associate-/l* 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\color{blue}{\frac{\sqrt{a} \cdot c}{b}} \cdot \left(\sqrt{a} \cdot \frac{c}{b}\right)\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    9. *-commutative 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\frac{\color{blue}{c \cdot \sqrt{a}}}{b} \cdot \left(\sqrt{a} \cdot \frac{c}{b}\right)\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    10. associate-/l* 97.7%

        \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\frac{c \cdot \sqrt{a}}{b} \cdot \color{blue}{\frac{\sqrt{a} \cdot c}{b}}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    11. *-commutative 97.7%

        \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\frac{c \cdot \sqrt{a}}{b} \cdot \frac{\color{blue}{c \cdot \sqrt{a}}}{b}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    12. unpow2 97.7%

        \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \color{blue}{{\left(\frac{c \cdot \sqrt{a}}{b}\right)}^{2}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

    13. associate-/l* 97.7%

        \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot {\color{blue}{\left(c \cdot \frac{\sqrt{a}}{b}\right)}}^{2} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

11. Simplified 97.7%

    \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(\color{blue}{-0.375 \cdot {\left(c \cdot \frac{\sqrt{a}}{b}\right)}^{2}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

12. Final simplification 97.7%

    \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot {\left(c \cdot \frac{\sqrt{a}}{b}\right)}^{2} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]
13. Add Preprocessing

Alternative 2: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (-
   (*
    c
    (fma
     -0.375
     (/ a (pow b 3.0))
     (*
      c
      (fma
       -0.5625
       (/ (pow a 2.0) (pow b 5.0))
       (*
        -0.16666666666666666
        (* c (/ (* (/ (pow a 4.0) (pow b 6.0)) 6.328125) (* a b))))))))
   (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((c * fma(-0.375, (a / pow(b, 3.0)), (c * fma(-0.5625, (pow(a, 2.0) / pow(b, 5.0)), (-0.16666666666666666 * (c * (((pow(a, 4.0) / pow(b, 6.0)) * 6.328125) / (a * b)))))))) - (0.5 / b));
}

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * fma(-0.375, Float64(a / (b ^ 3.0)), Float64(c * fma(-0.5625, Float64((a ^ 2.0) / (b ^ 5.0)), Float64(-0.16666666666666666 * Float64(c * Float64(Float64(Float64((a ^ 4.0) / (b ^ 6.0)) * 6.328125) / Float64(a * b)))))))) - Float64(0.5 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(c * N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(c * N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in c around 0 97.3%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

5. Simplified 97.3%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

6. Final simplification 97.3%

   \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right) \]
7. Add Preprocessing

Alternative 3: 96.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * (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)) + (a * (((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0))) + ((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0)))))
end function

Java

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

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (-0.375 * (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(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in a around 0 96.9%

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

4. Final simplification 96.9%

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

Alternative 4: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.5625 (/ (* (pow a 2.0) c) (pow b 5.0)))
     (* -0.375 (/ a (pow b 3.0)))))
   (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-0.5625 * ((pow(a, 2.0) * c) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((c * (((-0.5625d0) * (((a ** 2.0d0) * c) / (b ** 5.0d0))) + ((-0.375d0) * (a / (b ** 3.0d0))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-0.5625 * ((Math.pow(a, 2.0) * c) / Math.pow(b, 5.0))) + (-0.375 * (a / Math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((c * ((-0.5625 * ((math.pow(a, 2.0) * c) / math.pow(b, 5.0))) + (-0.375 * (a / math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-0.5625 * (((a ^ 2.0) * c) / (b ^ 5.0))) + (-0.375 * (a / (b ^ 3.0))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in c around 0 96.5%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]

4. Final simplification 96.5%

   \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right) \]
5. Add Preprocessing

Alternative 5: 95.2% 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 17.5%

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

3. Taylor expanded in a around 0 95.4%

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

4. Final simplification 95.4%

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

Alternative 6: 95.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in b around inf 95.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(\color{blue}{\frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

5. Applied egg-rr 95.4%

   \[\leadsto \frac{-0.5 \cdot c + \color{blue}{\frac{-0.375 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}}}{b} \]
6. Step-by-step derivation

   1. associate-*r/ 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(\color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   2. associate-/l* 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   3. rem-square-sqrt 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)} \cdot \frac{{c}^{2}}{{b}^{2}}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   4. unpow2 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\left(\sqrt{a} \cdot \sqrt{a}\right) \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   5. unpow2 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\left(\sqrt{a} \cdot \sqrt{a}\right) \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   6. times-frac 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\left(\sqrt{a} \cdot \sqrt{a}\right) \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   7. swap-sqr 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \color{blue}{\left(\left(\sqrt{a} \cdot \frac{c}{b}\right) \cdot \left(\sqrt{a} \cdot \frac{c}{b}\right)\right)} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   8. associate-/l* 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\color{blue}{\frac{\sqrt{a} \cdot c}{b}} \cdot \left(\sqrt{a} \cdot \frac{c}{b}\right)\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   9. *-commutative 97.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\frac{\color{blue}{c \cdot \sqrt{a}}}{b} \cdot \left(\sqrt{a} \cdot \frac{c}{b}\right)\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   10. associate-/l* 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\frac{c \cdot \sqrt{a}}{b} \cdot \color{blue}{\frac{\sqrt{a} \cdot c}{b}}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   11. *-commutative 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \left(\frac{c \cdot \sqrt{a}}{b} \cdot \frac{\color{blue}{c \cdot \sqrt{a}}}{b}\right) + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   12. unpow2 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \color{blue}{{\left(\frac{c \cdot \sqrt{a}}{b}\right)}^{2}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

   13. associate-/l* 97.7%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot {\color{blue}{\left(c \cdot \frac{\sqrt{a}}{b}\right)}}^{2} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b} \]

7. Simplified 95.4%

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

8. Final simplification 95.4%

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

Alternative 7: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in a around 0 94.9%

   \[\leadsto \frac{\color{blue}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]

4. Taylor expanded in c around 0 95.1%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
5. Step-by-step derivation

   1. associate-/l* 95.1%

      \[\leadsto c \cdot \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{3}}\right)} - 0.5 \cdot \frac{1}{b}\right) \]

   2. associate-*r/ 95.1%

      \[\leadsto c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   3. metadata-eval 95.1%

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

6. Simplified 95.1%

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

7. Final simplification 95.1%

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

Alternative 8: 94.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c}{b}\\ \frac{\frac{\left(a \cdot c\right) \cdot -1.5 + -1.125 \cdot \left(t\_0 \cdot t\_0\right)}{b}}{a \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* a c) b)))
   (/ (/ (+ (* (* a c) -1.5) (* -1.125 (* t_0 t_0))) b) (* a 3.0))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) / b;
	return ((((a * c) * -1.5) + (-1.125 * (t_0 * t_0))) / b) / (a * 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
    real(8) :: t_0
    t_0 = (a * c) / b
    code = ((((a * c) * (-1.5d0)) + ((-1.125d0) * (t_0 * t_0))) / b) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) / b;
	return ((((a * c) * -1.5) + (-1.125 * (t_0 * t_0))) / b) / (a * 3.0);
}

Python

def code(a, b, c):
	t_0 = (a * c) / b
	return ((((a * c) * -1.5) + (-1.125 * (t_0 * t_0))) / b) / (a * 3.0)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) / b)
	return Float64(Float64(Float64(Float64(Float64(a * c) * -1.5) + Float64(-1.125 * Float64(t_0 * t_0))) / b) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (a * c) / b;
	tmp = ((((a * c) * -1.5) + (-1.125 * (t_0 * t_0))) / b) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]}, N[(N[(N[(N[(N[(a * c), $MachinePrecision] * -1.5), $MachinePrecision] + N[(-1.125 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in b around inf 94.8%

   \[\leadsto \frac{\color{blue}{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}}{b}}}{3 \cdot a} \]
4. Step-by-step derivation

   1. pow-prod-down 94.8%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{2}}}{b}}{3 \cdot a} \]

   2. pow2 94.8%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{{b}^{2}}}{b}}{3 \cdot a} \]

   3. unpow2 94.8%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{\color{blue}{b \cdot b}}}{b}}{3 \cdot a} \]

   4. times-frac 94.8%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left(\frac{a \cdot c}{b} \cdot \frac{a \cdot c}{b}\right)}}{b}}{3 \cdot a} \]

5. Applied egg-rr 94.8%

   \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left(\frac{a \cdot c}{b} \cdot \frac{a \cdot c}{b}\right)}}{b}}{3 \cdot a} \]

6. Final simplification 94.8%

   \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -1.5 + -1.125 \cdot \left(\frac{a \cdot c}{b} \cdot \frac{a \cdot c}{b}\right)}{b}}{a \cdot 3} \]
7. Add Preprocessing

Alternative 9: 89.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.5%

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

3. Simplified 17.6%

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

5. Taylor expanded in a around 0 12.5%

   \[\leadsto \frac{\color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{3 \cdot a} \]

6. Taylor expanded in b around 0 90.5%

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

   1. associate-*r/ 90.5%

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

   2. *-commutative 90.5%

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

   3. associate-/l* 90.2%

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

8. Simplified 90.2%

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

9. Final simplification 90.2%

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

Alternative 10: 90.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.5%

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

3. Taylor expanded in b around inf 90.5%

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

   1. associate-*r/ 90.5%

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

   2. *-commutative 90.5%

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

5. Simplified 90.5%

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

6. Final simplification 90.5%

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

Alternative 11: 3.3% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 17.5%

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

3. Simplified 17.6%

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

5. Taylor expanded in a around 0 12.5%

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

   1. expm1-log1p-u 8.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right) - b}{3 \cdot a}\right)\right)} \]

   2. expm1-undefine 6.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right) - b}{3 \cdot a}\right)} - 1} \]

   3. +-commutative 6.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(-1.5 \cdot \frac{a \cdot c}{b} + b\right)} - b}{3 \cdot a}\right)} - 1 \]

   4. fma-define 6.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, b\right)} - b}{3 \cdot a}\right)} - 1 \]

   5. associate-/l* 6.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1.5, \color{blue}{a \cdot \frac{c}{b}}, b\right) - b}{3 \cdot a}\right)} - 1 \]

   6. *-commutative 6.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1.5, a \cdot \frac{c}{b}, b\right) - b}{\color{blue}{a \cdot 3}}\right)} - 1 \]

7. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1.5, a \cdot \frac{c}{b}, b\right) - b}{a \cdot 3}\right)} - 1} \]
8. Step-by-step derivation

   1. sub-neg 6.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1.5, a \cdot \frac{c}{b}, b\right) - b}{a \cdot 3}\right)} + \left(-1\right)} \]

   2. metadata-eval 6.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1.5, a \cdot \frac{c}{b}, b\right) - b}{a \cdot 3}\right)} + \color{blue}{-1} \]

   3. +-commutative 6.7%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1.5, a \cdot \frac{c}{b}, b\right) - b}{a \cdot 3}\right)}} \]

   4. log1p-undefine 6.7%

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

   5. rem-exp-log 10.6%

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

   6. +-commutative 10.6%

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

   7. *-lft-identity 10.6%

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

   8. *-commutative 10.6%

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

   9. times-frac 10.6%

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

   10. metadata-eval 10.6%

       \[\leadsto -1 + \left(\color{blue}{0.3333333333333333} \cdot \frac{\mathsf{fma}\left(-1.5, a \cdot \frac{c}{b}, b\right) - b}{a} + 1\right) \]

   11. fma-define 10.6%

       \[\leadsto -1 + \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\mathsf{fma}\left(-1.5, a \cdot \frac{c}{b}, b\right) - b}{a}, 1\right)} \]

9. Simplified 31.6%

   \[\leadsto \color{blue}{-1 + \mathsf{fma}\left(0.3333333333333333, \frac{\left(a \cdot c\right) \cdot \frac{-1.5}{b}}{a}, 1\right)} \]

10. Taylor expanded in c around 0 3.3%

    \[\leadsto -1 + \color{blue}{1} \]

11. Final simplification 3.3%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))