Cubic critical, medium range

Percentage Accurate: 30.9% → 95.4%

Time: 36.7s

Alternatives: 9

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: 30.9% 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

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

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)))
      (/ (* (* a -1.0546875) (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))) + (((a * -1.0546875) * 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))) + (((a * (-1.0546875d0)) * (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))) + (((a * -1.0546875) * 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))) + (((a * -1.0546875) * 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(Float64(Float64(a * -1.0546875) * (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))) + (((a * -1.0546875) * (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[(N[(N[(a * -1.0546875), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

3. Taylor expanded in a around 0 96.0%

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

4. Taylor expanded in c around 0 96.0%

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

   1. associate-*r/ 96.0%

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

   2. associate-*r* 96.0%

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

6. Simplified 96.0%

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

7. Final simplification 96.0%

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

Alternative 2: 95.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

3. Taylor expanded in c around 0 95.8%

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

5. Simplified 95.8%

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

6. Taylor expanded in c around 0 95.8%

   \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]

7. Taylor expanded in a around 0 95.8%

   \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right)} - \frac{0.5}{b}\right) \]

8. Taylor expanded in b around 0 95.8%

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

Alternative 3: 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 30.9%

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

3. Taylor expanded in a around 0 94.5%

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

4. Final simplification 94.5%

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

Alternative 4: 93.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (*
   b
   (-
    (fma
     -3.0
     (/ (* -0.375 (* c (pow a 2.0))) (pow b 4.0))
     (* 1.5 (/ a (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)), (1.5 * (a / 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(1.5 * Float64(a / (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[(1.5 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

   1. cancel-sign-sub-inv 30.9%

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

   2. *-commutative 30.9%

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

   3. distribute-rgt-neg-in 30.9%

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

   4. metadata-eval 30.9%

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

   5. associate-*r* 30.9%

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

   6. *-commutative 30.9%

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

   7. flip-+ 30.8%

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

   8. pow2 30.8%

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

   9. pow2 30.8%

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

   10. pow-prod-up 30.9%

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

   11. metadata-eval 30.9%

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

   12. pow2 30.9%

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

   13. pow2 30.9%

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

4. Applied egg-rr 30.9%

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

   1. add-cbrt-cube 30.9%

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

   2. pow3 30.9%

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

6. Applied egg-rr 30.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}{{b}^{2} - a \cdot \left(c \cdot -3\right)}}\right)}{a \cdot 3}\right)}^{3}}} \]
7. Step-by-step derivation

   1. rem-cbrt-cube 31.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}{{b}^{2} - a \cdot \left(c \cdot -3\right)}}\right)}{a \cdot 3}} \]

   2. clear-num 31.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}{{b}^{2} - a \cdot \left(c \cdot -3\right)}}\right)}}} \]

   3. associate-*r* 31.0%

      \[\leadsto \frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot -3}}}\right)}} \]

8. Applied egg-rr 31.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}{{b}^{2} - \left(a \cdot c\right) \cdot -3}}\right)}}} \]

9. Taylor expanded in b around inf 94.4%

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

    1. fma-define 94.4%

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

    2. distribute-rgt-out 94.4%

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

    3. *-commutative 94.4%

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

    4. metadata-eval 94.4%

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

    5. associate-*r/ 94.4%

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

    6. metadata-eval 94.4%

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

11. Simplified 94.4%

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

12. Final simplification 94.4%

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

Alternative 5: 93.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

3. Taylor expanded in c around 0 95.8%

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

5. Simplified 95.8%

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

6. Taylor expanded in c around 0 95.8%

   \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]

7. Taylor expanded in a around 0 94.2%

   \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right)} - \frac{0.5}{b}\right) \]
8. Step-by-step derivation

   1. associate-*r/ 94.2%

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

   2. associate-*r/ 94.2%

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

   3. metadata-eval 94.2%

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

9. Simplified 94.2%

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

10. Final simplification 94.2%

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

Alternative 6: 90.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

3. Taylor expanded in c around 0 90.7%

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

   1. clear-num 90.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}} \]

   2. inv-pow 90.7%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}\right)}^{-1}} \]

   3. *-commutative 90.7%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}\right)}^{-1} \]

   4. fma-define 90.6%

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

   5. associate-*r/ 90.6%

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

   6. *-commutative 90.6%

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

5. Applied egg-rr 90.6%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{c \cdot \mathsf{fma}\left(-1.5, \frac{a}{b}, \frac{-1.125 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}}\right)}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 90.6%

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

   2. times-frac 90.7%

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

   3. *-commutative 90.7%

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

   4. associate-*r/ 90.7%

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

   5. associate-*r/ 90.7%

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

   6. *-commutative 90.7%

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

   7. *-commutative 90.7%

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

   8. associate-*l* 90.7%

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

7. Simplified 90.7%

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

8. Taylor expanded in a around 0 91.4%

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
9. Add Preprocessing

Alternative 7: 81.6% 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 30.9%

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

3. Taylor expanded in b around inf 81.6%

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

   1. associate-*r/ 81.6%

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

   2. *-commutative 81.6%

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

5. Simplified 81.6%

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

6. Final simplification 81.6%

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

Alternative 8: 81.4% 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 30.9%

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

3. Taylor expanded in c around 0 91.0%

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

4. Taylor expanded in a around 0 81.4%

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

Alternative 9: 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 30.9%

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

   1. cancel-sign-sub-inv 30.9%

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

   2. *-commutative 30.9%

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

   3. distribute-rgt-neg-in 30.9%

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

   4. metadata-eval 30.9%

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

   5. associate-*r* 30.9%

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

   6. *-commutative 30.9%

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

   7. flip-+ 30.8%

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

   8. pow2 30.8%

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

   9. pow2 30.8%

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

   10. pow-prod-up 30.9%

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

   11. metadata-eval 30.9%

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

   12. pow2 30.9%

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

   13. pow2 30.9%

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

4. Applied egg-rr 30.9%

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

   1. add-cbrt-cube 30.9%

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

   2. pow3 30.9%

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

6. Applied egg-rr 30.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}{{b}^{2} - a \cdot \left(c \cdot -3\right)}}\right)}{a \cdot 3}\right)}^{3}}} \]

7. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
8. 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%

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

9. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))