Cubic critical, narrow range

Percentage Accurate: 54.9% → 97.1%

Time: 16.5s

Alternatives: 13

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: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0 \cdot \left(b + b\right) - c \cdot \left(a \cdot 3\right)}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (- (* 0.0 (+ b b)) (* c (* a 3.0)))
   (+ b (sqrt (fma b b (* -3.0 (* c a))))))
  (pow (pow (* a 3.0) 3.0) 0.3333333333333333)))

C

double code(double a, double b, double c) {
	return (((0.0 * (b + b)) - (c * (a * 3.0))) / (b + sqrt(fma(b, b, (-3.0 * (c * a)))))) / pow(pow((a * 3.0), 3.0), 0.3333333333333333);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(0.0 * Float64(b + b)) - Float64(c * Float64(a * 3.0))) / Float64(b + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a)))))) / ((Float64(a * 3.0) ^ 3.0) ^ 0.3333333333333333))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(0.0 * N[(b + b), $MachinePrecision]), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[N[(a * 3.0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

   1. add-cbrt-cube 55.2%

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

   2. pow1/3 55.0%

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

   3. pow3 55.0%

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

4. Applied egg-rr 55.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. add-cbrt-cube 54.1%

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

   2. cbrt-prod 53.5%

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

   3. distribute-rgt-neg-in 53.5%

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

   4. cbrt-prod 52.2%

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

   5. pow2 52.2%

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

6. Applied egg-rr 52.2%

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

   1. flip-+ 52.1%

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

8. Applied egg-rr 56.3%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]
9. Step-by-step derivation

   1. associate--r- 96.9%

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

   2. unpow2 96.9%

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

   3. unpow2 96.9%

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

   4. difference-of-squares 96.9%

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

   5. +-commutative 96.9%

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

   6. neg-mul-1 96.9%

      \[\leadsto \frac{\frac{\left(b + \color{blue}{-1 \cdot b}\right) \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. distribute-rgt1-in 96.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(-1 + 1\right) \cdot b\right)} \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   8. metadata-eval 96.9%

      \[\leadsto \frac{\frac{\left(\color{blue}{0} \cdot b\right) \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   9. mul0-lft 96.9%

      \[\leadsto \frac{\frac{\color{blue}{0} \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   10. unpow2 96.9%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} - c \cdot \left(a \cdot 3\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   11. fmm-def 97.0%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(a \cdot 3\right)\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   12. associate-*r* 97.0%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(c \cdot a\right) \cdot 3}\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   13. *-commutative 97.0%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right)} \cdot 3\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   14. distribute-rgt-neg-in 97.0%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-3\right)}\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   15. *-commutative 97.0%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right)} \cdot \left(-3\right)\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   16. metadata-eval 97.0%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot \color{blue}{-3}\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

10. Simplified 97.0%

    \[\leadsto \frac{\color{blue}{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

11. Final simplification 97.0%

    \[\leadsto \frac{\frac{0 \cdot \left(b + b\right) - c \cdot \left(a \cdot 3\right)}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]
12. Add Preprocessing

Alternative 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0245:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot {\left(\sqrt[3]{3}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0245)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a (pow (cbrt 3.0) 3.0)))
   (/
    (+
     (* c -0.5)
     (*
      a
      (+
       (* -0.5625 (/ (* a (pow c 3.0)) (pow b 4.0)))
       (* -0.375 (/ (pow c 2.0) (pow b 2.0))))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0245) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * pow(cbrt(3.0), 3.0));
	} else {
		tmp = ((c * -0.5) + (a * ((-0.5625 * ((a * pow(c, 3.0)) / pow(b, 4.0))) + (-0.375 * (pow(c, 2.0) / pow(b, 2.0)))))) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0245) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * Math.pow(Math.cbrt(3.0), 3.0));
	} else {
		tmp = ((c * -0.5) + (a * ((-0.5625 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 4.0))) + (-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 2.0)))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0245)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * (cbrt(3.0) ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 2.0)))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0245], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * N[Power[N[Power[3.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.024500000000000001

   1. Initial program 85.3%

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

      1. add-cube-cbrt 85.2%

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

      2. pow3 85.1%

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

   4. Applied egg-rr 85.1%

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

      1. cbrt-prod 85.2%

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

      2. unpow-prod-down 85.3%

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

      3. pow3 85.2%

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

      4. add-cube-cbrt 85.5%

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

   6. Applied egg-rr 85.5%

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

   if 0.024500000000000001 < b

   1. Initial program 52.0%

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

   3. Simplified 52.1%

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

      1. pow1/2 52.1%

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

      2. pow-to-exp 49.1%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   6. Applied egg-rr 49.1%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   7. Taylor expanded in b around inf 94.4%

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

   9. Simplified 94.4%

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

   10. Taylor expanded in a around 0 92.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0245:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot {\left(\sqrt[3]{3}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.021:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot {\left(\sqrt[3]{3}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} + 0.375 \cdot \frac{-1}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.021)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a (pow (cbrt 3.0) 3.0)))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (*
      (pow c 3.0)
      (+
       (* -0.5625 (/ a (pow b 5.0)))
       (* 0.375 (/ -1.0 (* c (pow b 3.0))))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.021) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * pow(cbrt(3.0), 3.0));
	} else {
		tmp = (-0.5 * (c / b)) + (a * (pow(c, 3.0) * ((-0.5625 * (a / pow(b, 5.0))) + (0.375 * (-1.0 / (c * pow(b, 3.0)))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.021) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * Math.pow(Math.cbrt(3.0), 3.0));
	} else {
		tmp = (-0.5 * (c / b)) + (a * (Math.pow(c, 3.0) * ((-0.5625 * (a / Math.pow(b, 5.0))) + (0.375 * (-1.0 / (c * Math.pow(b, 3.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.021)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * (cbrt(3.0) ^ 3.0)));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64((c ^ 3.0) * Float64(Float64(-0.5625 * Float64(a / (b ^ 5.0))) + Float64(0.375 * Float64(-1.0 / Float64(c * (b ^ 3.0))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.021], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * N[Power[N[Power[3.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(-0.5625 * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.375 * N[(-1.0 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0210000000000000013

   1. Initial program 85.3%

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

      1. add-cube-cbrt 85.2%

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

      2. pow3 85.1%

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

   4. Applied egg-rr 85.1%

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

      1. cbrt-prod 85.2%

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

      2. unpow-prod-down 85.3%

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

      3. pow3 85.2%

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

      4. add-cube-cbrt 85.5%

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

   6. Applied egg-rr 85.5%

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

   if 0.0210000000000000013 < b

   1. Initial program 52.0%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in a around 0 92.0%

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

   6. Taylor expanded in c around inf 92.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.021:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot {\left(\sqrt[3]{3}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} + 0.375 \cdot \frac{-1}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.024:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot {\left(\sqrt[3]{3}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{4}} + -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.024)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a (pow (cbrt 3.0) 3.0)))
   (/
    (*
     c
     (-
      (*
       c
       (+
        (* -0.5625 (/ (* c (pow a 2.0)) (pow b 4.0)))
        (* -0.375 (/ a (pow b 2.0)))))
      0.5))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.024) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * pow(cbrt(3.0), 3.0));
	} else {
		tmp = (c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 4.0))) + (-0.375 * (a / pow(b, 2.0))))) - 0.5)) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.024) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * Math.pow(Math.cbrt(3.0), 3.0));
	} else {
		tmp = (c * ((c * ((-0.5625 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 4.0))) + (-0.375 * (a / Math.pow(b, 2.0))))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.024)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * (cbrt(3.0) ^ 3.0)));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 4.0))) + Float64(-0.375 * Float64(a / (b ^ 2.0))))) - 0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.024], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * N[Power[N[Power[3.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.024

   1. Initial program 85.3%

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

      1. add-cube-cbrt 85.2%

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

      2. pow3 85.1%

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

   4. Applied egg-rr 85.1%

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

      1. cbrt-prod 85.2%

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

      2. unpow-prod-down 85.3%

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

      3. pow3 85.2%

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

      4. add-cube-cbrt 85.5%

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

   6. Applied egg-rr 85.5%

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

   if 0.024 < b

   1. Initial program 52.0%

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

   3. Simplified 52.1%

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

      1. pow1/2 52.1%

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

      2. pow-to-exp 49.1%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   6. Applied egg-rr 49.1%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   7. Taylor expanded in b around inf 94.4%

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

   9. Simplified 94.4%

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

   10. Taylor expanded in c around 0 92.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.024:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot {\left(\sqrt[3]{3}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{4}} + -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.66:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot {\left(\sqrt[3]{3}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.66)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a (pow (cbrt 3.0) 3.0)))
   (/ (fma -0.5 c (* -0.375 (* a (pow (/ c b) 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.66) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * pow(cbrt(3.0), 3.0));
	} else {
		tmp = fma(-0.5, c, (-0.375 * (a * pow((c / b), 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.66)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * (cbrt(3.0) ^ 3.0)));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.66], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * N[Power[N[Power[3.0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c + N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.660000000000000031

   1. Initial program 83.0%

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

      1. add-cube-cbrt 82.8%

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

      2. pow3 82.8%

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

   4. Applied egg-rr 82.8%

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

      1. cbrt-prod 83.0%

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

      2. unpow-prod-down 83.0%

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

      3. pow3 82.9%

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

      4. add-cube-cbrt 83.2%

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

   6. Applied egg-rr 83.2%

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

   if 0.660000000000000031 < b

   1. Initial program 50.7%

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

   3. Simplified 50.8%

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

      1. pow1/2 50.8%

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

      2. pow-to-exp 47.8%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   6. Applied egg-rr 47.8%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   7. Taylor expanded in b around inf 86.3%

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

      1. fma-define 86.3%

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

      2. associate-/l* 86.3%

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

      3. unpow2 86.3%

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

      4. unpow2 86.3%

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

      5. times-frac 86.3%

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

      6. unpow1 86.3%

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

      7. pow-plus 86.3%

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

      8. metadata-eval 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.66:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot {\left(\sqrt[3]{3}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.66:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.66)
   (/ 1.0 (* a (/ 3.0 (fma -1.0 b (sqrt (fma b b (* -3.0 (* c a))))))))
   (/ (fma -0.5 c (* -0.375 (* a (pow (/ c b) 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.66) {
		tmp = 1.0 / (a * (3.0 / fma(-1.0, b, sqrt(fma(b, b, (-3.0 * (c * a)))))));
	} else {
		tmp = fma(-0.5, c, (-0.375 * (a * pow((c / b), 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.66)
		tmp = Float64(1.0 / Float64(a * Float64(3.0 / fma(-1.0, b, sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))))));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.66], N[(1.0 / N[(a * N[(3.0 / N[(-1.0 * b + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c + N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.660000000000000031

   1. Initial program 83.0%

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

      1. add-cbrt-cube 83.1%

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

      2. pow1/3 82.6%

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

      3. pow3 82.6%

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

   4. Applied egg-rr 82.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}}} \]
   5. Step-by-step derivation

      1. pow-pow 83.0%

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

      2. metadata-eval 83.0%

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

      3. pow1 83.0%

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

      4. log1p-expm1-u 83.0%

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

      5. log1p-undefine 77.2%

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

      6. *-commutative 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. clear-num 77.2%

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

      2. log1p-define 83.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(a \cdot 3\right)\right)}}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}} \]

      3. log1p-expm1-u 83.0%

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

      4. *-commutative 83.0%

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

      5. add-cube-cbrt 78.9%

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

      6. unpow2 78.9%

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

      7. distribute-rgt-neg-out 78.9%

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

      8. inv-pow 78.9%

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

   8. Applied egg-rr 83.0%

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

      1. unpow-1 83.0%

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

      2. associate-/l* 83.1%

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

      3. unpow2 83.1%

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

      4. fmm-def 83.2%

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

      5. associate-*r* 83.1%

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

      6. *-commutative 83.1%

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

      7. distribute-rgt-neg-in 83.1%

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

      8. *-commutative 83.1%

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

      9. metadata-eval 83.1%

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

   10. Simplified 83.1%

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

   if 0.660000000000000031 < b

   1. Initial program 50.7%

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

   3. Simplified 50.8%

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

      1. pow1/2 50.8%

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

      2. pow-to-exp 47.8%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   6. Applied egg-rr 47.8%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   7. Taylor expanded in b around inf 86.3%

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

      1. fma-define 86.3%

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

      2. associate-/l* 86.3%

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

      3. unpow2 86.3%

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

      4. unpow2 86.3%

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

      5. times-frac 86.3%

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

      6. unpow1 86.3%

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

      7. pow-plus 86.3%

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

      8. metadata-eval 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.66:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.54:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.54)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (fma -0.5 c (* -0.375 (* a (pow (/ c b) 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.54) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.5, c, (-0.375 * (a * pow((c / b), 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.54)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.54], 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[(-0.5 * c + N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.54000000000000004

   1. Initial program 83.0%

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

   3. Simplified 83.1%

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

   if 0.54000000000000004 < b

   1. Initial program 50.7%

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

   3. Simplified 50.8%

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

      1. pow1/2 50.8%

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

      2. pow-to-exp 47.8%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   6. Applied egg-rr 47.8%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   7. Taylor expanded in b around inf 86.3%

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

      1. fma-define 86.3%

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

      2. associate-/l* 86.3%

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

      3. unpow2 86.3%

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

      4. unpow2 86.3%

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

      5. times-frac 86.3%

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

      6. unpow1 86.3%

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

      7. pow-plus 86.3%

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

      8. metadata-eval 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.54:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.54:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.54)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (/ (fma -0.5 c (* -0.375 (* a (pow (/ c b) 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.54) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.5, c, (-0.375 * (a * pow((c / b), 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.54)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.54], 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[(-0.5 * c + N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.54000000000000004

   1. Initial program 83.0%

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

   if 0.54000000000000004 < b

   1. Initial program 50.7%

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

   3. Simplified 50.8%

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

      1. pow1/2 50.8%

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

      2. pow-to-exp 47.8%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   6. Applied egg-rr 47.8%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   7. Taylor expanded in b around inf 86.3%

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

      1. fma-define 86.3%

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

      2. associate-/l* 86.3%

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

      3. unpow2 86.3%

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

      4. unpow2 86.3%

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

      5. times-frac 86.3%

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

      6. unpow1 86.3%

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

      7. pow-plus 86.3%

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

      8. metadata-eval 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.54:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.7)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (/ (* c (- (* -0.375 (/ (* c a) (pow b 2.0))) 0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.7) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * ((-0.375 * ((c * a) / pow(b, 2.0))) - 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 <= 0.7d0) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (((-0.375d0) * ((c * a) / (b ** 2.0d0))) - 0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.7) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * ((-0.375 * ((c * a) / Math.pow(b, 2.0))) - 0.5)) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.7:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c * ((-0.375 * ((c * a) / math.pow(b, 2.0))) - 0.5)) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.7)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(c * a) / (b ^ 2.0))) - 0.5)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.7)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c * ((-0.375 * ((c * a) / (b ^ 2.0))) - 0.5)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.7], 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[(c * N[(N[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.69999999999999996

   1. Initial program 83.0%

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

   if 0.69999999999999996 < b

   1. Initial program 50.7%

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

   3. Simplified 50.8%

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

      1. pow1/2 50.8%

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

      2. pow-to-exp 47.8%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   6. Applied egg-rr 47.8%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

   7. Taylor expanded in b around inf 95.0%

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

   9. Simplified 95.0%

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

   10. Taylor expanded in c around 0 86.3%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Simplified 55.2%

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

   1. pow1/2 55.2%

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

   2. pow-to-exp 52.3%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

6. Applied egg-rr 52.3%

   \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right) \cdot 0.5}} - b}{3 \cdot a} \]

7. Taylor expanded in b around inf 92.1%

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

9. Simplified 92.1%

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

10. Taylor expanded in c around 0 82.4%

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

11. Final simplification 82.4%

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

Alternative 11: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Simplified 55.2%

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

5. Taylor expanded in c around 0 82.3%

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

   1. associate-*r/ 82.3%

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

   2. metadata-eval 82.3%

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

7. Simplified 82.3%

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

8. Final simplification 82.3%

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

Alternative 12: 64.9% 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 55.1%

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

3. Simplified 55.2%

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

5. Taylor expanded in b around inf 64.6%

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

Alternative 13: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

   1. add-cbrt-cube 55.2%

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

   2. pow1/3 55.0%

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

   3. pow3 55.0%

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

4. Applied egg-rr 55.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. add-cbrt-cube 54.1%

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

   2. cbrt-prod 53.5%

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

   3. distribute-rgt-neg-in 53.5%

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

   4. cbrt-prod 52.2%

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

   5. pow2 52.2%

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

6. Applied egg-rr 52.2%

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

7. Taylor expanded in a around 0 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

9. Simplified 3.2%

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

Reproduce

herbie shell --seed 2024184 
(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)))