Cubic critical, narrow range

Percentage Accurate: 55.4% → 91.8%

Time: 17.4s

Alternatives: 15

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0115:\\ \;\;\;\;\frac{\sqrt{a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right)}}{a \cdot 3} - \frac{b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0115)
   (-
    (/ (sqrt (* a (fma c -3.0 (/ (pow b 2.0) a)))) (* a 3.0))
    (/ b (* a 3.0)))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (+
      (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
      (*
       a
       (+
        (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
        (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0115) {
		tmp = (sqrt((a * fma(c, -3.0, (pow(b, 2.0) / a)))) / (a * 3.0)) - (b / (a * 3.0));
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 7.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0115)
		tmp = Float64(Float64(sqrt(Float64(a * fma(c, -3.0, Float64((b ^ 2.0) / a)))) / Float64(a * 3.0)) - Float64(b / Float64(a * 3.0)));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0115], N[(N[(N[Sqrt[N[(a * N[(c * -3.0 + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision] - N[(b / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0115

   1. Initial program 92.5%

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

      1. /-rgt-identity 92.5%

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

      2. metadata-eval 92.5%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in a around inf 92.3%

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

      1. div-sub 93.3%

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

      2. *-commutative 93.3%

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

      3. fma-define 93.3%

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

      4. *-commutative 93.3%

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

      5. *-commutative 93.3%

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

   7. Applied egg-rr 93.3%

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

   if 0.0115 < b

   1. Initial program 53.6%

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

      1. /-rgt-identity 53.6%

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

      2. metadata-eval 53.6%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in a around 0 94.2%

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

   6. Taylor expanded in c around 0 94.2%

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

Alternative 2: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.012:\\ \;\;\;\;\frac{\sqrt{a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right)}}{a \cdot 3} - \frac{b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) + 0.375 \cdot \frac{-1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.012)
   (-
    (/ (sqrt (* a (fma c -3.0 (/ (pow b 2.0) a)))) (* a 3.0))
    (/ b (* a 3.0)))
   (*
    c
    (-
     (*
      c
      (*
       a
       (+
        (*
         a
         (+
          (* -1.0546875 (/ (* a (pow c 2.0)) (pow b 7.0)))
          (* -0.5625 (/ c (pow b 5.0)))))
        (* 0.375 (/ -1.0 (pow b 3.0))))))
     (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.012) {
		tmp = (sqrt((a * fma(c, -3.0, (pow(b, 2.0) / a)))) / (a * 3.0)) - (b / (a * 3.0));
	} else {
		tmp = c * ((c * (a * ((a * ((-1.0546875 * ((a * pow(c, 2.0)) / pow(b, 7.0))) + (-0.5625 * (c / pow(b, 5.0))))) + (0.375 * (-1.0 / pow(b, 3.0)))))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.012)
		tmp = Float64(Float64(sqrt(Float64(a * fma(c, -3.0, Float64((b ^ 2.0) / a)))) / Float64(a * 3.0)) - Float64(b / Float64(a * 3.0)));
	else
		tmp = 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 * Float64(-1.0 / (b ^ 3.0)))))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.012], N[(N[(N[Sqrt[N[(a * N[(c * -3.0 + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision] - N[(b / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.012

   1. Initial program 92.5%

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

      1. /-rgt-identity 92.5%

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

      2. metadata-eval 92.5%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in a around inf 92.3%

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

      1. div-sub 93.3%

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

      2. *-commutative 93.3%

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

      3. fma-define 93.3%

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

      4. *-commutative 93.3%

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

      5. *-commutative 93.3%

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

   7. Applied egg-rr 93.3%

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

   if 0.012 < b

   1. Initial program 53.6%

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

      1. /-rgt-identity 53.6%

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

      2. metadata-eval 53.6%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in c around 0 94.0%

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

   7. Simplified 94.0%

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

   8. Taylor expanded in a around 0 94.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.012:\\ \;\;\;\;\frac{\sqrt{a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right)}}{a \cdot 3} - \frac{b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) + 0.375 \cdot \frac{-1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{a \cdot 3}\\ \mathbf{if}\;b \leq 0.35:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{t\_0 \cdot {t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (cbrt (* a 3.0))))
   (if (<= b 0.35)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* t_0 (pow t_0 2.0)))
     (+
      (* -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) {
	double t_0 = cbrt((a * 3.0));
	double tmp;
	if (b <= 0.35) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (t_0 * pow(t_0, 2.0));
	} else {
		tmp = (-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)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.cbrt((a * 3.0));
	double tmp;
	if (b <= 0.35) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (t_0 * Math.pow(t_0, 2.0));
	} else {
		tmp = (-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)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = cbrt(Float64(a * 3.0))
	tmp = 0.0
	if (b <= 0.35)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(t_0 * (t_0 ^ 2.0)));
	else
		tmp = 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
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * 3.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[b, 0.35], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 0.34999999999999998

   1. Initial program 86.8%

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

      1. add-cbrt-cube 86.9%

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

      2. pow3 86.9%

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

   4. Applied egg-rr 86.9%

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

      1. rem-cbrt-cube 86.8%

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

      2. add-cube-cbrt 87.0%

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

      3. pow2 87.0%

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

      4. *-commutative 87.0%

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

      5. *-commutative 87.0%

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

   6. Applied egg-rr 87.0%

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

   if 0.34999999999999998 < b

   1. Initial program 50.3%

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

      1. /-rgt-identity 50.3%

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

      2. metadata-eval 50.3%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in a around 0 93.1%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.35:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{\sqrt[3]{a \cdot 3} \cdot {\left(\sqrt[3]{a \cdot 3}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{a \cdot 3}\\ \mathbf{if}\;b \leq 0.35:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{t\_0 \cdot {t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (cbrt (* a 3.0))))
   (if (<= b 0.35)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* t_0 (pow t_0 2.0)))
     (*
      c
      (-
       (*
        c
        (* a (- (/ (* -0.5625 (* a c)) (pow b 5.0)) (/ 0.375 (pow b 3.0)))))
       (/ 0.5 b))))))

C

double code(double a, double b, double c) {
	double t_0 = cbrt((a * 3.0));
	double tmp;
	if (b <= 0.35) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (t_0 * pow(t_0, 2.0));
	} else {
		tmp = c * ((c * (a * (((-0.5625 * (a * c)) / pow(b, 5.0)) - (0.375 / pow(b, 3.0))))) - (0.5 / b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.cbrt((a * 3.0));
	double tmp;
	if (b <= 0.35) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (t_0 * Math.pow(t_0, 2.0));
	} else {
		tmp = c * ((c * (a * (((-0.5625 * (a * c)) / Math.pow(b, 5.0)) - (0.375 / Math.pow(b, 3.0))))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = cbrt(Float64(a * 3.0))
	tmp = 0.0
	if (b <= 0.35)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(t_0 * (t_0 ^ 2.0)));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(Float64(-0.5625 * Float64(a * c)) / (b ^ 5.0)) - Float64(0.375 / (b ^ 3.0))))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * 3.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[b, 0.35], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(a * N[(N[(N[(-0.5625 * N[(a * c), $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. Split input into 2 regimes

2. if b < 0.34999999999999998

   1. Initial program 86.8%

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

      1. add-cbrt-cube 86.9%

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

      2. pow3 86.9%

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

   4. Applied egg-rr 86.9%

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

      1. rem-cbrt-cube 86.8%

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

      2. add-cube-cbrt 87.0%

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

      3. pow2 87.0%

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

      4. *-commutative 87.0%

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

      5. *-commutative 87.0%

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

   6. Applied egg-rr 87.0%

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

   if 0.34999999999999998 < b

   1. Initial program 50.3%

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

      1. /-rgt-identity 50.3%

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

      2. metadata-eval 50.3%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in c around 0 92.9%

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

   6. Taylor expanded in a around 0 92.9%

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

      1. associate-*r/ 92.9%

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

      2. associate-*r/ 92.9%

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

      3. metadata-eval 92.9%

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

   8. Simplified 92.9%

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

   9. Taylor expanded in b around 0 92.9%

      \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \color{blue}{\frac{0.5}{b}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.35:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{\sqrt[3]{a \cdot 3} \cdot {\left(\sqrt[3]{a \cdot 3}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.45:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{\sqrt[3]{{\left(a \cdot 3\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.45)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (cbrt (pow (* a 3.0) 3.0)))
   (*
    c
    (-
     (* c (* a (- (/ (* -0.5625 (* a c)) (pow b 5.0)) (/ 0.375 (pow b 3.0)))))
     (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.45) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / cbrt(pow((a * 3.0), 3.0));
	} else {
		tmp = c * ((c * (a * (((-0.5625 * (a * c)) / pow(b, 5.0)) - (0.375 / pow(b, 3.0))))) - (0.5 / b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.45) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / Math.cbrt(Math.pow((a * 3.0), 3.0));
	} else {
		tmp = c * ((c * (a * (((-0.5625 * (a * c)) / Math.pow(b, 5.0)) - (0.375 / Math.pow(b, 3.0))))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.45)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / cbrt((Float64(a * 3.0) ^ 3.0)));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(Float64(-0.5625 * Float64(a * c)) / (b ^ 5.0)) - Float64(0.375 / (b ^ 3.0))))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.45], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[Power[N[(a * 3.0), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(a * N[(N[(N[(-0.5625 * N[(a * c), $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. Split input into 2 regimes

2. if b < 0.450000000000000011

   1. Initial program 86.8%

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

      1. add-cbrt-cube 86.9%

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

      2. pow3 86.9%

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

   4. Applied egg-rr 86.9%

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

   if 0.450000000000000011 < b

   1. Initial program 50.3%

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

      1. /-rgt-identity 50.3%

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

      2. metadata-eval 50.3%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in c around 0 92.9%

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

   6. Taylor expanded in a around 0 92.9%

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

      1. associate-*r/ 92.9%

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

      2. associate-*r/ 92.9%

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

      3. metadata-eval 92.9%

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

   8. Simplified 92.9%

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

   9. Taylor expanded in b around 0 92.9%

      \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \color{blue}{\frac{0.5}{b}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.45:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{\sqrt[3]{{\left(a \cdot 3\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.36:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{\sqrt[3]{{a}^{3} \cdot 27}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.36)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (cbrt (* (pow a 3.0) 27.0)))
   (*
    c
    (-
     (* c (* a (- (/ (* -0.5625 (* a c)) (pow b 5.0)) (/ 0.375 (pow b 3.0)))))
     (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.36) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / cbrt((pow(a, 3.0) * 27.0));
	} else {
		tmp = c * ((c * (a * (((-0.5625 * (a * c)) / pow(b, 5.0)) - (0.375 / pow(b, 3.0))))) - (0.5 / b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.36) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / Math.cbrt((Math.pow(a, 3.0) * 27.0));
	} else {
		tmp = c * ((c * (a * (((-0.5625 * (a * c)) / Math.pow(b, 5.0)) - (0.375 / Math.pow(b, 3.0))))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.36)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / cbrt(Float64((a ^ 3.0) * 27.0)));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(Float64(-0.5625 * Float64(a * c)) / (b ^ 5.0)) - Float64(0.375 / (b ^ 3.0))))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.36], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[(N[Power[a, 3.0], $MachinePrecision] * 27.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(a * N[(N[(N[(-0.5625 * N[(a * c), $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. Split input into 2 regimes

2. if b < 0.35999999999999999

   1. Initial program 86.8%

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

      1. add-cbrt-cube 86.9%

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

      2. pow3 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in a around 0 86.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\sqrt[3]{\color{blue}{27 \cdot {a}^{3}}}} \]
   6. Step-by-step derivation

      1. *-commutative 86.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\sqrt[3]{\color{blue}{{a}^{3} \cdot 27}}} \]

   7. Simplified 86.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\sqrt[3]{\color{blue}{{a}^{3} \cdot 27}}} \]

   if 0.35999999999999999 < b

   1. Initial program 50.3%

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

      1. /-rgt-identity 50.3%

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

      2. metadata-eval 50.3%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in c around 0 92.9%

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

   6. Taylor expanded in a around 0 92.9%

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

      1. associate-*r/ 92.9%

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

      2. associate-*r/ 92.9%

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

      3. metadata-eval 92.9%

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

   8. Simplified 92.9%

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

   9. Taylor expanded in b around 0 92.9%

      \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \color{blue}{\frac{0.5}{b}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.36:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{\sqrt[3]{{a}^{3} \cdot 27}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.4:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.4)
   (* (- (sqrt (fma b b (* -3.0 (* a c)))) b) (/ 1.0 (* a 3.0)))
   (*
    c
    (-
     (* c (* a (- (/ (* -0.5625 (* a c)) (pow b 5.0)) (/ 0.375 (pow b 3.0)))))
     (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.4) {
		tmp = (sqrt(fma(b, b, (-3.0 * (a * c)))) - b) * (1.0 / (a * 3.0));
	} else {
		tmp = c * ((c * (a * (((-0.5625 * (a * c)) / pow(b, 5.0)) - (0.375 / pow(b, 3.0))))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))) - b) * Float64(1.0 / Float64(a * 3.0)));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(Float64(-0.5625 * Float64(a * c)) / (b ^ 5.0)) - Float64(0.375 / (b ^ 3.0))))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.4], N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(1.0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(a * N[(N[(N[(-0.5625 * N[(a * c), $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. Split input into 2 regimes

2. if b < 0.40000000000000002

   1. Initial program 86.8%

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

      1. add-cbrt-cube 86.9%

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

      2. pow3 86.9%

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

   4. Applied egg-rr 86.9%

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

      1. rem-cbrt-cube 86.8%

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

      2. div-inv 86.8%

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

      3. neg-mul-1 86.8%

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

      4. fma-define 86.8%

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

      5. pow2 86.8%

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

      6. *-commutative 86.8%

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

      7. *-commutative 86.8%

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

      8. *-commutative 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. fma-undefine 86.8%

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

      2. associate-*r* 86.8%

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

   8. Applied egg-rr 86.8%

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

      1. +-commutative 86.8%

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

      2. neg-mul-1 86.8%

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

      3. unsub-neg 86.8%

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

      4. unpow2 86.8%

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

      5. fma-neg 86.9%

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

      6. *-commutative 86.9%

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

      7. distribute-rgt-neg-in 86.9%

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

      8. metadata-eval 86.9%

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

   10. Simplified 86.9%

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

   if 0.40000000000000002 < b

   1. Initial program 50.3%

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

      1. /-rgt-identity 50.3%

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

      2. metadata-eval 50.3%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in c around 0 92.9%

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

   6. Taylor expanded in a around 0 92.9%

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

      1. associate-*r/ 92.9%

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

      2. associate-*r/ 92.9%

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

      3. metadata-eval 92.9%

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

   8. Simplified 92.9%

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

   9. Taylor expanded in b around 0 92.9%

      \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \color{blue}{\frac{0.5}{b}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.4:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.0)
   (* (- (sqrt (fma b b (* -3.0 (* a c)))) b) (/ 1.0 (* a 3.0)))
   (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.0) {
		tmp = (sqrt(fma(b, b, (-3.0 * (a * c)))) - b) * (1.0 / (a * 3.0));
	} else {
		tmp = fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(a * c)))) - b) * Float64(1.0 / Float64(a * 3.0)));
	else
		tmp = Float64(fma(-0.375, Float64(a * (Float64(c / b) ^ 2.0)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.0], N[(N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(1.0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4

   1. Initial program 83.5%

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

      1. add-cbrt-cube 83.6%

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

      2. pow3 83.6%

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

   4. Applied egg-rr 83.6%

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

      1. rem-cbrt-cube 83.5%

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

      2. div-inv 83.5%

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

      3. neg-mul-1 83.5%

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

      4. fma-define 83.5%

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

      5. pow2 83.5%

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

      6. *-commutative 83.5%

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

      7. *-commutative 83.5%

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

      8. *-commutative 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. fma-undefine 83.5%

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

      2. associate-*r* 83.5%

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

   8. Applied egg-rr 83.5%

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

      1. +-commutative 83.5%

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

      2. neg-mul-1 83.5%

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

      3. unsub-neg 83.5%

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

      4. unpow2 83.5%

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

      5. fma-neg 83.6%

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

      6. *-commutative 83.6%

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

      7. distribute-rgt-neg-in 83.6%

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

      8. metadata-eval 83.6%

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

   10. Simplified 83.6%

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

   if 4 < b

   1. Initial program 48.9%

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

      1. add-cbrt-cube 48.9%

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

      2. pow3 48.9%

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

   4. Applied egg-rr 48.9%

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

   5. Taylor expanded in b around inf 88.4%

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

      1. +-commutative 88.4%

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

      2. fma-define 88.4%

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

      3. associate-/l* 88.4%

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

      4. unpow2 88.4%

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

      5. unpow2 88.4%

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

      6. times-frac 88.4%

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

      7. unpow1 88.4%

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

      8. pow-plus 88.4%

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

      9. metadata-eval 88.4%

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

      10. *-commutative 88.4%

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

   7. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.9:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.9)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.9) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.9)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(-0.375, Float64(a * (Float64(c / b) ^ 2.0)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.9], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.89999999999999991

   1. Initial program 83.5%

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

      1. /-rgt-identity 83.5%

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

      2. metadata-eval 83.5%

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

   3. Simplified 83.5%

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

   if 3.89999999999999991 < b

   1. Initial program 48.9%

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

      1. add-cbrt-cube 48.9%

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

      2. pow3 48.9%

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

   4. Applied egg-rr 48.9%

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

   5. Taylor expanded in b around inf 88.4%

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

      1. +-commutative 88.4%

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

      2. fma-define 88.4%

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

      3. associate-/l* 88.4%

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

      4. unpow2 88.4%

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

      5. unpow2 88.4%

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

      6. times-frac 88.4%

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

      7. unpow1 88.4%

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

      8. pow-plus 88.4%

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

      9. metadata-eval 88.4%

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

      10. *-commutative 88.4%

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

   7. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.9:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.6)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.6) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(-0.375, Float64(a * (Float64(c / b) ^ 2.0)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5999999999999996

   1. Initial program 83.5%

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

   if 4.5999999999999996 < b

   1. Initial program 48.9%

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

      1. add-cbrt-cube 48.9%

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

      2. pow3 48.9%

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

   4. Applied egg-rr 48.9%

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

   5. Taylor expanded in b around inf 88.4%

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

      1. +-commutative 88.4%

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

      2. fma-define 88.4%

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

      3. associate-/l* 88.4%

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

      4. unpow2 88.4%

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

      5. unpow2 88.4%

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

      6. times-frac 88.4%

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

      7. unpow1 88.4%

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

      8. pow-plus 88.4%

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

      9. metadata-eval 88.4%

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

      10. *-commutative 88.4%

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

   7. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.9:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{c \cdot \left(a \cdot -0.375\right)}{{b}^{3}} - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.9)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (* c (- (/ (* c (* a -0.375)) (pow b 3.0)) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.9) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * (((c * (a * -0.375)) / pow(b, 3.0)) - (0.5 / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.9d0) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = c * (((c * (a * (-0.375d0))) / (b ** 3.0d0)) - (0.5d0 / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.9) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * (((c * (a * -0.375)) / Math.pow(b, 3.0)) - (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.9:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = c * (((c * (a * -0.375)) / math.pow(b, 3.0)) - (0.5 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.9)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(Float64(c * Float64(a * -0.375)) / (b ^ 3.0)) - Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.9)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = c * (((c * (a * -0.375)) / (b ^ 3.0)) - (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.9], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(N[(c * N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.89999999999999991

   1. Initial program 83.5%

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

   if 3.89999999999999991 < b

   1. Initial program 48.9%

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

      1. /-rgt-identity 48.9%

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

      2. metadata-eval 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in c around 0 95.4%

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

   7. Simplified 95.4%

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

   8. Taylor expanded in c around 0 88.1%

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

      1. associate-*r/ 88.1%

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

      2. associate-*r* 88.1%

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

   10. Simplified 88.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.9:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{c \cdot \left(a \cdot -0.375\right)}{{b}^{3}} - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. /-rgt-identity 55.9%

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

   2. metadata-eval 55.9%

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

3. Simplified 56.0%

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

5. Taylor expanded in c around 0 91.9%

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

7. Simplified 91.9%

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

8. Taylor expanded in c around 0 82.0%

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

   1. associate-*r/ 82.0%

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

   2. associate-*r* 82.0%

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

10. Simplified 82.0%

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

11. Final simplification 82.0%

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

Alternative 13: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. /-rgt-identity 55.9%

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

   2. metadata-eval 55.9%

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

3. Simplified 56.0%

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

5. Taylor expanded in c around 0 82.0%

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

   1. associate-*r/ 82.0%

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

   2. metadata-eval 82.0%

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

7. Simplified 82.0%

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

Alternative 14: 64.4% 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 55.9%

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

   1. /-rgt-identity 55.9%

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

   2. metadata-eval 55.9%

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

3. Simplified 56.0%

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

5. Taylor expanded in b around inf 63.9%

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

Alternative 15: 3.2% 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 55.9%

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

   1. /-rgt-identity 55.9%

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

   2. metadata-eval 55.9%

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

3. Simplified 56.0%

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

5. Taylor expanded in a around inf 55.6%

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

   1. *-un-lft-identity 55.6%

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

   2. add-sqr-sqrt 54.5%

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

   3. prod-diff 55.4%

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

   4. add-sqr-sqrt 56.0%

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

   5. fma-neg 56.0%

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

   6. *-un-lft-identity 56.0%

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

   7. *-commutative 56.0%

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

   8. fma-define 56.0%

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

   9. add-sqr-sqrt 55.4%

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

7. Applied egg-rr 55.4%

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

8. Taylor expanded in a around 0 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

10. Simplified 3.2%

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

11. Taylor expanded in a around 0 3.2%

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

Reproduce

herbie shell --seed 2024144 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))