Cubic critical, narrow range

Percentage Accurate: 55.5% → 91.7%

Time: 18.5s

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: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)\\ \mathbf{if}\;b \leq 1.5:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* -3.0 a)))))
   (if (<= b 1.5)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.3333333333333333 a))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma
       -0.16666666666666666
       (* (/ (pow (* c a) 4.0) (pow b 7.0)) (/ 6.328125 a))
       (fma -0.5 (/ c b) (* -0.375 (/ (* a (* c c)) (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (-3.0 * a)));
	double tmp;
	if (b <= 1.5) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.3333333333333333 / a);
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.16666666666666666, ((pow((c * a), 4.0) / pow(b, 7.0)) * (6.328125 / a)), fma(-0.5, (c / b), (-0.375 * ((a * (c * c)) / pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(-3.0 * a)))
	tmp = 0.0
	if (b <= 1.5)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.3333333333333333 / a));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.16666666666666666, Float64(Float64((Float64(c * a) ^ 4.0) / (b ^ 7.0)) * Float64(6.328125 / a)), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.5], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5

   1. Initial program 86.0%

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

      1. neg-sub0 86.0%

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

      2. associate-+l- 86.0%

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

      3. sub0-neg 86.0%

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

      4. neg-mul-1 86.0%

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

      5. associate-*r/ 86.0%

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

      6. *-commutative 86.0%

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

      7. metadata-eval 86.0%

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

      8. metadata-eval 86.0%

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

      9. times-frac 86.0%

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

      10. *-commutative 86.0%

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

      11. times-frac 85.9%

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

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 86.1%

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

      2. pow3 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. flip-- 85.8%

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

      2. add-sqr-sqrt 86.8%

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

      3. unpow3 86.8%

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

      4. add-cube-cbrt 87.0%

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

      5. unpow3 86.9%

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

      6. add-cube-cbrt 86.9%

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

   7. Applied egg-rr 86.9%

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

      1. *-commutative 86.9%

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

      2. associate-*l* 86.9%

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

      3. +-commutative 86.9%

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

      4. *-commutative 86.9%

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

      5. associate-*l* 86.9%

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

   9. Simplified 86.9%

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

   if 1.5 < b

   1. Initial program 50.8%

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

      1. /-rgt-identity 50.8%

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

      2. metadata-eval 50.8%

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

      3. associate-/l* 50.8%

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

      4. associate-*r/ 50.8%

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

      5. *-commutative 50.8%

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

      6. associate-*l/ 50.8%

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

      7. associate-*r/ 50.8%

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

      8. metadata-eval 50.8%

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

      9. metadata-eval 50.8%

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

      10. times-frac 50.8%

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

      11. neg-mul-1 50.8%

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

      12. distribute-rgt-neg-in 50.8%

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

      13. times-frac 50.8%

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

      14. metadata-eval 50.8%

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

      15. neg-mul-1 50.8%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in b around inf 94.4%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 94.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

      2. associate-/l* 94.4%

         \[\leadsto \mathsf{fma}\left(-0.5625, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

      3. unpow2 94.4%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

      4. fma-def 94.4%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)}\right) \]

   6. Simplified 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right)} \]

   7. Taylor expanded in c around 0 94.4%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative 94.4%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{\color{blue}{{b}^{7} \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right) \]

      2. distribute-rgt-out 94.4%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{{b}^{7} \cdot a}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right) \]

      3. associate-*r* 94.4%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{{b}^{7} \cdot a}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right) \]

      4. times-frac 94.4%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{c}^{4} \cdot {a}^{4}}{{b}^{7}} \cdot \frac{1.265625 + 5.0625}{a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right) \]

   9. Simplified 94.4%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\right)\\ \end{array} \]

Alternative 2: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)\\ \mathbf{if}\;b \leq 8:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \frac{c \cdot -0.5}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* -3.0 a)))))
   (if (<= b 8.0)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.3333333333333333 a))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma -0.375 (/ (* c c) (/ (pow b 3.0) a)) (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (-3.0 * a)));
	double tmp;
	if (b <= 8.0) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.3333333333333333 / a);
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.375, ((c * c) / (pow(b, 3.0) / a)), ((c * -0.5) / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(-3.0 * a)))
	tmp = 0.0
	if (b <= 8.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.3333333333333333 / a));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.375, Float64(Float64(c * c) / Float64((b ^ 3.0) / a)), Float64(Float64(c * -0.5) / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 8.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8

   1. Initial program 83.8%

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

      1. neg-sub0 83.8%

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

      2. associate-+l- 83.8%

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

      3. sub0-neg 83.8%

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

      4. neg-mul-1 83.8%

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

      5. associate-*r/ 83.8%

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

      6. *-commutative 83.8%

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

      7. metadata-eval 83.8%

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

      8. metadata-eval 83.8%

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

      9. times-frac 83.8%

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

      10. *-commutative 83.8%

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

      11. times-frac 83.7%

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

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 83.8%

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

      2. pow3 83.9%

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

   5. Applied egg-rr 83.9%

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

      1. flip-- 83.6%

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

      2. add-sqr-sqrt 84.7%

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

      3. unpow3 84.7%

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

      4. add-cube-cbrt 84.9%

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

      5. unpow3 84.9%

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

      6. add-cube-cbrt 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. *-commutative 84.9%

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

      2. associate-*l* 84.9%

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

      3. +-commutative 84.9%

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

      4. *-commutative 84.9%

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

      5. associate-*l* 84.9%

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

   9. Simplified 84.9%

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

   if 8 < b

   1. Initial program 49.0%

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

      1. /-rgt-identity 49.0%

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

      2. metadata-eval 49.0%

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

      3. associate-/l* 49.0%

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

      4. associate-*r/ 49.0%

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

      5. *-commutative 49.0%

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

      6. associate-*l/ 49.0%

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

      7. associate-*r/ 49.0%

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

      8. metadata-eval 49.0%

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

      9. metadata-eval 49.0%

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

      10. times-frac 49.0%

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

      11. neg-mul-1 49.0%

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

      12. distribute-rgt-neg-in 49.0%

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

      13. times-frac 49.0%

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

      14. metadata-eval 49.0%

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

      15. neg-mul-1 49.0%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in b around inf 93.1%

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

      1. fma-def 93.1%

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

      2. associate-/l* 93.1%

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

      3. unpow2 93.1%

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

      4. +-commutative 93.1%

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

      5. fma-def 93.1%

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

      6. associate-/l* 93.1%

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

      7. unpow2 93.1%

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

      8. associate-*r/ 93.1%

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

   6. Simplified 93.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \frac{c \cdot -0.5}{b}\right)\right)\\ \end{array} \]

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)\\ \mathbf{if}\;b \leq 8:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{c \cdot a}{b}, -0.375 \cdot \left(\left(a \cdot a\right) \cdot \frac{c \cdot c}{{b}^{3}}\right) + -0.5625 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* -3.0 a)))))
   (if (<= b 8.0)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.3333333333333333 a))
     (/
      (fma
       -0.5
       (/ (* c a) b)
       (+
        (* -0.375 (* (* a a) (/ (* c c) (pow b 3.0))))
        (* -0.5625 (/ (pow (* c a) 3.0) (pow b 5.0)))))
      a))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (-3.0 * a)));
	double tmp;
	if (b <= 8.0) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.3333333333333333 / a);
	} else {
		tmp = fma(-0.5, ((c * a) / b), ((-0.375 * ((a * a) * ((c * c) / pow(b, 3.0)))) + (-0.5625 * (pow((c * a), 3.0) / pow(b, 5.0))))) / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(-3.0 * a)))
	tmp = 0.0
	if (b <= 8.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(fma(-0.5, Float64(Float64(c * a) / b), Float64(Float64(-0.375 * Float64(Float64(a * a) * Float64(Float64(c * c) / (b ^ 3.0)))) + Float64(-0.5625 * Float64((Float64(c * a) ^ 3.0) / (b ^ 5.0))))) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 8.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8

   1. Initial program 83.8%

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

      1. neg-sub0 83.8%

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

      2. associate-+l- 83.8%

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

      3. sub0-neg 83.8%

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

      4. neg-mul-1 83.8%

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

      5. associate-*r/ 83.8%

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

      6. *-commutative 83.8%

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

      7. metadata-eval 83.8%

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

      8. metadata-eval 83.8%

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

      9. times-frac 83.8%

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

      10. *-commutative 83.8%

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

      11. times-frac 83.7%

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

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 83.8%

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

      2. pow3 83.9%

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

   5. Applied egg-rr 83.9%

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

      1. flip-- 83.6%

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

      2. add-sqr-sqrt 84.7%

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

      3. unpow3 84.7%

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

      4. add-cube-cbrt 84.9%

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

      5. unpow3 84.9%

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

      6. add-cube-cbrt 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. *-commutative 84.9%

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

      2. associate-*l* 84.9%

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

      3. +-commutative 84.9%

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

      4. *-commutative 84.9%

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

      5. associate-*l* 84.9%

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

   9. Simplified 84.9%

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

   if 8 < b

   1. Initial program 49.0%

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

      1. /-rgt-identity 49.0%

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

      2. metadata-eval 49.0%

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

      3. associate-/r/ 49.0%

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

      4. metadata-eval 49.0%

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

      5. metadata-eval 49.0%

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

      6. times-frac 49.0%

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

      7. *-commutative 49.0%

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

      8. times-frac 49.0%

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

      9. *-commutative 49.0%

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

      10. associate-/r* 49.0%

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

      11. associate-*l/ 49.0%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in b around inf 92.9%

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

      1. fma-def 92.9%

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

      2. *-commutative 92.9%

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

      3. fma-def 92.9%

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

      4. associate-/l* 92.9%

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

      5. unpow2 92.9%

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

      6. unpow2 92.9%

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

      7. *-commutative 92.9%

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

      8. cube-prod 92.9%

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

   6. Simplified 92.9%

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

      1. fma-udef 92.9%

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

      2. associate-/r/ 92.9%

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

   8. Applied egg-rr 92.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{c \cdot a}{b}, -0.375 \cdot \left(\left(a \cdot a\right) \cdot \frac{c \cdot c}{{b}^{3}}\right) + -0.5625 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)}{a}\\ \end{array} \]

Alternative 4: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)\\ \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* -3.0 a)))))
   (if (<= b 8.2)
     (* (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (/ 0.3333333333333333 a))
     (fma -0.375 (/ (* c c) (/ (pow b 3.0) a)) (/ (* c -0.5) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (-3.0 * a)));
	double tmp;
	if (b <= 8.2) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) * (0.3333333333333333 / a);
	} else {
		tmp = fma(-0.375, ((c * c) / (pow(b, 3.0) / a)), ((c * -0.5) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(-3.0 * a)))
	tmp = 0.0
	if (b <= 8.2)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) * Float64(0.3333333333333333 / a));
	else
		tmp = fma(-0.375, Float64(Float64(c * c) / Float64((b ^ 3.0) / a)), Float64(Float64(c * -0.5) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 8.2], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.1999999999999993

   1. Initial program 83.7%

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

      1. neg-sub0 83.7%

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

      2. associate-+l- 83.7%

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

      3. sub0-neg 83.7%

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

      4. neg-mul-1 83.7%

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

      5. associate-*r/ 83.7%

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

      6. *-commutative 83.7%

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

      7. metadata-eval 83.7%

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

      8. metadata-eval 83.7%

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

      9. times-frac 83.7%

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

      10. *-commutative 83.7%

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

      11. times-frac 83.7%

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

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 83.8%

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

      2. pow3 83.8%

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

   5. Applied egg-rr 83.8%

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

      1. flip-- 83.5%

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

      2. add-sqr-sqrt 84.6%

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

      3. unpow3 84.6%

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

      4. add-cube-cbrt 84.8%

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

      5. unpow3 84.8%

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

      6. add-cube-cbrt 84.8%

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

   7. Applied egg-rr 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 84.8%

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

      3. +-commutative 84.8%

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

      4. *-commutative 84.8%

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

      5. associate-*l* 84.8%

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

   9. Simplified 84.8%

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

   if 8.1999999999999993 < b

   1. Initial program 48.8%

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

      1. /-rgt-identity 48.8%

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

      2. metadata-eval 48.8%

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

      3. associate-/l* 48.8%

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

      4. associate-*r/ 48.8%

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

      5. *-commutative 48.8%

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

      6. associate-*l/ 48.8%

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

      7. associate-*r/ 48.8%

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

      8. metadata-eval 48.8%

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

      9. metadata-eval 48.8%

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

      10. times-frac 48.8%

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

      11. neg-mul-1 48.8%

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

      12. distribute-rgt-neg-in 48.8%

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

      13. times-frac 48.8%

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

      14. metadata-eval 48.8%

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

      15. neg-mul-1 48.8%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in b around inf 88.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   5. Step-by-step derivation

      1. +-commutative 88.3%

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

      2. fma-def 88.3%

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

      3. associate-/l* 88.3%

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

      4. unpow2 88.3%

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

      5. associate-*r/ 88.3%

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

   6. Simplified 88.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \]

Alternative 5: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \sqrt{\frac{0.1111111111111111}{a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.2)
   (*
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (sqrt (/ 0.1111111111111111 (* a a))))
   (fma -0.375 (/ (* c c) (/ (pow b 3.0) a)) (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * sqrt((0.1111111111111111 / (a * a)));
	} else {
		tmp = fma(-0.375, ((c * c) / (pow(b, 3.0) / a)), ((c * -0.5) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * sqrt(Float64(0.1111111111111111 / Float64(a * a))));
	else
		tmp = fma(-0.375, Float64(Float64(c * c) / Float64((b ^ 3.0) / a)), Float64(Float64(c * -0.5) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.2], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[Sqrt[N[(0.1111111111111111 / N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.1999999999999993

   1. Initial program 83.7%

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

      1. neg-sub0 83.7%

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

      2. associate-+l- 83.7%

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

      3. sub0-neg 83.7%

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

      4. neg-mul-1 83.7%

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

      5. associate-*r/ 83.7%

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

      6. *-commutative 83.7%

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

      7. metadata-eval 83.7%

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

      8. metadata-eval 83.7%

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

      9. times-frac 83.7%

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

      10. *-commutative 83.7%

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

      11. times-frac 83.7%

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

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 84.0%

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

      2. sqrt-unprod 84.0%

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

      3. frac-times 84.1%

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

      4. metadata-eval 84.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \sqrt{\frac{\color{blue}{0.1111111111111111}}{a \cdot a}} \]

   5. Applied egg-rr 84.1%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{\sqrt{\frac{0.1111111111111111}{a \cdot a}}} \]

   if 8.1999999999999993 < b

   1. Initial program 48.8%

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

      1. /-rgt-identity 48.8%

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

      2. metadata-eval 48.8%

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

      3. associate-/l* 48.8%

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

      4. associate-*r/ 48.8%

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

      5. *-commutative 48.8%

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

      6. associate-*l/ 48.8%

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

      7. associate-*r/ 48.8%

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

      8. metadata-eval 48.8%

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

      9. metadata-eval 48.8%

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

      10. times-frac 48.8%

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

      11. neg-mul-1 48.8%

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

      12. distribute-rgt-neg-in 48.8%

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

      13. times-frac 48.8%

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

      14. metadata-eval 48.8%

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

      15. neg-mul-1 48.8%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in b around inf 88.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   5. Step-by-step derivation

      1. +-commutative 88.3%

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

      2. fma-def 88.3%

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

      3. associate-/l* 88.3%

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

      4. unpow2 88.3%

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

      5. associate-*r/ 88.3%

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

   6. Simplified 88.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \sqrt{\frac{0.1111111111111111}{a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \]

Alternative 6: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.2)
   (* -0.3333333333333333 (/ (- b (sqrt (fma b b (* a (* c -3.0))))) a))
   (fma -0.375 (/ (* c c) (/ (pow b 3.0) a)) (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2) {
		tmp = -0.3333333333333333 * ((b - sqrt(fma(b, b, (a * (c * -3.0))))) / a);
	} else {
		tmp = fma(-0.375, ((c * c) / (pow(b, 3.0) / a)), ((c * -0.5) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.2)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / a));
	else
		tmp = fma(-0.375, Float64(Float64(c * c) / Float64((b ^ 3.0) / a)), Float64(Float64(c * -0.5) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.2], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.1999999999999993

   1. Initial program 83.7%

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

      1. /-rgt-identity 83.7%

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

      2. metadata-eval 83.7%

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

      3. associate-/l* 83.7%

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

      4. associate-*r/ 83.7%

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

      5. *-commutative 83.7%

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

      6. associate-*l/ 83.7%

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

      7. associate-*r/ 83.7%

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

      8. metadata-eval 83.7%

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

      9. metadata-eval 83.7%

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

      10. times-frac 83.7%

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

      11. neg-mul-1 83.7%

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

      12. distribute-rgt-neg-in 83.7%

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

      13. times-frac 83.7%

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

      14. metadata-eval 83.7%

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

      15. neg-mul-1 83.7%

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

   3. Simplified 84.1%

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

   if 8.1999999999999993 < b

   1. Initial program 48.8%

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

      1. /-rgt-identity 48.8%

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

      2. metadata-eval 48.8%

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

      3. associate-/l* 48.8%

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

      4. associate-*r/ 48.8%

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

      5. *-commutative 48.8%

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

      6. associate-*l/ 48.8%

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

      7. associate-*r/ 48.8%

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

      8. metadata-eval 48.8%

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

      9. metadata-eval 48.8%

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

      10. times-frac 48.8%

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

      11. neg-mul-1 48.8%

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

      12. distribute-rgt-neg-in 48.8%

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

      13. times-frac 48.8%

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

      14. metadata-eval 48.8%

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

      15. neg-mul-1 48.8%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in b around inf 88.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   5. Step-by-step derivation

      1. +-commutative 88.3%

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

      2. fma-def 88.3%

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

      3. associate-/l* 88.3%

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

      4. unpow2 88.3%

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

      5. associate-*r/ 88.3%

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

   6. Simplified 88.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \]

Alternative 7: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \cdot 1.125 + 1.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.2)
   (* -0.3333333333333333 (/ (- b (sqrt (fma b b (* a (* c -3.0))))) a))
   (*
    -0.3333333333333333
    (+ (* (/ (* a (* c c)) (pow b 3.0)) 1.125) (* 1.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2) {
		tmp = -0.3333333333333333 * ((b - sqrt(fma(b, b, (a * (c * -3.0))))) / a);
	} else {
		tmp = -0.3333333333333333 * ((((a * (c * c)) / pow(b, 3.0)) * 1.125) + (1.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.2)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / a));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(Float64(a * Float64(c * c)) / (b ^ 3.0)) * 1.125) + Float64(1.5 * Float64(c / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.2], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 1.125), $MachinePrecision] + N[(1.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.1999999999999993

   1. Initial program 83.7%

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

      1. /-rgt-identity 83.7%

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

      2. metadata-eval 83.7%

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

      3. associate-/l* 83.7%

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

      4. associate-*r/ 83.7%

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

      5. *-commutative 83.7%

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

      6. associate-*l/ 83.7%

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

      7. associate-*r/ 83.7%

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

      8. metadata-eval 83.7%

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

      9. metadata-eval 83.7%

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

      10. times-frac 83.7%

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

      11. neg-mul-1 83.7%

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

      12. distribute-rgt-neg-in 83.7%

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

      13. times-frac 83.7%

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

      14. metadata-eval 83.7%

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

      15. neg-mul-1 83.7%

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

   3. Simplified 84.1%

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

   if 8.1999999999999993 < b

   1. Initial program 48.8%

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

      1. /-rgt-identity 48.8%

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

      2. metadata-eval 48.8%

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

      3. associate-/l* 48.8%

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

      4. associate-*r/ 48.8%

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

      5. *-commutative 48.8%

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

      6. associate-*l/ 48.8%

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

      7. associate-*r/ 48.8%

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

      8. metadata-eval 48.8%

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

      9. metadata-eval 48.8%

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

      10. times-frac 48.8%

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

      11. neg-mul-1 48.8%

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

      12. distribute-rgt-neg-in 48.8%

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

      13. times-frac 48.8%

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

      14. metadata-eval 48.8%

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

      15. neg-mul-1 48.8%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in b around inf 88.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. fma-def 88.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]

      2. *-commutative 88.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

      3. unpow2 88.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

   6. Simplified 88.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]
   7. Step-by-step derivation

      1. fma-udef 88.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]

      2. *-commutative 88.0%

         \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \frac{\color{blue}{\left(c \cdot c\right) \cdot a}}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right) \]

   8. Applied egg-rr 88.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \cdot 1.125 + 1.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 8: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \cdot 1.125 + 1.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.2)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (*
    -0.3333333333333333
    (+ (* (/ (* a (* c c)) (pow b 3.0)) 1.125) (* 1.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 * ((((a * (c * c)) / pow(b, 3.0)) * 1.125) + (1.5 * (c / 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 <= 8.2d0) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.3333333333333333d0) * ((((a * (c * c)) / (b ** 3.0d0)) * 1.125d0) + (1.5d0 * (c / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 * ((((a * (c * c)) / Math.pow(b, 3.0)) * 1.125) + (1.5 * (c / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.2:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = -0.3333333333333333 * ((((a * (c * c)) / math.pow(b, 3.0)) * 1.125) + (1.5 * (c / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.2)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(Float64(a * Float64(c * c)) / (b ^ 3.0)) * 1.125) + Float64(1.5 * Float64(c / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.2)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	else
		tmp = -0.3333333333333333 * ((((a * (c * c)) / (b ^ 3.0)) * 1.125) + (1.5 * (c / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.2], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 1.125), $MachinePrecision] + N[(1.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.1999999999999993

   1. Initial program 83.7%

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

      1. neg-sub0 83.7%

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

      2. associate-+l- 83.7%

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

      3. sub0-neg 83.7%

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

      4. neg-mul-1 83.7%

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

      5. associate-*r/ 83.7%

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

      6. metadata-eval 83.7%

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

      7. metadata-eval 83.7%

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

      8. times-frac 83.7%

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

      9. *-commutative 83.7%

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

      10. times-frac 83.7%

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

      11. associate-*l/ 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in a around 0 83.7%

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

      1. associate-*r* 83.8%

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

   6. Simplified 83.8%

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

   if 8.1999999999999993 < b

   1. Initial program 48.8%

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

      1. /-rgt-identity 48.8%

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

      2. metadata-eval 48.8%

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

      3. associate-/l* 48.8%

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

      4. associate-*r/ 48.8%

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

      5. *-commutative 48.8%

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

      6. associate-*l/ 48.8%

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

      7. associate-*r/ 48.8%

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

      8. metadata-eval 48.8%

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

      9. metadata-eval 48.8%

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

      10. times-frac 48.8%

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

      11. neg-mul-1 48.8%

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

      12. distribute-rgt-neg-in 48.8%

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

      13. times-frac 48.8%

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

      14. metadata-eval 48.8%

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

      15. neg-mul-1 48.8%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in b around inf 88.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. fma-def 88.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]

      2. *-commutative 88.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

      3. unpow2 88.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

   6. Simplified 88.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]
   7. Step-by-step derivation

      1. fma-udef 88.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]

      2. *-commutative 88.0%

         \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \frac{\color{blue}{\left(c \cdot c\right) \cdot a}}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right) \]

   8. Applied egg-rr 88.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} \cdot 1.125 + 1.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 9: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(c \cdot \left(\frac{1.5}{b} + 1.125 \cdot \left(c \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.2)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (*
    -0.3333333333333333
    (* c (+ (/ 1.5 b) (* 1.125 (* c (* a (pow b -3.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * pow(b, -3.0))))));
	}
	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 <= 8.2d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.3333333333333333d0) * (c * ((1.5d0 / b) + (1.125d0 * (c * (a * (b ** (-3.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * Math.pow(b, -3.0))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.2:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * math.pow(b, -3.0))))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.2)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.3333333333333333 * Float64(c * Float64(Float64(1.5 / b) + Float64(1.125 * Float64(c * Float64(a * (b ^ -3.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.2)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * (b ^ -3.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.2], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(c * N[(N[(1.5 / b), $MachinePrecision] + N[(1.125 * N[(c * N[(a * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.1999999999999993

   1. Initial program 83.7%

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

      1. neg-sub0 83.7%

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

      2. associate-+l- 83.7%

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

      3. sub0-neg 83.7%

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

      4. neg-mul-1 83.7%

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

      5. associate-*r/ 83.7%

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

      6. metadata-eval 83.7%

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

      7. metadata-eval 83.7%

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

      8. times-frac 83.7%

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

      9. *-commutative 83.7%

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

      10. times-frac 83.7%

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

      11. associate-*l/ 83.7%

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

   3. Simplified 83.7%

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

   if 8.1999999999999993 < b

   1. Initial program 48.8%

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

      1. /-rgt-identity 48.8%

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

      2. metadata-eval 48.8%

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

      3. associate-/l* 48.8%

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

      4. associate-*r/ 48.8%

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

      5. *-commutative 48.8%

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

      6. associate-*l/ 48.8%

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

      7. associate-*r/ 48.8%

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

      8. metadata-eval 48.8%

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

      9. metadata-eval 48.8%

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

      10. times-frac 48.8%

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

      11. neg-mul-1 48.8%

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

      12. distribute-rgt-neg-in 48.8%

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

      13. times-frac 48.8%

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

      14. metadata-eval 48.8%

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

      15. neg-mul-1 48.8%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in b around inf 88.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. fma-def 88.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]

      2. *-commutative 88.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

      3. unpow2 88.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

   6. Simplified 88.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]
   7. Step-by-step derivation

      1. add-exp-log 86.4%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)\right)}} \]

      2. *-commutative 86.4%

         \[\leadsto -0.3333333333333333 \cdot e^{\log \left(\mathsf{fma}\left(1.125, \frac{\color{blue}{\left(c \cdot c\right) \cdot a}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)\right)} \]

   8. Applied egg-rr 86.4%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(1.125, \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)\right)}} \]
   9. Step-by-step derivation

      1. add-exp-log 88.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]

      2. fma-udef 88.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]

      3. div-inv 88.0%

         \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \color{blue}{\left(\left(\left(c \cdot c\right) \cdot a\right) \cdot \frac{1}{{b}^{3}}\right)} + 1.5 \cdot \frac{c}{b}\right) \]

      4. associate-*l* 88.0%

         \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \left(\color{blue}{\left(c \cdot \left(c \cdot a\right)\right)} \cdot \frac{1}{{b}^{3}}\right) + 1.5 \cdot \frac{c}{b}\right) \]

      5. pow-flip 88.0%

         \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right) + 1.5 \cdot \frac{c}{b}\right) \]

      6. metadata-eval 88.0%

         \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{\color{blue}{-3}}\right) + 1.5 \cdot \frac{c}{b}\right) \]

   10. Applied egg-rr 88.0%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right) + 1.5 \cdot \frac{c}{b}\right)} \]
   11. Step-by-step derivation

       1. +-commutative 88.0%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right)} \]

       2. *-commutative 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\frac{c}{b} \cdot 1.5} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

       3. metadata-eval 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(\frac{c}{b} \cdot \color{blue}{\frac{1}{0.6666666666666666}} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

       4. times-frac 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\frac{c \cdot 1}{b \cdot 0.6666666666666666}} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

       5. associate-*r/ 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{c \cdot \frac{1}{b \cdot 0.6666666666666666}} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

       6. *-commutative 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \frac{1}{b \cdot 0.6666666666666666} + \color{blue}{\left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right) \cdot 1.125}\right) \]

       7. associate-*l* 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \frac{1}{b \cdot 0.6666666666666666} + \color{blue}{\left(c \cdot \left(\left(c \cdot a\right) \cdot {b}^{-3}\right)\right)} \cdot 1.125\right) \]

       8. associate-*l* 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \frac{1}{b \cdot 0.6666666666666666} + \color{blue}{c \cdot \left(\left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)}\right) \]

       9. distribute-lft-out 88.0%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(c \cdot \left(\frac{1}{b \cdot 0.6666666666666666} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right)} \]

       10. *-commutative 88.0%

           \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\frac{1}{\color{blue}{0.6666666666666666 \cdot b}} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right) \]

       11. associate-/r* 88.0%

           \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\color{blue}{\frac{\frac{1}{0.6666666666666666}}{b}} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right) \]

       12. metadata-eval 88.0%

           \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\frac{\color{blue}{1.5}}{b} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right) \]

       13. associate-*l* 88.0%

           \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\frac{1.5}{b} + \color{blue}{\left(c \cdot \left(a \cdot {b}^{-3}\right)\right)} \cdot 1.125\right)\right) \]

   12. Simplified 88.0%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(c \cdot \left(\frac{1.5}{b} + \left(c \cdot \left(a \cdot {b}^{-3}\right)\right) \cdot 1.125\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(c \cdot \left(\frac{1.5}{b} + 1.125 \cdot \left(c \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(c \cdot \left(\frac{1.5}{b} + 1.125 \cdot \left(c \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.2)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (*
    -0.3333333333333333
    (* c (+ (/ 1.5 b) (* 1.125 (* c (* a (pow b -3.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * pow(b, -3.0))))));
	}
	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 <= 8.2d0) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.3333333333333333d0) * (c * ((1.5d0 / b) + (1.125d0 * (c * (a * (b ** (-3.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * Math.pow(b, -3.0))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.2:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * math.pow(b, -3.0))))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.2)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.3333333333333333 * Float64(c * Float64(Float64(1.5 / b) + Float64(1.125 * Float64(c * Float64(a * (b ^ -3.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.2)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	else
		tmp = -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * (b ^ -3.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.2], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(c * N[(N[(1.5 / b), $MachinePrecision] + N[(1.125 * N[(c * N[(a * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.1999999999999993

   1. Initial program 83.7%

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

      1. neg-sub0 83.7%

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

      2. associate-+l- 83.7%

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

      3. sub0-neg 83.7%

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

      4. neg-mul-1 83.7%

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

      5. associate-*r/ 83.7%

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

      6. metadata-eval 83.7%

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

      7. metadata-eval 83.7%

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

      8. times-frac 83.7%

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

      9. *-commutative 83.7%

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

      10. times-frac 83.7%

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

      11. associate-*l/ 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in a around 0 83.7%

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

      1. associate-*r* 83.8%

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

   6. Simplified 83.8%

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

   if 8.1999999999999993 < b

   1. Initial program 48.8%

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

      1. /-rgt-identity 48.8%

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

      2. metadata-eval 48.8%

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

      3. associate-/l* 48.8%

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

      4. associate-*r/ 48.8%

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

      5. *-commutative 48.8%

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

      6. associate-*l/ 48.8%

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

      7. associate-*r/ 48.8%

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

      8. metadata-eval 48.8%

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

      9. metadata-eval 48.8%

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

      10. times-frac 48.8%

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

      11. neg-mul-1 48.8%

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

      12. distribute-rgt-neg-in 48.8%

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

      13. times-frac 48.8%

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

      14. metadata-eval 48.8%

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

      15. neg-mul-1 48.8%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in b around inf 88.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. fma-def 88.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]

      2. *-commutative 88.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

      3. unpow2 88.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

   6. Simplified 88.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]
   7. Step-by-step derivation

      1. add-exp-log 86.4%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)\right)}} \]

      2. *-commutative 86.4%

         \[\leadsto -0.3333333333333333 \cdot e^{\log \left(\mathsf{fma}\left(1.125, \frac{\color{blue}{\left(c \cdot c\right) \cdot a}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)\right)} \]

   8. Applied egg-rr 86.4%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(1.125, \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)\right)}} \]
   9. Step-by-step derivation

      1. add-exp-log 88.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]

      2. fma-udef 88.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]

      3. div-inv 88.0%

         \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \color{blue}{\left(\left(\left(c \cdot c\right) \cdot a\right) \cdot \frac{1}{{b}^{3}}\right)} + 1.5 \cdot \frac{c}{b}\right) \]

      4. associate-*l* 88.0%

         \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \left(\color{blue}{\left(c \cdot \left(c \cdot a\right)\right)} \cdot \frac{1}{{b}^{3}}\right) + 1.5 \cdot \frac{c}{b}\right) \]

      5. pow-flip 88.0%

         \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right) + 1.5 \cdot \frac{c}{b}\right) \]

      6. metadata-eval 88.0%

         \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{\color{blue}{-3}}\right) + 1.5 \cdot \frac{c}{b}\right) \]

   10. Applied egg-rr 88.0%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right) + 1.5 \cdot \frac{c}{b}\right)} \]
   11. Step-by-step derivation

       1. +-commutative 88.0%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right)} \]

       2. *-commutative 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\frac{c}{b} \cdot 1.5} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

       3. metadata-eval 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(\frac{c}{b} \cdot \color{blue}{\frac{1}{0.6666666666666666}} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

       4. times-frac 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\frac{c \cdot 1}{b \cdot 0.6666666666666666}} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

       5. associate-*r/ 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{c \cdot \frac{1}{b \cdot 0.6666666666666666}} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

       6. *-commutative 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \frac{1}{b \cdot 0.6666666666666666} + \color{blue}{\left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right) \cdot 1.125}\right) \]

       7. associate-*l* 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \frac{1}{b \cdot 0.6666666666666666} + \color{blue}{\left(c \cdot \left(\left(c \cdot a\right) \cdot {b}^{-3}\right)\right)} \cdot 1.125\right) \]

       8. associate-*l* 88.0%

          \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \frac{1}{b \cdot 0.6666666666666666} + \color{blue}{c \cdot \left(\left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)}\right) \]

       9. distribute-lft-out 88.0%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(c \cdot \left(\frac{1}{b \cdot 0.6666666666666666} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right)} \]

       10. *-commutative 88.0%

           \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\frac{1}{\color{blue}{0.6666666666666666 \cdot b}} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right) \]

       11. associate-/r* 88.0%

           \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\color{blue}{\frac{\frac{1}{0.6666666666666666}}{b}} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right) \]

       12. metadata-eval 88.0%

           \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\frac{\color{blue}{1.5}}{b} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right) \]

       13. associate-*l* 88.0%

           \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\frac{1.5}{b} + \color{blue}{\left(c \cdot \left(a \cdot {b}^{-3}\right)\right)} \cdot 1.125\right)\right) \]

   12. Simplified 88.0%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(c \cdot \left(\frac{1.5}{b} + \left(c \cdot \left(a \cdot {b}^{-3}\right)\right) \cdot 1.125\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(c \cdot \left(\frac{1.5}{b} + 1.125 \cdot \left(c \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right)\\ \end{array} \]

Alternative 11: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \left(c \cdot \left(\frac{1.5}{b} + 1.125 \cdot \left(c \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  -0.3333333333333333
  (* c (+ (/ 1.5 b) (* 1.125 (* c (* a (pow b -3.0))))))))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * pow(b, -3.0))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.3333333333333333d0) * (c * ((1.5d0 / b) + (1.125d0 * (c * (a * (b ** (-3.0d0)))))))
end function

Java

public static double code(double a, double b, double c) {
	return -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * Math.pow(b, -3.0))))));
}

Python

def code(a, b, c):
	return -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * math.pow(b, -3.0))))))

Julia

function code(a, b, c)
	return Float64(-0.3333333333333333 * Float64(c * Float64(Float64(1.5 / b) + Float64(1.125 * Float64(c * Float64(a * (b ^ -3.0)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 * (c * ((1.5 / b) + (1.125 * (c * (a * (b ^ -3.0))))));
end

Wolfram

code[a_, b_, c_] := N[(-0.3333333333333333 * N[(c * N[(N[(1.5 / b), $MachinePrecision] + N[(1.125 * N[(c * N[(a * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

   1. /-rgt-identity 57.2%

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

   2. metadata-eval 57.2%

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

   3. associate-/l* 57.2%

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

   4. associate-*r/ 57.1%

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

   5. *-commutative 57.1%

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

   6. associate-*l/ 57.2%

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

   7. associate-*r/ 57.2%

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

   8. metadata-eval 57.2%

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

   9. metadata-eval 57.2%

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

   10. times-frac 57.2%

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

   11. neg-mul-1 57.2%

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

   12. distribute-rgt-neg-in 57.2%

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

   13. times-frac 57.1%

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

   14. metadata-eval 57.1%

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

   15. neg-mul-1 57.1%

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

3. Simplified 57.3%

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

4. Taylor expanded in b around inf 80.2%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
5. Step-by-step derivation

   1. fma-def 80.2%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]

   2. *-commutative 80.2%

      \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

   3. unpow2 80.2%

      \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

6. Simplified 80.2%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]
7. Step-by-step derivation

   1. add-exp-log 78.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)\right)}} \]

   2. *-commutative 78.9%

      \[\leadsto -0.3333333333333333 \cdot e^{\log \left(\mathsf{fma}\left(1.125, \frac{\color{blue}{\left(c \cdot c\right) \cdot a}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)\right)} \]

8. Applied egg-rr 78.9%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(1.125, \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)\right)}} \]
9. Step-by-step derivation

   1. add-exp-log 80.2%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]

   2. fma-udef 80.2%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{\left(c \cdot c\right) \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]

   3. div-inv 80.2%

      \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \color{blue}{\left(\left(\left(c \cdot c\right) \cdot a\right) \cdot \frac{1}{{b}^{3}}\right)} + 1.5 \cdot \frac{c}{b}\right) \]

   4. associate-*l* 80.2%

      \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \left(\color{blue}{\left(c \cdot \left(c \cdot a\right)\right)} \cdot \frac{1}{{b}^{3}}\right) + 1.5 \cdot \frac{c}{b}\right) \]

   5. pow-flip 80.2%

      \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right) + 1.5 \cdot \frac{c}{b}\right) \]

   6. metadata-eval 80.2%

      \[\leadsto -0.3333333333333333 \cdot \left(1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{\color{blue}{-3}}\right) + 1.5 \cdot \frac{c}{b}\right) \]

10. Applied egg-rr 80.2%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right) + 1.5 \cdot \frac{c}{b}\right)} \]
11. Step-by-step derivation

    1. +-commutative 80.2%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right)} \]

    2. *-commutative 80.2%

       \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\frac{c}{b} \cdot 1.5} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

    3. metadata-eval 80.2%

       \[\leadsto -0.3333333333333333 \cdot \left(\frac{c}{b} \cdot \color{blue}{\frac{1}{0.6666666666666666}} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

    4. times-frac 80.2%

       \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\frac{c \cdot 1}{b \cdot 0.6666666666666666}} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

    5. associate-*r/ 80.2%

       \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{c \cdot \frac{1}{b \cdot 0.6666666666666666}} + 1.125 \cdot \left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right)\right) \]

    6. *-commutative 80.2%

       \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \frac{1}{b \cdot 0.6666666666666666} + \color{blue}{\left(\left(c \cdot \left(c \cdot a\right)\right) \cdot {b}^{-3}\right) \cdot 1.125}\right) \]

    7. associate-*l* 80.2%

       \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \frac{1}{b \cdot 0.6666666666666666} + \color{blue}{\left(c \cdot \left(\left(c \cdot a\right) \cdot {b}^{-3}\right)\right)} \cdot 1.125\right) \]

    8. associate-*l* 80.2%

       \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \frac{1}{b \cdot 0.6666666666666666} + \color{blue}{c \cdot \left(\left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)}\right) \]

    9. distribute-lft-out 80.2%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(c \cdot \left(\frac{1}{b \cdot 0.6666666666666666} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right)} \]

    10. *-commutative 80.2%

        \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\frac{1}{\color{blue}{0.6666666666666666 \cdot b}} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right) \]

    11. associate-/r* 80.1%

        \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\color{blue}{\frac{\frac{1}{0.6666666666666666}}{b}} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right) \]

    12. metadata-eval 80.1%

        \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\frac{\color{blue}{1.5}}{b} + \left(\left(c \cdot a\right) \cdot {b}^{-3}\right) \cdot 1.125\right)\right) \]

    13. associate-*l* 80.1%

        \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\frac{1.5}{b} + \color{blue}{\left(c \cdot \left(a \cdot {b}^{-3}\right)\right)} \cdot 1.125\right)\right) \]

12. Simplified 80.1%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(c \cdot \left(\frac{1.5}{b} + \left(c \cdot \left(a \cdot {b}^{-3}\right)\right) \cdot 1.125\right)\right)} \]

13. Final simplification 80.1%

    \[\leadsto -0.3333333333333333 \cdot \left(c \cdot \left(\frac{1.5}{b} + 1.125 \cdot \left(c \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) \]

Alternative 12: 64.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

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) / (b / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

   1. /-rgt-identity 57.2%

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

   2. metadata-eval 57.2%

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

   3. associate-/l* 57.2%

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

   4. associate-*r/ 57.1%

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

   5. *-commutative 57.1%

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

   6. associate-*l/ 57.2%

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

   7. associate-*r/ 57.2%

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

   8. metadata-eval 57.2%

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

   9. metadata-eval 57.2%

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

   10. times-frac 57.2%

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

   11. neg-mul-1 57.2%

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

   12. distribute-rgt-neg-in 57.2%

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

   13. times-frac 57.1%

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

   14. metadata-eval 57.1%

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

   15. neg-mul-1 57.1%

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

3. Simplified 57.3%

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

4. Taylor expanded in b around inf 80.2%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
5. Step-by-step derivation

   1. fma-def 80.2%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]

   2. *-commutative 80.2%

      \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

   3. unpow2 80.2%

      \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.125, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right) \]

6. Simplified 80.2%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.125, \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}, 1.5 \cdot \frac{c}{b}\right)} \]

7. Taylor expanded in a around 0 62.8%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 62.7%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1.5 \cdot c}{b}} \]

   2. associate-/l* 62.7%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1.5}{\frac{b}{c}}} \]

   3. associate-/r/ 62.7%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{1.5}{b} \cdot c\right)} \]

9. Simplified 62.7%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{1.5}{b} \cdot c\right)} \]

10. Taylor expanded in b around 0 62.8%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b}\right)} \]
11. Step-by-step derivation

    1. associate-*r/ 62.7%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1.5 \cdot c}{b}} \]

    2. *-commutative 62.7%

       \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{c \cdot 1.5}}{b} \]

    3. associate-/l* 62.8%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{c}{\frac{b}{1.5}}} \]

12. Simplified 62.8%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{c}{\frac{b}{1.5}}} \]

13. Taylor expanded in c around 0 62.8%

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

    1. associate-*r/ 62.8%

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

    2. associate-/l* 62.7%

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

15. Simplified 62.7%

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

16. Final simplification 62.7%

    \[\leadsto \frac{-0.5}{\frac{b}{c}} \]

Alternative 13: 64.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

   1. /-rgt-identity 57.2%

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

   2. metadata-eval 57.2%

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

   3. associate-/l* 57.2%

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

   4. associate-*r/ 57.1%

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

   5. *-commutative 57.1%

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

   6. associate-*l/ 57.2%

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

   7. associate-*r/ 57.2%

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

   8. metadata-eval 57.2%

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

   9. metadata-eval 57.2%

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

   10. times-frac 57.2%

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

   11. neg-mul-1 57.2%

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

   12. distribute-rgt-neg-in 57.2%

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

   13. times-frac 57.1%

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

   14. metadata-eval 57.1%

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

   15. neg-mul-1 57.1%

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

3. Simplified 57.3%

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

4. Taylor expanded in b around inf 62.8%

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

   1. associate-*r/ 62.8%

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

6. Simplified 62.8%

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

7. Final simplification 62.8%

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

Reproduce

herbie shell --seed 2023200 
(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)))