Cubic critical, medium range

Percentage Accurate: 31.4% → 95.4%

Time: 15.1s

Alternatives: 12

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: 31.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (*
   a
   (+
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
    (*
     a
     (+
      (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
      (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0)))))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (-1.0546875 * ((a * pow(c, 4.0)) / 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 = ((-0.5d0) * (c / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * (((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-1.0546875d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0)))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * ((-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-1.0546875 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0)))))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * ((-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-1.0546875 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0)))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((-0.5625 * ((c ^ 3.0) / (b ^ 5.0))) + (-1.0546875 * ((a * (c ^ 4.0)) / (b ^ 7.0)))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.3%

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

   1. sqr-neg 29.3%

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

   2. sqr-neg 29.3%

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

   3. associate-*l* 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in a around 0 95.4%

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

6. Taylor expanded in c around 0 95.4%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
7. Add Preprocessing

Alternative 2: 93.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

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 * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + ((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.3%

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

   1. sqr-neg 29.3%

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

   2. sqr-neg 29.3%

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

   3. associate-*l* 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in a around 0 94.2%

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

6. Final simplification 94.2%

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

Alternative 3: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{-0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \frac{2}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (*
   b
   (-
    (fma
     -3.0
     (/ (* -0.375 (* c (pow a 2.0))) (pow b 4.0))
     (/ (* a 1.5) (pow b 2.0)))
    (/ 2.0 c)))))

C

double code(double a, double b, double c) {
	return 1.0 / (b * (fma(-3.0, ((-0.375 * (c * pow(a, 2.0))) / pow(b, 4.0)), ((a * 1.5) / pow(b, 2.0))) - (2.0 / c)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(b * Float64(fma(-3.0, Float64(Float64(-0.375 * Float64(c * (a ^ 2.0))) / (b ^ 4.0)), Float64(Float64(a * 1.5) / (b ^ 2.0))) - Float64(2.0 / c))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(b * N[(N[(-3.0 * N[(N[(-0.375 * N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.3%

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

   1. sqr-neg 29.3%

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

   2. sqr-neg 29.3%

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

   3. associate-*l* 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in a around inf 29.3%

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

   1. clear-num 29.3%

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

   2. inv-pow 29.3%

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

   3. neg-mul-1 29.3%

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

   4. fma-define 29.3%

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

   5. cancel-sign-sub-inv 29.3%

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

   6. metadata-eval 29.3%

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

7. Applied egg-rr 29.3%

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

   1. unpow-1 29.3%

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

   2. associate-/l* 29.3%

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

   3. +-commutative 29.3%

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

   4. fma-define 29.3%

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

9. Simplified 29.3%

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

10. Taylor expanded in b around inf 94.1%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)}} \]
11. Step-by-step derivation

    1. fma-define 94.1%

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

    2. distribute-rgt-out 94.1%

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

    3. *-commutative 94.1%

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

    4. metadata-eval 94.1%

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

    5. associate-*r/ 94.1%

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

    6. *-commutative 94.1%

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

    7. associate-*r/ 94.1%

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

    8. metadata-eval 94.1%

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

12. Simplified 94.1%

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

13. Final simplification 94.1%

    \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{-0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \frac{2}{c}\right)} \]
14. Add Preprocessing

Alternative 4: 93.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{b \cdot \left(\frac{-3 \cdot \left(-0.375 \cdot \left(c \cdot {a}^{2}\right)\right)}{{b}^{4}} + \mathsf{fma}\left(1.5, \frac{a}{{b}^{2}}, \frac{-2}{c}\right)\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (*
   b
   (+
    (/ (* -3.0 (* -0.375 (* c (pow a 2.0)))) (pow b 4.0))
    (fma 1.5 (/ a (pow b 2.0)) (/ -2.0 c))))))

C

double code(double a, double b, double c) {
	return 1.0 / (b * (((-3.0 * (-0.375 * (c * pow(a, 2.0)))) / pow(b, 4.0)) + fma(1.5, (a / pow(b, 2.0)), (-2.0 / c))));
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(b * Float64(Float64(Float64(-3.0 * Float64(-0.375 * Float64(c * (a ^ 2.0)))) / (b ^ 4.0)) + fma(1.5, Float64(a / (b ^ 2.0)), Float64(-2.0 / c)))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(b * N[(N[(N[(-3.0 * N[(-0.375 * N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.3%

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

   1. sqr-neg 29.3%

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

   2. sqr-neg 29.3%

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

   3. associate-*l* 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in a around inf 29.3%

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

   1. clear-num 29.3%

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

   2. inv-pow 29.3%

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

   3. neg-mul-1 29.3%

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

   4. fma-define 29.3%

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

   5. cancel-sign-sub-inv 29.3%

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

   6. metadata-eval 29.3%

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

7. Applied egg-rr 29.3%

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

   1. unpow-1 29.3%

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

   2. associate-/l* 29.3%

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

   3. +-commutative 29.3%

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

   4. fma-define 29.3%

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

9. Simplified 29.3%

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

10. Taylor expanded in b around inf 94.1%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)}} \]
11. Step-by-step derivation

    1. associate--l+ 94.1%

       \[\leadsto \frac{1}{b \cdot \color{blue}{\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)}} \]

    2. associate-*r/ 94.1%

       \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\frac{-3 \cdot \left(-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)\right)}{{b}^{4}}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)} \]

    3. distribute-rgt-out 94.1%

       \[\leadsto \frac{1}{b \cdot \left(\frac{-3 \cdot \color{blue}{\left(\left({a}^{2} \cdot c\right) \cdot \left(-0.75 + 0.375\right)\right)}}{{b}^{4}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)} \]

    4. *-commutative 94.1%

       \[\leadsto \frac{1}{b \cdot \left(\frac{-3 \cdot \left(\color{blue}{\left(c \cdot {a}^{2}\right)} \cdot \left(-0.75 + 0.375\right)\right)}{{b}^{4}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)} \]

    5. metadata-eval 94.1%

       \[\leadsto \frac{1}{b \cdot \left(\frac{-3 \cdot \left(\left(c \cdot {a}^{2}\right) \cdot \color{blue}{-0.375}\right)}{{b}^{4}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)} \]

    6. fma-neg 94.1%

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

    7. associate-*r/ 94.1%

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

    8. metadata-eval 94.1%

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

    9. distribute-neg-frac 94.1%

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

    10. metadata-eval 94.1%

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

12. Simplified 94.1%

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

13. Final simplification 94.1%

    \[\leadsto \frac{1}{b \cdot \left(\frac{-3 \cdot \left(-0.375 \cdot \left(c \cdot {a}^{2}\right)\right)}{{b}^{4}} + \mathsf{fma}\left(1.5, \frac{a}{{b}^{2}}, \frac{-2}{c}\right)\right)} \]
14. Add Preprocessing

Alternative 5: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -22000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -22000.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (/ (+ (* -0.5 c) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -22000.0) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = ((-0.5 * c) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0)))) / 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)) <= (-22000.0d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = (((-0.5d0) * c) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 2.0d0)))) / 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)) <= -22000.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = ((-0.5 * c) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 2.0)))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -22000.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = ((-0.5 * c) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 2.0)))) / 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)) <= -22000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(-0.5 * c) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / 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)) <= -22000.0)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = ((-0.5 * c) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 2.0)))) / 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], -22000.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * c), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -22000

   1. Initial program 82.7%

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

      1. sqr-neg 82.7%

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

      2. sqr-neg 82.7%

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

      3. associate-*l* 82.8%

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

   3. Simplified 82.8%

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

   if -22000 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 25.5%

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

      1. sqr-neg 25.5%

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

      2. sqr-neg 25.5%

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

      3. associate-*l* 25.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in b around inf 94.1%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -22000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -22000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -22000.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -22000.0) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	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)) <= (-22000.0d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
    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)) <= -22000.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -22000.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))
	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)) <= -22000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	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)) <= -22000.0)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
	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], -22000.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -22000

   1. Initial program 82.7%

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

      1. sqr-neg 82.7%

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

      2. sqr-neg 82.7%

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

      3. associate-*l* 82.8%

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

   3. Simplified 82.8%

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

   if -22000 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 25.5%

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

      1. sqr-neg 25.5%

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

      2. sqr-neg 25.5%

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

      3. associate-*l* 25.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in a around 0 94.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -22000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.5625 (/ (* c (pow a 2.0)) (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 * ((c * pow(a, 2.0)) / 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) * ((c * (a ** 2.0d0)) / (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 * ((c * Math.pow(a, 2.0)) / 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 * ((c * math.pow(a, 2.0)) / 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(c * (a ^ 2.0)) / (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 * ((c * (a ^ 2.0)) / (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[(c * N[Power[a, 2.0], $MachinePrecision]), $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 29.3%

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

   1. sqr-neg 29.3%

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

   2. sqr-neg 29.3%

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

   3. associate-*l* 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in c around 0 93.9%

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

6. Final simplification 93.9%

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

Alternative 8: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -22000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -22000.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (/ 1.0 (* b (- (/ (* a 1.5) (pow b 2.0)) (/ 2.0 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -22000.0) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / pow(b, 2.0)) - (2.0 / c)));
	}
	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)) <= (-22000.0d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (b * (((a * 1.5d0) / (b ** 2.0d0)) - (2.0d0 / c)))
    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)) <= -22000.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / Math.pow(b, 2.0)) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -22000.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / (b * (((a * 1.5) / math.pow(b, 2.0)) - (2.0 / c)))
	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)) <= -22000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(Float64(a * 1.5) / (b ^ 2.0)) - Float64(2.0 / c))));
	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)) <= -22000.0)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = 1.0 / (b * (((a * 1.5) / (b ^ 2.0)) - (2.0 / c)));
	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], -22000.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -22000

   1. Initial program 82.7%

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

      1. sqr-neg 82.7%

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

      2. sqr-neg 82.7%

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

      3. associate-*l* 82.8%

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

   3. Simplified 82.8%

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

   if -22000 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 25.5%

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

      1. sqr-neg 25.5%

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

      2. sqr-neg 25.5%

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

      3. associate-*l* 25.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in a around inf 25.5%

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

      1. clear-num 25.5%

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

      2. inv-pow 25.5%

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

      3. neg-mul-1 25.5%

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

      4. fma-define 25.5%

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

      5. cancel-sign-sub-inv 25.5%

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

      6. metadata-eval 25.5%

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

   7. Applied egg-rr 25.5%

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

      1. unpow-1 25.5%

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

      2. associate-/l* 25.5%

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

      3. +-commutative 25.5%

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

      4. fma-define 25.5%

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

   9. Simplified 25.5%

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

   10. Taylor expanded in b around inf 93.9%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 93.9%

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

       2. *-commutative 93.9%

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

       3. associate-*r/ 93.9%

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

       4. metadata-eval 93.9%

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

   12. Simplified 93.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -22000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (b * (((a * 1.5d0) / (b ** 2.0d0)) - (2.0d0 / c)))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (b * (((a * 1.5) / Math.pow(b, 2.0)) - (2.0 / c)));
}

Python

def code(a, b, c):
	return 1.0 / (b * (((a * 1.5) / math.pow(b, 2.0)) - (2.0 / c)))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (b * (((a * 1.5) / (b ^ 2.0)) - (2.0 / c)));
end

Wolfram

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

Derivation

1. Initial program 29.3%

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

   1. sqr-neg 29.3%

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

   2. sqr-neg 29.3%

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

   3. associate-*l* 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in a around inf 29.3%

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

   1. clear-num 29.3%

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

   2. inv-pow 29.3%

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

   3. neg-mul-1 29.3%

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

   4. fma-define 29.3%

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

   5. cancel-sign-sub-inv 29.3%

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

   6. metadata-eval 29.3%

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

7. Applied egg-rr 29.3%

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

   1. unpow-1 29.3%

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

   2. associate-/l* 29.3%

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

   3. +-commutative 29.3%

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

   4. fma-define 29.3%

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

9. Simplified 29.3%

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

10. Taylor expanded in b around inf 91.5%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
11. Step-by-step derivation

    1. associate-*r/ 91.5%

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

    2. *-commutative 91.5%

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

    3. associate-*r/ 91.5%

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

    4. metadata-eval 91.5%

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

12. Simplified 91.5%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}} \]
13. Add Preprocessing

Alternative 10: 90.4% 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 29.3%

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

   1. sqr-neg 29.3%

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

   2. sqr-neg 29.3%

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

   3. associate-*l* 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in a around inf 29.3%

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

6. Taylor expanded in c around 0 91.3%

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

   1. associate-/l* 91.3%

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

   2. associate-*r/ 91.3%

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

   3. metadata-eval 91.3%

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

8. Simplified 91.3%

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

Alternative 11: 81.3% accurate, 23.2× speedup

Math

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

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(Float64(-0.5 * c) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.3%

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

   1. sqr-neg 29.3%

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

   2. sqr-neg 29.3%

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

   3. associate-*l* 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in b around inf 82.6%

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

   1. associate-*r/ 82.6%

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

   2. *-commutative 82.6%

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

7. Simplified 82.6%

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

8. Final simplification 82.6%

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

Alternative 12: 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 29.3%

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

   1. sqr-neg 29.3%

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

   2. sqr-neg 29.3%

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

   3. associate-*l* 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in a around inf 29.3%

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

   1. add-log-exp 21.7%

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

   2. neg-mul-1 21.7%

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

   3. fma-define 21.7%

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

   4. cancel-sign-sub-inv 21.7%

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

   5. metadata-eval 21.7%

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

7. Applied egg-rr 21.7%

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

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. Add Preprocessing

Reproduce

herbie shell --seed 2024092 
(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)))