Cubic critical, wide range

Percentage Accurate: 17.8% → 97.8%

Time: 17.2s

Alternatives: 6

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% 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 15.3%

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

3. Simplified 15.3%

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

5. Taylor expanded in a around 0 98.2%

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

6. Taylor expanded in c around 0 98.2%

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

   1. associate-*r/ 98.2%

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

   2. associate-*r* 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 2: 96.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.3%

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

3. Simplified 15.3%

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

5. Taylor expanded in a around 0 97.5%

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

6. Taylor expanded in c around inf 97.5%

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

   1. associate-*r/ 97.5%

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

   2. associate-*r/ 97.5%

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

   3. metadata-eval 97.5%

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

   4. *-commutative 97.5%

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

8. Simplified 97.5%

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

9. Final simplification 97.5%

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

Alternative 3: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{c \cdot a}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (+
   (* -2.0 (/ b c))
   (* a (+ (* 1.125 (/ (* c a) (pow b 3.0))) (* 1.5 (/ 1.0 b)))))))

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / pow(b, 3.0))) + (1.5 * (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 = 1.0d0 / (((-2.0d0) * (b / c)) + (a * ((1.125d0 * ((c * a) / (b ** 3.0d0))) + (1.5d0 * (1.0d0 / b)))))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / Math.pow(b, 3.0))) + (1.5 * (1.0 / b)))));
}

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / math.pow(b, 3.0))) + (1.5 * (1.0 / b)))))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(a * Float64(Float64(1.125 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(1.5 * Float64(1.0 / b))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / (b ^ 3.0))) + (1.5 * (1.0 / b)))));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(1.125 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.3%

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

   1. expm1-log1p-u 15.3%

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

   2. expm1-undefine 10.9%

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

4. Applied egg-rr 10.9%

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

   1. expm1-define 15.3%

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

6. Simplified 15.3%

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

   1. expm1-log1p-u 15.3%

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

   2. clear-num 15.3%

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

   3. inv-pow 15.3%

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

   4. *-commutative 15.3%

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

   5. neg-mul-1 15.3%

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

   6. fma-define 15.3%

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

   7. pow2 15.3%

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

   8. *-commutative 15.3%

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

   9. *-commutative 15.3%

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

8. Applied egg-rr 15.3%

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

   1. unpow-1 15.3%

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

   2. unpow2 15.3%

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

   3. fmm-def 15.3%

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

   4. associate-*r* 15.3%

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

   5. *-commutative 15.3%

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

   6. distribute-rgt-neg-in 15.3%

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

   7. metadata-eval 15.3%

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

10. Simplified 15.3%

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

11. Taylor expanded in b around inf 97.1%

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

    1. fma-define 97.1%

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

    2. distribute-rgt-out 97.1%

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

    3. *-commutative 97.1%

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

    4. metadata-eval 97.1%

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

    5. associate-*r/ 97.1%

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

    6. metadata-eval 97.1%

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

13. Simplified 97.1%

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

14. Taylor expanded in a around 0 97.3%

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

15. Final simplification 97.3%

    \[\leadsto \frac{1}{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{c \cdot a}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)} \]
16. Add Preprocessing

Alternative 4: 95.1% 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 15.3%

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

   1. expm1-log1p-u 15.3%

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

   2. expm1-undefine 10.9%

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

4. Applied egg-rr 10.9%

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

   1. expm1-define 15.3%

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

6. Simplified 15.3%

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

   1. expm1-log1p-u 15.3%

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

   2. clear-num 15.3%

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

   3. inv-pow 15.3%

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

   4. *-commutative 15.3%

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

   5. neg-mul-1 15.3%

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

   6. fma-define 15.3%

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

   7. pow2 15.3%

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

   8. *-commutative 15.3%

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

   9. *-commutative 15.3%

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

8. Applied egg-rr 15.3%

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

   1. unpow-1 15.3%

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

   2. unpow2 15.3%

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

   3. fmm-def 15.3%

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

   4. associate-*r* 15.3%

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

   5. *-commutative 15.3%

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

   6. distribute-rgt-neg-in 15.3%

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

   7. metadata-eval 15.3%

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

10. Simplified 15.3%

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

11. Taylor expanded in b around inf 97.1%

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

    1. fma-define 97.1%

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

    2. distribute-rgt-out 97.1%

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

    3. *-commutative 97.1%

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

    4. metadata-eval 97.1%

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

    5. associate-*r/ 97.1%

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

    6. metadata-eval 97.1%

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

13. Simplified 97.1%

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

14. Taylor expanded in a around 0 96.0%

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

Alternative 5: 90.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.3%

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

3. Simplified 15.3%

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

5. Taylor expanded in b around inf 92.1%

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

Alternative 6: 3.3% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 15.3%

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

   1. expm1-log1p-u 15.3%

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

   2. expm1-undefine 10.9%

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

4. Applied egg-rr 10.9%

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

   1. expm1-define 15.3%

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

6. Simplified 15.3%

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

   1. expm1-log1p-u 15.3%

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

   2. clear-num 15.3%

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

   3. inv-pow 15.3%

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

   4. *-commutative 15.3%

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

   5. neg-mul-1 15.3%

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

   6. fma-define 15.3%

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

   7. pow2 15.3%

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

   8. *-commutative 15.3%

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

   9. *-commutative 15.3%

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

8. Applied egg-rr 15.3%

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

   1. unpow-1 15.3%

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

   2. unpow2 15.3%

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

   3. fmm-def 15.3%

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

   4. associate-*r* 15.3%

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

   5. *-commutative 15.3%

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

   6. distribute-rgt-neg-in 15.3%

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

   7. metadata-eval 15.3%

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

10. Simplified 15.3%

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

11. Taylor expanded in a around 0 3.3%

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

    1. associate-*r/ 3.3%

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

    2. distribute-rgt1-in 3.3%

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

    3. metadata-eval 3.3%

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

    4. mul0-lft 3.3%

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

    5. metadata-eval 3.3%

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

13. Simplified 3.3%

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

14. Taylor expanded in a around 0 3.3%

    \[\leadsto \color{blue}{0} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))