Cubic critical, narrow range

Percentage Accurate: 54.7% → 92.1%

Time: 22.0s

Alternatives: 10

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% 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: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.45:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{\frac{{\left(a \cdot c\right)}^{4}}{a}}{\frac{{b}^{7}}{-1.0546875}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (* b b))))
   (if (<= b 0.45)
     (* 0.3333333333333333 (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) a))
     (fma
      -0.5625
      (/ (* a a) (/ (pow b 5.0) (pow c 3.0)))
      (fma
       -0.5
       (/ c b)
       (fma
        -0.375
        (/ a (/ (pow b 3.0) (* c c)))
        (/ (/ (pow (* a c) 4.0) a) (/ (pow b 7.0) -1.0546875))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), (b * b));
	double tmp;
	if (b <= 0.45) {
		tmp = 0.3333333333333333 * (((t_0 - (b * b)) / (b + sqrt(t_0))) / a);
	} else {
		tmp = fma(-0.5625, ((a * a) / (pow(b, 5.0) / pow(c, 3.0))), fma(-0.5, (c / b), fma(-0.375, (a / (pow(b, 3.0) / (c * c))), ((pow((a * c), 4.0) / a) / (pow(b, 7.0) / -1.0546875)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 0.45)
		tmp = Float64(0.3333333333333333 * Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / a));
	else
		tmp = fma(-0.5625, Float64(Float64(a * a) / Float64((b ^ 5.0) / (c ^ 3.0))), fma(-0.5, Float64(c / b), fma(-0.375, Float64(a / Float64((b ^ 3.0) / Float64(c * c))), Float64(Float64((Float64(a * c) ^ 4.0) / a) / Float64((b ^ 7.0) / -1.0546875)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.45], N[(0.3333333333333333 * N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / -1.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.450000000000000011

   1. Initial program 86.2%

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

      1. neg-sub0 86.2%

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

      2. sqr-neg 86.2%

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

      3. associate-+l- 86.2%

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

      4. sub0-neg 86.2%

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

      5. neg-mul-1 86.2%

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

   3. Simplified 86.4%

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

      1. div-inv 86.4%

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

      2. metadata-eval 86.4%

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

      3. *-commutative 86.4%

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

      4. add-cbrt-cube 86.5%

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

      5. pow3 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. rem-cbrt-cube 86.4%

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

      2. div-sub 85.6%

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

      3. *-commutative 85.6%

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

      4. *-commutative 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. div-sub 86.4%

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

      2. *-lft-identity 86.4%

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

      3. *-commutative 86.4%

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

      4. times-frac 86.4%

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

      5. metadata-eval 86.4%

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

      6. fma-udef 86.2%

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

      7. unpow2 86.2%

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

      8. associate-*r* 86.1%

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

      9. *-commutative 86.1%

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

      10. +-commutative 86.1%

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

      11. *-commutative 86.1%

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

      12. associate-*r* 86.2%

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

      13. fma-def 86.2%

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

      14. unpow2 86.2%

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

   9. Simplified 86.2%

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

       1. flip-- 86.1%

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

       2. add-sqr-sqrt 87.5%

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

   11. Applied egg-rr 87.5%

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

   if 0.450000000000000011 < b

   1. Initial program 52.0%

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

      1. sqr-neg 52.0%

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

      2. sqr-neg 52.0%

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

      3. associate-*l* 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in b around inf 93.1%

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

      1. fma-def 93.1%

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

      2. associate-/l* 93.1%

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

      3. unpow2 93.1%

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

      4. fma-def 93.1%

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

      5. fma-def 93.1%

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

   6. Simplified 93.1%

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

   7. Taylor expanded in c around 0 93.1%

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

   9. Simplified 93.1%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{\frac{a}{6.328125}} \cdot \frac{-0.16666666666666666}{{b}^{7}}}\right)\right)\right) \]
   10. Step-by-step derivation

       1. frac-times 93.1%

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

       2. div-inv 93.1%

          \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{\color{blue}{\left(a \cdot \frac{1}{6.328125}\right)} \cdot {b}^{7}}\right)\right)\right) \]

       3. metadata-eval 93.1%

          \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{\left(a \cdot \color{blue}{0.1580246913580247}\right) \cdot {b}^{7}}\right)\right)\right) \]

   11. Applied egg-rr 93.1%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{\left(a \cdot 0.1580246913580247\right) \cdot {b}^{7}}}\right)\right)\right) \]
   12. Step-by-step derivation

       1. associate-/r* 93.1%

          \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \color{blue}{\frac{\frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{a \cdot 0.1580246913580247}}{{b}^{7}}}\right)\right)\right) \]

       2. times-frac 93.1%

          \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{\color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{-0.16666666666666666}{0.1580246913580247}}}{{b}^{7}}\right)\right)\right) \]

       3. metadata-eval 93.1%

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

       4. associate-/l* 93.1%

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

   13. Simplified 93.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.45:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{\frac{{\left(a \cdot c\right)}^{4}}{a}}{\frac{{b}^{7}}{-1.0546875}}\right)\right)\right)\\ \end{array} \]

Alternative 2: 90.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.54:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (* b b))))
   (if (<= b 0.54)
     (* 0.3333333333333333 (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) a))
     (fma
      -0.5625
      (/ (* a a) (/ (pow b 5.0) (pow c 3.0)))
      (fma -0.5 (/ c b) (* -0.375 (/ a (/ (pow b 3.0) (* c c)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), (b * b));
	double tmp;
	if (b <= 0.54) {
		tmp = 0.3333333333333333 * (((t_0 - (b * b)) / (b + sqrt(t_0))) / a);
	} else {
		tmp = fma(-0.5625, ((a * a) / (pow(b, 5.0) / pow(c, 3.0))), fma(-0.5, (c / b), (-0.375 * (a / (pow(b, 3.0) / (c * c))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 0.54)
		tmp = Float64(0.3333333333333333 * Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / a));
	else
		tmp = fma(-0.5625, Float64(Float64(a * a) / Float64((b ^ 5.0) / (c ^ 3.0))), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(a / Float64((b ^ 3.0) / Float64(c * c))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.54], N[(0.3333333333333333 * N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.54000000000000004

   1. Initial program 86.1%

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

      1. neg-sub0 86.1%

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

      2. sqr-neg 86.1%

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

      3. associate-+l- 86.1%

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

      4. sub0-neg 86.1%

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

      5. neg-mul-1 86.1%

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

   3. Simplified 86.3%

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

      1. div-inv 86.3%

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

      2. metadata-eval 86.3%

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

      3. *-commutative 86.3%

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

      4. add-cbrt-cube 86.4%

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

      5. pow3 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. rem-cbrt-cube 86.3%

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

      2. div-sub 85.5%

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

      3. *-commutative 85.5%

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

      4. *-commutative 85.5%

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

   7. Applied egg-rr 85.5%

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

      1. div-sub 86.3%

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

      2. *-lft-identity 86.3%

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

      3. *-commutative 86.3%

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

      4. times-frac 86.3%

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

      5. metadata-eval 86.3%

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

      6. fma-udef 86.1%

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

      7. unpow2 86.1%

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

      8. associate-*r* 86.0%

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

      9. *-commutative 86.0%

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

      10. +-commutative 86.0%

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

      11. *-commutative 86.0%

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

      12. associate-*r* 86.1%

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

      13. fma-def 86.1%

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

      14. unpow2 86.1%

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

   9. Simplified 86.1%

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

       1. flip-- 86.0%

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

       2. add-sqr-sqrt 87.5%

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

   11. Applied egg-rr 87.5%

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

   if 0.54000000000000004 < b

   1. Initial program 51.8%

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

      1. sqr-neg 51.8%

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

      2. sqr-neg 51.8%

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

      3. associate-*l* 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in b around inf 90.5%

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

      1. fma-def 90.5%

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

      2. associate-/l* 90.5%

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

      3. unpow2 90.5%

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

      4. fma-def 90.5%

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

      5. associate-/l* 90.5%

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

      6. unpow2 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.54:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)\\ \end{array} \]

Alternative 3: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 235:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (* b b))))
   (if (<= b 235.0)
     (* 0.3333333333333333 (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) a))
     (+ (* -0.375 (* (* c c) (/ a (pow b 3.0)))) (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), (b * b));
	double tmp;
	if (b <= 235.0) {
		tmp = 0.3333333333333333 * (((t_0 - (b * b)) / (b + sqrt(t_0))) / a);
	} else {
		tmp = (-0.375 * ((c * c) * (a / pow(b, 3.0)))) + (-0.5 * (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 235.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / a));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(c * c) * Float64(a / (b ^ 3.0)))) + Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 235.0], N[(0.3333333333333333 * N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 235

   1. Initial program 79.7%

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

      1. neg-sub0 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-+l- 79.7%

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

      4. sub0-neg 79.7%

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

      5. neg-mul-1 79.7%

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

   3. Simplified 79.9%

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

      1. div-inv 79.9%

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

      2. metadata-eval 79.9%

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

      3. *-commutative 79.9%

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

      4. add-cbrt-cube 79.9%

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

      5. pow3 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. rem-cbrt-cube 79.9%

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

      2. div-sub 78.9%

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

      3. *-commutative 78.9%

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

      4. *-commutative 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. div-sub 79.9%

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

      2. *-lft-identity 79.9%

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

      3. *-commutative 79.9%

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

      4. times-frac 79.9%

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

      5. metadata-eval 79.9%

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

      6. fma-udef 79.7%

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

      7. unpow2 79.7%

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

      8. associate-*r* 79.7%

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

      9. *-commutative 79.7%

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

      10. +-commutative 79.7%

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

      11. *-commutative 79.7%

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

      12. associate-*r* 79.7%

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

      13. fma-def 79.7%

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

      14. unpow2 79.7%

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

   9. Simplified 79.7%

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

       1. flip-- 79.7%

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

       2. add-sqr-sqrt 81.4%

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

   11. Applied egg-rr 81.4%

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

   if 235 < b

   1. Initial program 44.5%

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

      1. sqr-neg 44.5%

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

      2. sqr-neg 44.5%

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

      3. associate-*l* 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in b around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. fma-def 90.0%

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

      3. associate-/l* 90.0%

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

      4. associate-/r/ 90.0%

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

      5. unpow2 90.0%

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

   6. Simplified 90.0%

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

      1. fma-udef 90.0%

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

      2. *-commutative 90.0%

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

   8. Applied egg-rr 90.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\sqrt[3]{\left(a \cdot 3\right) \cdot \left(\left(a \cdot 3\right) \cdot \left(a \cdot 3\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 235.0)
   (/
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (cbrt (* (* a 3.0) (* (* a 3.0) (* a 3.0)))))
   (+ (* -0.375 (* (* c c) (/ a (pow b 3.0)))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 235.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / cbrt(((a * 3.0) * ((a * 3.0) * (a * 3.0))));
	} else {
		tmp = (-0.375 * ((c * c) * (a / pow(b, 3.0)))) + (-0.5 * (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 235.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / cbrt(Float64(Float64(a * 3.0) * Float64(Float64(a * 3.0) * Float64(a * 3.0)))));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(c * c) * Float64(a / (b ^ 3.0)))) + Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 235.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[(N[(a * 3.0), $MachinePrecision] * N[(N[(a * 3.0), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 235

   1. Initial program 79.7%

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

      1. neg-sub0 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-+l- 79.7%

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

      4. sub0-neg 79.7%

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

      5. neg-mul-1 79.7%

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

   3. Simplified 79.9%

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

      1. div-inv 79.9%

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

      2. metadata-eval 79.9%

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

      3. *-commutative 79.9%

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

      4. add-cbrt-cube 79.9%

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

      5. pow3 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. add-cube-cbrt 79.9%

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

      2. rem-cbrt-cube 79.9%

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

      3. *-commutative 79.9%

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

      4. rem-cbrt-cube 79.9%

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

      5. *-commutative 79.9%

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

      6. rem-cbrt-cube 79.9%

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

      7. *-commutative 79.9%

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

   7. Applied egg-rr 79.9%

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

   if 235 < b

   1. Initial program 44.5%

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

      1. sqr-neg 44.5%

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

      2. sqr-neg 44.5%

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

      3. associate-*l* 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in b around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. fma-def 90.0%

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

      3. associate-/l* 90.0%

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

      4. associate-/r/ 90.0%

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

      5. unpow2 90.0%

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

   6. Simplified 90.0%

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

      1. fma-udef 90.0%

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

      2. *-commutative 90.0%

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

   8. Applied egg-rr 90.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\sqrt[3]{\left(a \cdot 3\right) \cdot \left(\left(a \cdot 3\right) \cdot \left(a \cdot 3\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\sqrt[3]{-27} \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 235.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* (cbrt -27.0) (- a)))
   (+ (* -0.375 (* (* c c) (/ a (pow b 3.0)))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 235.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (cbrt(-27.0) * -a);
	} else {
		tmp = (-0.375 * ((c * c) * (a / pow(b, 3.0)))) + (-0.5 * (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 235.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(cbrt(-27.0) * Float64(-a)));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(c * c) * Float64(a / (b ^ 3.0)))) + Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 235.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(N[Power[-27.0, 1/3], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 235

   1. Initial program 79.7%

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

      1. neg-sub0 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-+l- 79.7%

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

      4. sub0-neg 79.7%

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

      5. neg-mul-1 79.7%

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

   3. Simplified 79.9%

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

      1. div-inv 79.9%

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

      2. metadata-eval 79.9%

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

      3. *-commutative 79.9%

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

      4. add-cbrt-cube 79.9%

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

      5. pow3 79.9%

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

   5. Applied egg-rr 79.9%

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

   6. Taylor expanded in a around -inf 79.9%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{-1 \cdot \left(a \cdot \sqrt[3]{-27}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 79.9%

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

      2. distribute-rgt-neg-in 79.9%

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

   8. Simplified 79.9%

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

   if 235 < b

   1. Initial program 44.5%

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

      1. sqr-neg 44.5%

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

      2. sqr-neg 44.5%

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

      3. associate-*l* 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in b around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. fma-def 90.0%

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

      3. associate-/l* 90.0%

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

      4. associate-/r/ 90.0%

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

      5. unpow2 90.0%

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

   6. Simplified 90.0%

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

      1. fma-udef 90.0%

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

      2. *-commutative 90.0%

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

   8. Applied egg-rr 90.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\sqrt[3]{-27} \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 235.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ a 0.3333333333333333))
   (+ (* -0.375 (* (* c c) (/ a (pow b 3.0)))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 235.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a / 0.3333333333333333);
	} else {
		tmp = (-0.375 * ((c * c) * (a / pow(b, 3.0)))) + (-0.5 * (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 235.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a / 0.3333333333333333));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(c * c) * Float64(a / (b ^ 3.0)))) + Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 235.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 235

   1. Initial program 79.7%

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

      1. neg-sub0 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-+l- 79.7%

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

      4. sub0-neg 79.7%

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

      5. neg-mul-1 79.7%

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

   3. Simplified 79.9%

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

   if 235 < b

   1. Initial program 44.5%

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

      1. sqr-neg 44.5%

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

      2. sqr-neg 44.5%

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

      3. associate-*l* 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in b around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. fma-def 90.0%

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

      3. associate-/l* 90.0%

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

      4. associate-/r/ 90.0%

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

      5. unpow2 90.0%

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

   6. Simplified 90.0%

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

      1. fma-udef 90.0%

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

      2. *-commutative 90.0%

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

   8. Applied egg-rr 90.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 235.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (+ (* -0.375 (* (* c c) (/ a (pow b 3.0)))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 235.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.375 * ((c * c) * (a / pow(b, 3.0)))) + (-0.5 * (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 235.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(c * c) * Float64(a / (b ^ 3.0)))) + Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 235.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 235

   1. Initial program 79.7%

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

      1. neg-sub0 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-+l- 79.7%

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

      4. sub0-neg 79.7%

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

   3. Simplified 79.9%

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

   if 235 < b

   1. Initial program 44.5%

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

      1. sqr-neg 44.5%

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

      2. sqr-neg 44.5%

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

      3. associate-*l* 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in b around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. fma-def 90.0%

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

      3. associate-/l* 90.0%

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

      4. associate-/r/ 90.0%

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

      5. unpow2 90.0%

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

   6. Simplified 90.0%

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

      1. fma-udef 90.0%

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

      2. *-commutative 90.0%

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

   8. Applied egg-rr 90.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 235.0)
   (* 0.3333333333333333 (/ (- (sqrt (+ (* b b) (* a (* c -3.0)))) b) a))
   (+ (* -0.375 (* (* c c) (/ a (pow b 3.0)))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 235.0) {
		tmp = 0.3333333333333333 * ((sqrt(((b * b) + (a * (c * -3.0)))) - b) / a);
	} else {
		tmp = (-0.375 * ((c * c) * (a / pow(b, 3.0)))) + (-0.5 * (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 <= 235.0d0) then
        tmp = 0.3333333333333333d0 * ((sqrt(((b * b) + (a * (c * (-3.0d0))))) - b) / a)
    else
        tmp = ((-0.375d0) * ((c * c) * (a / (b ** 3.0d0)))) + ((-0.5d0) * (c / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 235.0) {
		tmp = 0.3333333333333333 * ((Math.sqrt(((b * b) + (a * (c * -3.0)))) - b) / a);
	} else {
		tmp = (-0.375 * ((c * c) * (a / Math.pow(b, 3.0)))) + (-0.5 * (c / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 235.0:
		tmp = 0.3333333333333333 * ((math.sqrt(((b * b) + (a * (c * -3.0)))) - b) / a)
	else:
		tmp = (-0.375 * ((c * c) * (a / math.pow(b, 3.0)))) + (-0.5 * (c / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 235.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0)))) - b) / a));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(c * c) * Float64(a / (b ^ 3.0)))) + Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 235.0)
		tmp = 0.3333333333333333 * ((sqrt(((b * b) + (a * (c * -3.0)))) - b) / a);
	else
		tmp = (-0.375 * ((c * c) * (a / (b ^ 3.0)))) + (-0.5 * (c / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 235.0], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 235

   1. Initial program 79.7%

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

      1. neg-sub0 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-+l- 79.7%

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

      4. sub0-neg 79.7%

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

      5. neg-mul-1 79.7%

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

   3. Simplified 79.9%

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

      1. div-inv 79.9%

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

      2. metadata-eval 79.9%

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

      3. *-commutative 79.9%

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

      4. add-cbrt-cube 79.9%

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

      5. pow3 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. rem-cbrt-cube 79.9%

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

      2. div-sub 78.9%

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

      3. *-commutative 78.9%

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

      4. *-commutative 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. div-sub 79.9%

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

      2. *-lft-identity 79.9%

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

      3. *-commutative 79.9%

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

      4. times-frac 79.9%

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

      5. metadata-eval 79.9%

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

      6. fma-udef 79.7%

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

      7. unpow2 79.7%

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

      8. associate-*r* 79.7%

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

      9. *-commutative 79.7%

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

      10. +-commutative 79.7%

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

      11. *-commutative 79.7%

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

      12. associate-*r* 79.7%

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

      13. fma-def 79.7%

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

      14. unpow2 79.7%

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

   9. Simplified 79.7%

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

       1. fma-udef 79.7%

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

   11. Applied egg-rr 79.7%

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

   if 235 < b

   1. Initial program 44.5%

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

      1. sqr-neg 44.5%

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

      2. sqr-neg 44.5%

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

      3. associate-*l* 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in b around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. fma-def 90.0%

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

      3. associate-/l* 90.0%

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

      4. associate-/r/ 90.0%

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

      5. unpow2 90.0%

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

   6. Simplified 90.0%

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

      1. fma-udef 90.0%

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

      2. *-commutative 90.0%

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

   8. Applied egg-rr 90.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 280:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 280.0)
   (* 0.3333333333333333 (/ (- (sqrt (+ (* b b) (* a (* c -3.0)))) b) a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 280.0) {
		tmp = 0.3333333333333333 * ((sqrt(((b * b) + (a * (c * -3.0)))) - b) / a);
	} else {
		tmp = -0.5 * (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 <= 280.0d0) then
        tmp = 0.3333333333333333d0 * ((sqrt(((b * b) + (a * (c * (-3.0d0))))) - b) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 280.0) {
		tmp = 0.3333333333333333 * ((Math.sqrt(((b * b) + (a * (c * -3.0)))) - b) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 280.0:
		tmp = 0.3333333333333333 * ((math.sqrt(((b * b) + (a * (c * -3.0)))) - b) / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 280.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0)))) - b) / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 280.0)
		tmp = 0.3333333333333333 * ((sqrt(((b * b) + (a * (c * -3.0)))) - b) / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 280.0], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 280

   1. Initial program 79.6%

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

      1. neg-sub0 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-+l- 79.6%

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

      4. sub0-neg 79.6%

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

      5. neg-mul-1 79.6%

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

   3. Simplified 79.8%

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

      1. div-inv 79.8%

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

      2. metadata-eval 79.8%

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

      3. *-commutative 79.8%

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

      4. add-cbrt-cube 79.8%

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

      5. pow3 79.8%

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

   5. Applied egg-rr 79.8%

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

      1. rem-cbrt-cube 79.8%

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

      2. div-sub 78.8%

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

      3. *-commutative 78.8%

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

      4. *-commutative 78.8%

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

   7. Applied egg-rr 78.8%

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

      1. div-sub 79.8%

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

      2. *-lft-identity 79.8%

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

      3. *-commutative 79.8%

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

      4. times-frac 79.8%

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

      5. metadata-eval 79.8%

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

      6. fma-udef 79.6%

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

      7. unpow2 79.6%

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

      8. associate-*r* 79.6%

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

      9. *-commutative 79.6%

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

      10. +-commutative 79.6%

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

      11. *-commutative 79.6%

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

      12. associate-*r* 79.6%

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

      13. fma-def 79.6%

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

      14. unpow2 79.6%

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

   9. Simplified 79.6%

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

       1. fma-udef 79.6%

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

   11. Applied egg-rr 79.6%

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

   if 280 < b

   1. Initial program 44.4%

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

      1. sqr-neg 44.4%

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

      2. sqr-neg 44.4%

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

      3. associate-*l* 44.4%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in b around inf 73.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 280:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 64.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

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

   1. sqr-neg 57.7%

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

   2. sqr-neg 57.7%

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

   3. associate-*l* 57.7%

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

3. Simplified 57.7%

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

4. Taylor expanded in b around inf 62.1%

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

5. Final simplification 62.1%

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

Reproduce

herbie shell --seed 2023274 
(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)))