Cubic critical, medium range

Percentage Accurate: 31.5% → 95.5%

Time: 14.9s

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: 31.5% 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: 95.5% accurate, 0.2× speedup

Math

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

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

Derivation

1. Initial program 31.0%

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

3. Taylor expanded in a around 0 95.2%

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

4. Taylor expanded in c around 0 95.2%

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

   1. associate-/l* 95.2%

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

   2. associate-*r* 95.2%

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

6. Simplified 95.2%

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

7. Final simplification 95.2%

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

Alternative 2: 95.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.0%

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

3. Taylor expanded in c around 0 94.9%

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

5. Simplified 94.9%

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

6. Taylor expanded in c around 0 94.9%

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

7. Final simplification 94.9%

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

Alternative 3: 94.0% 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 31.0%

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

3. Taylor expanded in a around 0 93.6%

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

4. Final simplification 93.6%

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

Alternative 4: 93.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.0%

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

3. Taylor expanded in c around 0 94.9%

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

5. Simplified 94.9%

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

6. Taylor expanded in c around 0 93.3%

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

7. Final simplification 93.3%

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

Alternative 5: 90.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 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. Initial program 31.0%

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

3. Taylor expanded in b around inf 90.7%

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

Alternative 6: 90.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.0%

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

3. Taylor expanded in a around 0 90.7%

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

Alternative 7: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.0%

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

   1. log1p-expm1-u 20.7%

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

   2. log1p-undefine 17.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

4. Applied egg-rr 17.9%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
5. Step-by-step derivation

   1. log1p-define 20.7%

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

   2. log1p-expm1-u 31.0%

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

   3. rem-cube-cbrt 31.0%

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

   4. add-sqr-sqrt 31.0%

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

   5. unpow-prod-down 31.0%

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

   6. pow1/3 31.0%

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

   7. sqrt-pow1 31.0%

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

   8. metadata-eval 31.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{3 \cdot a}}\right)}^{3}} \]

   9. pow1/3 31.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt{\color{blue}{{\left(3 \cdot a\right)}^{0.3333333333333333}}}\right)}^{3}} \]

   10. sqrt-pow1 31.0%

       \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot {\color{blue}{\left({\left(3 \cdot a\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3}} \]

   11. metadata-eval 31.0%

       \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(3 \cdot a\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3}} \]

6. Applied egg-rr 31.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3}}} \]
7. Step-by-step derivation

   1. add-log-exp 23.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot \color{blue}{\log \left(e^{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3}}\right)}} \]

   2. pow-pow 23.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot \log \left(e^{\color{blue}{{\left(3 \cdot a\right)}^{\left(0.16666666666666666 \cdot 3\right)}}}\right)} \]

   3. metadata-eval 23.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot \log \left(e^{{\left(3 \cdot a\right)}^{\color{blue}{0.5}}}\right)} \]

   4. pow1/2 23.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot \log \left(e^{\color{blue}{\sqrt{3 \cdot a}}}\right)} \]

8. Applied egg-rr 23.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot \color{blue}{\log \left(e^{\sqrt{3 \cdot a}}\right)}} \]

9. Taylor expanded in c around 0 89.8%

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

    1. associate-*r/ 89.8%

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

    2. *-commutative 89.8%

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

    3. times-frac 89.8%

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

    4. unpow2 89.8%

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

    5. rem-square-sqrt 89.8%

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

    6. metadata-eval 89.8%

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

    7. *-commutative 89.8%

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

    8. associate-/l* 89.8%

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

    9. associate-*r* 89.8%

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

    10. *-commutative 89.8%

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

    11. *-commutative 89.8%

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

    12. associate-*l/ 89.8%

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

    13. associate-*r/ 89.7%

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

    14. metadata-eval 89.7%

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

11. Simplified 90.4%

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

Alternative 8: 81.2% 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 31.0%

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

3. Taylor expanded in b around inf 81.4%

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

   1. associate-*r/ 81.4%

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

   2. *-commutative 81.4%

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

5. Simplified 81.4%

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

6. Final simplification 81.4%

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

Alternative 9: 81.0% 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 31.0%

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

   1. log1p-expm1-u 20.7%

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

   2. log1p-undefine 17.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

4. Applied egg-rr 17.9%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
5. Step-by-step derivation

   1. log1p-define 20.7%

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

   2. log1p-expm1-u 31.0%

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

   3. rem-cube-cbrt 31.0%

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

   4. add-sqr-sqrt 31.0%

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

   5. unpow-prod-down 31.0%

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

   6. pow1/3 31.0%

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

   7. sqrt-pow1 31.0%

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

   8. metadata-eval 31.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{3 \cdot a}}\right)}^{3}} \]

   9. pow1/3 31.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt{\color{blue}{{\left(3 \cdot a\right)}^{0.3333333333333333}}}\right)}^{3}} \]

   10. sqrt-pow1 31.0%

       \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot {\color{blue}{\left({\left(3 \cdot a\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3}} \]

   11. metadata-eval 31.0%

       \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(3 \cdot a\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3}} \]

6. Applied egg-rr 31.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(3 \cdot a\right)}^{0.16666666666666666}\right)}^{3}}} \]

7. Taylor expanded in b around inf 80.7%

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

   1. associate-*r/ 80.7%

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

   2. *-commutative 80.7%

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

   3. unpow2 80.7%

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

   4. rem-square-sqrt 81.1%

      \[\leadsto \frac{-1.5 \cdot c}{\color{blue}{3} \cdot b} \]

   5. times-frac 81.4%

      \[\leadsto \color{blue}{\frac{-1.5}{3} \cdot \frac{c}{b}} \]

   6. metadata-eval 81.4%

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

   7. associate-*r/ 81.4%

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

   8. *-commutative 81.4%

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

   9. associate-/l* 81.2%

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

9. Simplified 81.2%

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

Reproduce

herbie shell --seed 2024111 
(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)))