Cubic critical, narrow range

Percentage Accurate: 55.3% → 90.8%

Time: 13.1s

Alternatives: 9

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% 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: 90.8% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5625
  (* (* a a) (/ (pow c 3.0) (pow b 5.0)))
  (fma
   -0.16666666666666666
   (* (/ (pow (* a c) 4.0) (pow b 7.0)) (/ 6.328125 a))
   (fma -0.5 (/ c b) (* -0.375 (* c (* c (/ a (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	return fma(-0.5625, ((a * a) * (pow(c, 3.0) / pow(b, 5.0))), fma(-0.16666666666666666, ((pow((a * c), 4.0) / pow(b, 7.0)) * (6.328125 / a)), fma(-0.5, (c / b), (-0.375 * (c * (c * (a / pow(b, 3.0))))))));
}

Julia

function code(a, b, c)
	return fma(-0.5625, Float64(Float64(a * a) * Float64((c ^ 3.0) / (b ^ 5.0))), fma(-0.16666666666666666, Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * Float64(6.328125 / a)), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(c * Float64(c * Float64(a / (b ^ 3.0))))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(c * N[(c * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

   1. /-rgt-identity 52.4%

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

   2. metadata-eval 52.4%

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

   3. associate-/l* 52.4%

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

   4. associate-*r/ 52.4%

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

   5. *-commutative 52.4%

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

   6. associate-*l/ 52.4%

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

   7. associate-*r/ 52.4%

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

   8. metadata-eval 52.4%

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

   9. metadata-eval 52.4%

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

   10. times-frac 52.4%

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

   11. neg-mul-1 52.4%

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

   12. distribute-rgt-neg-in 52.4%

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

   13. times-frac 52.3%

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

   14. metadata-eval 52.3%

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

   15. neg-mul-1 52.3%

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

3. Simplified 52.3%

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

4. Taylor expanded in b around inf 93.6%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.6875 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)\right)} \]
5. Step-by-step derivation

   1. fma-def 93.6%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)} \]

   2. associate-/l* 93.6%

      \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

   3. unpow2 93.6%

      \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

   4. fma-def 93.6%

      \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot a}{{b}^{3}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)}\right) \]

   5. associate-/l* 93.6%

      \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

   6. unpow2 93.6%

      \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}}, 1.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

   7. fma-def 93.6%

      \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)}\right)\right) \]

6. Simplified 93.6%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(1.125, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \mathsf{fma}\left(1.5, \frac{c}{b}, 0.5 \cdot \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)\right)} \]

7. Taylor expanded in c around 0 93.8%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

9. Simplified 93.8%

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

10. Final simplification 93.8%

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

Alternative 2: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq 0.9:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(c \cdot \left(a \cdot c\right)\right)}{{b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 3.0 (* a c))))
   (if (<= b 0.9)
     (/
      (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) (sqrt (- (* b b) t_0))))
      (* a 3.0))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma -0.5 (/ c b) (/ (* -0.375 (* c (* a c))) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = 3.0 * (a * c);
	double tmp;
	if (b <= 0.9) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 3.0);
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.5, (c / b), ((-0.375 * (c * (a * c))) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(3.0 * Float64(a * c))
	tmp = 0.0
	if (b <= 0.9)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - t_0)))) / Float64(a * 3.0));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(c * Float64(a * c))) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.9], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.900000000000000022

   1. Initial program 82.1%

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

      1. flip-+ 82.6%

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

      2. pow2 82.6%

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

      3. add-sqr-sqrt 84.4%

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

      4. associate-*l* 84.4%

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

      5. associate-*l* 84.4%

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

   3. Applied egg-rr 84.4%

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

   if 0.900000000000000022 < b

   1. Initial program 47.3%

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

   2. Taylor expanded in b around inf 93.9%

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

      1. fma-def 93.9%

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

      2. associate-/l* 93.9%

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

      3. unpow2 93.9%

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

      4. fma-def 93.9%

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

      5. associate-*r/ 93.9%

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

      6. unpow2 93.9%

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

      7. associate-*l* 93.9%

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

   4. Simplified 93.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.9:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(3 \cdot \left(a \cdot c\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(c \cdot \left(a \cdot c\right)\right)}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 3: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq 0.96:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{c}{\frac{b}{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 3.0 (* a c))))
   (if (<= b 0.96)
     (/
      (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) (sqrt (- (* b b) t_0))))
      (* a 3.0))
     (+ (* -0.375 (* (/ a (pow b 3.0)) (* c c))) (/ c (/ b -0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = 3.0 * (a * c);
	double tmp;
	if (b <= 0.96) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 3.0);
	} else {
		tmp = (-0.375 * ((a / pow(b, 3.0)) * (c * c))) + (c / (b / -0.5));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 * (a * c)
    if (b <= 0.96d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 3.0d0)
    else
        tmp = ((-0.375d0) * ((a / (b ** 3.0d0)) * (c * c))) + (c / (b / (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 3.0 * (a * c);
	double tmp;
	if (b <= 0.96) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - Math.sqrt(((b * b) - t_0)))) / (a * 3.0);
	} else {
		tmp = (-0.375 * ((a / Math.pow(b, 3.0)) * (c * c))) + (c / (b / -0.5));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = 3.0 * (a * c)
	tmp = 0
	if b <= 0.96:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - math.sqrt(((b * b) - t_0)))) / (a * 3.0)
	else:
		tmp = (-0.375 * ((a / math.pow(b, 3.0)) * (c * c))) + (c / (b / -0.5))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(3.0 * Float64(a * c))
	tmp = 0.0
	if (b <= 0.96)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - t_0)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(a / (b ^ 3.0)) * Float64(c * c))) + Float64(c / Float64(b / -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = 3.0 * (a * c);
	tmp = 0.0;
	if (b <= 0.96)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - sqrt(((b * b) - t_0)))) / (a * 3.0);
	else
		tmp = (-0.375 * ((a / (b ^ 3.0)) * (c * c))) + (c / (b / -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.96], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / N[(b / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.95999999999999996

   1. Initial program 82.1%

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

      1. flip-+ 82.6%

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

      2. pow2 82.6%

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

      3. add-sqr-sqrt 84.4%

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

      4. associate-*l* 84.4%

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

      5. associate-*l* 84.4%

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

   3. Applied egg-rr 84.4%

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

   if 0.95999999999999996 < b

   1. Initial program 47.3%

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

      1. neg-sub0 47.3%

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

      2. associate-+l- 47.3%

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

      3. sub0-neg 47.3%

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

      4. neg-mul-1 47.3%

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

      5. associate-*r/ 47.3%

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

      6. *-commutative 47.3%

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

      7. metadata-eval 47.3%

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

      8. metadata-eval 47.3%

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

      9. times-frac 47.3%

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

      10. *-commutative 47.3%

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

      11. times-frac 47.3%

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

   3. Simplified 47.3%

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

      1. clear-num 47.3%

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

      2. inv-pow 47.3%

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

   5. Applied egg-rr 47.3%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in b around inf 88.7%

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

      1. +-commutative 88.7%

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

      2. unpow2 88.7%

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

      3. associate-*l/ 88.7%

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

      4. fma-def 88.7%

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

      5. associate-*l/ 88.7%

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

      6. *-commutative 88.7%

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

      7. associate-/l* 88.7%

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

      8. *-commutative 88.7%

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

      9. associate-*l/ 88.7%

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

   10. Simplified 88.7%

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

       1. fma-udef 88.7%

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

       2. associate-/r/ 88.7%

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

       3. associate-/l* 88.7%

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

   12. Applied egg-rr 88.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.96:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(3 \cdot \left(a \cdot c\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.96:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \left(0.3333333333333333 \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{c}{\frac{b}{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.96)
   (* (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 0.3333333333333333 (/ 1.0 a)))
   (+ (* -0.375 (* (/ a (pow b 3.0)) (* c c))) (/ c (/ b -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.96) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * (0.3333333333333333 * (1.0 / a));
	} else {
		tmp = (-0.375 * ((a / pow(b, 3.0)) * (c * c))) + (c / (b / -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.96)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * Float64(0.3333333333333333 * Float64(1.0 / a)));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(a / (b ^ 3.0)) * Float64(c * c))) + Float64(c / Float64(b / -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.96], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / N[(b / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.95999999999999996

   1. Initial program 82.1%

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

      1. neg-sub0 82.1%

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

      2. associate-+l- 82.1%

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

      3. sub0-neg 82.1%

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

      4. neg-mul-1 82.1%

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

      5. associate-*r/ 82.1%

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

      6. *-commutative 82.1%

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

      7. metadata-eval 82.1%

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

      8. metadata-eval 82.1%

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

      9. times-frac 82.1%

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

      10. *-commutative 82.1%

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

      11. times-frac 82.1%

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

   3. Simplified 82.4%

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

      1. div-inv 82.4%

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

   5. Applied egg-rr 82.4%

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

   if 0.95999999999999996 < b

   1. Initial program 47.3%

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

      1. neg-sub0 47.3%

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

      2. associate-+l- 47.3%

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

      3. sub0-neg 47.3%

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

      4. neg-mul-1 47.3%

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

      5. associate-*r/ 47.3%

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

      6. *-commutative 47.3%

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

      7. metadata-eval 47.3%

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

      8. metadata-eval 47.3%

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

      9. times-frac 47.3%

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

      10. *-commutative 47.3%

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

      11. times-frac 47.3%

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

   3. Simplified 47.3%

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

      1. clear-num 47.3%

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

      2. inv-pow 47.3%

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

   5. Applied egg-rr 47.3%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in b around inf 88.7%

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

      1. +-commutative 88.7%

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

      2. unpow2 88.7%

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

      3. associate-*l/ 88.7%

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

      4. fma-def 88.7%

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

      5. associate-*l/ 88.7%

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

      6. *-commutative 88.7%

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

      7. associate-/l* 88.7%

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

      8. *-commutative 88.7%

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

      9. associate-*l/ 88.7%

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

   10. Simplified 88.7%

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

       1. fma-udef 88.7%

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

       2. associate-/r/ 88.7%

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

       3. associate-/l* 88.7%

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

   12. Applied egg-rr 88.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.96:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \left(0.3333333333333333 \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

Alternative 5: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{c}{\frac{b}{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.0)
   (* -0.3333333333333333 (/ (- b (sqrt (fma b b (* a (* c -3.0))))) a))
   (+ (* -0.375 (* (/ a (pow b 3.0)) (* c c))) (/ c (/ b -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.0) {
		tmp = -0.3333333333333333 * ((b - sqrt(fma(b, b, (a * (c * -3.0))))) / a);
	} else {
		tmp = (-0.375 * ((a / pow(b, 3.0)) * (c * c))) + (c / (b / -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.0)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / a));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(a / (b ^ 3.0)) * Float64(c * c))) + Float64(c / Float64(b / -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.0], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / N[(b / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1

   1. Initial program 82.1%

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

      1. /-rgt-identity 82.1%

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

      2. metadata-eval 82.1%

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

      3. associate-/l* 82.1%

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

      4. associate-*r/ 82.1%

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

      5. *-commutative 82.1%

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

      6. associate-*l/ 82.1%

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

      7. associate-*r/ 82.1%

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

      8. metadata-eval 82.1%

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

      9. metadata-eval 82.1%

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

      10. times-frac 82.1%

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

      11. neg-mul-1 82.1%

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

      12. distribute-rgt-neg-in 82.1%

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

      13. times-frac 82.1%

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

      14. metadata-eval 82.1%

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

      15. neg-mul-1 82.1%

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

   3. Simplified 82.4%

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

   if 1 < b

   1. Initial program 47.3%

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

      1. neg-sub0 47.3%

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

      2. associate-+l- 47.3%

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

      3. sub0-neg 47.3%

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

      4. neg-mul-1 47.3%

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

      5. associate-*r/ 47.3%

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

      6. *-commutative 47.3%

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

      7. metadata-eval 47.3%

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

      8. metadata-eval 47.3%

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

      9. times-frac 47.3%

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

      10. *-commutative 47.3%

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

      11. times-frac 47.3%

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

   3. Simplified 47.3%

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

      1. clear-num 47.3%

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

      2. inv-pow 47.3%

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

   5. Applied egg-rr 47.3%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in b around inf 88.7%

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

      1. +-commutative 88.7%

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

      2. unpow2 88.7%

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

      3. associate-*l/ 88.7%

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

      4. fma-def 88.7%

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

      5. associate-*l/ 88.7%

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

      6. *-commutative 88.7%

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

      7. associate-/l* 88.7%

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

      8. *-commutative 88.7%

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

      9. associate-*l/ 88.7%

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

   10. Simplified 88.7%

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

       1. fma-udef 88.7%

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

       2. associate-/r/ 88.7%

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

       3. associate-/l* 88.7%

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

   12. Applied egg-rr 88.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

Alternative 6: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.96:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{c}{\frac{b}{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.96)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (+ (* -0.375 (* (/ a (pow b 3.0)) (* c c))) (/ c (/ b -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.96) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.375 * ((a / pow(b, 3.0)) * (c * c))) + (c / (b / -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.96)
		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(a / (b ^ 3.0)) * Float64(c * c))) + Float64(c / Float64(b / -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.96], 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[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / N[(b / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.95999999999999996

   1. Initial program 82.1%

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

      1. neg-sub0 82.1%

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

      2. associate-+l- 82.1%

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

      3. sub0-neg 82.1%

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

      4. neg-mul-1 82.1%

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

      5. associate-*r/ 82.1%

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

      6. metadata-eval 82.1%

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

      7. metadata-eval 82.1%

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

      8. times-frac 82.1%

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

      9. *-commutative 82.1%

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

      10. times-frac 82.1%

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

      11. associate-*l/ 82.1%

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

   3. Simplified 82.4%

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

   if 0.95999999999999996 < b

   1. Initial program 47.3%

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

      1. neg-sub0 47.3%

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

      2. associate-+l- 47.3%

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

      3. sub0-neg 47.3%

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

      4. neg-mul-1 47.3%

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

      5. associate-*r/ 47.3%

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

      6. *-commutative 47.3%

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

      7. metadata-eval 47.3%

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

      8. metadata-eval 47.3%

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

      9. times-frac 47.3%

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

      10. *-commutative 47.3%

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

      11. times-frac 47.3%

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

   3. Simplified 47.3%

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

      1. clear-num 47.3%

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

      2. inv-pow 47.3%

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

   5. Applied egg-rr 47.3%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in b around inf 88.7%

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

      1. +-commutative 88.7%

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

      2. unpow2 88.7%

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

      3. associate-*l/ 88.7%

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

      4. fma-def 88.7%

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

      5. associate-*l/ 88.7%

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

      6. *-commutative 88.7%

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

      7. associate-/l* 88.7%

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

      8. *-commutative 88.7%

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

      9. associate-*l/ 88.7%

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

   10. Simplified 88.7%

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

       1. fma-udef 88.7%

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

       2. associate-/r/ 88.7%

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

       3. associate-/l* 88.7%

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

   12. Applied egg-rr 88.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.96:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

Alternative 7: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.98:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{c}{\frac{b}{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.98)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (+ (* -0.375 (* (/ a (pow b 3.0)) (* c c))) (/ c (/ b -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.98) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.375 * ((a / pow(b, 3.0)) * (c * c))) + (c / (b / -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.98d0) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.375d0) * ((a / (b ** 3.0d0)) * (c * c))) + (c / (b / (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.98) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.375 * ((a / Math.pow(b, 3.0)) * (c * c))) + (c / (b / -0.5));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.98:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (-0.375 * ((a / math.pow(b, 3.0)) * (c * c))) + (c / (b / -0.5))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.98)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.375 * Float64(Float64(a / (b ^ 3.0)) * Float64(c * c))) + Float64(c / Float64(b / -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.98)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (-0.375 * ((a / (b ^ 3.0)) * (c * c))) + (c / (b / -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.98], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / N[(b / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.97999999999999998

   1. Initial program 82.1%

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

   if 0.97999999999999998 < b

   1. Initial program 47.3%

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

      1. neg-sub0 47.3%

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

      2. associate-+l- 47.3%

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

      3. sub0-neg 47.3%

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

      4. neg-mul-1 47.3%

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

      5. associate-*r/ 47.3%

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

      6. *-commutative 47.3%

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

      7. metadata-eval 47.3%

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

      8. metadata-eval 47.3%

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

      9. times-frac 47.3%

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

      10. *-commutative 47.3%

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

      11. times-frac 47.3%

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

   3. Simplified 47.3%

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

      1. clear-num 47.3%

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

      2. inv-pow 47.3%

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

   5. Applied egg-rr 47.3%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in b around inf 88.7%

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

      1. +-commutative 88.7%

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

      2. unpow2 88.7%

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

      3. associate-*l/ 88.7%

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

      4. fma-def 88.7%

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

      5. associate-*l/ 88.7%

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

      6. *-commutative 88.7%

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

      7. associate-/l* 88.7%

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

      8. *-commutative 88.7%

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

      9. associate-*l/ 88.7%

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

   10. Simplified 88.7%

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

       1. fma-udef 88.7%

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

       2. associate-/r/ 88.7%

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

       3. associate-/l* 88.7%

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

   12. Applied egg-rr 88.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.98:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) + \frac{c}{\frac{b}{-0.5}}\\ \end{array} \]

Alternative 8: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

   1. neg-sub0 52.4%

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

   2. associate-+l- 52.4%

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

   3. sub0-neg 52.4%

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

   4. neg-mul-1 52.4%

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

   5. associate-*r/ 52.4%

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

   6. *-commutative 52.4%

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

   7. metadata-eval 52.4%

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

   8. metadata-eval 52.4%

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

   9. times-frac 52.4%

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

   10. *-commutative 52.4%

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

   11. times-frac 52.4%

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

3. Simplified 52.3%

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

   1. clear-num 52.3%

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

   2. inv-pow 52.3%

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

5. Applied egg-rr 52.3%

   \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 52.3%

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

7. Simplified 52.3%

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

8. Taylor expanded in b around inf 84.5%

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

   1. +-commutative 84.5%

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

   2. unpow2 84.5%

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

   3. associate-*l/ 84.5%

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

   4. fma-def 84.5%

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

   5. associate-*l/ 84.5%

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

   6. *-commutative 84.5%

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

   7. associate-/l* 84.5%

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

   8. *-commutative 84.5%

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

   9. associate-*l/ 84.5%

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

10. Simplified 84.5%

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

    1. fma-udef 84.5%

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

    2. associate-/r/ 84.5%

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

    3. associate-/l* 84.5%

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

12. Applied egg-rr 84.5%

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

13. Final simplification 84.5%

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

Alternative 9: 64.6% 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 52.4%

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

2. Taylor expanded in b around inf 66.7%

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

3. Final simplification 66.7%

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

Reproduce

herbie shell --seed 2023182 
(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)))