Cubic critical, wide range

Percentage Accurate: 18.5% → 97.7%

Time: 11.7s

Alternatives: 8

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 18.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot -1.125\right)\\ \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.16666666666666666, \frac{t_0 \cdot t_0 + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) (* (* a a) -1.125))))
   (fma
    -0.5625
    (/ (* (pow c 3.0) (* a a)) (pow b 5.0))
    (fma
     -0.16666666666666666
     (/ (+ (* t_0 t_0) (* 5.0625 (pow (* c a) 4.0))) (* a (pow b 7.0)))
     (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * ((a * a) * -1.125);
	return fma(-0.5625, ((pow(c, 3.0) * (a * a)) / pow(b, 5.0)), fma(-0.16666666666666666, (((t_0 * t_0) + (5.0625 * pow((c * a), 4.0))) / (a * pow(b, 7.0))), fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0)))));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * Float64(Float64(a * a) * -1.125))
	return fma(-0.5625, Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0)), fma(-0.16666666666666666, Float64(Float64(Float64(t_0 * t_0) + Float64(5.0625 * (Float64(c * a) ^ 4.0))) / Float64(a * (b ^ 7.0))), fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -1.125), $MachinePrecision]), $MachinePrecision]}, N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(5.0625 * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 20.7%

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

   1. /-rgt-identity 20.7%

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

   2. metadata-eval 20.7%

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

   3. associate-/l* 20.7%

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

   4. associate-*r/ 20.7%

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

   5. *-commutative 20.7%

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

   6. associate-*l/ 20.7%

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

   7. associate-*r/ 20.7%

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

   8. metadata-eval 20.7%

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

   9. metadata-eval 20.7%

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

   10. times-frac 20.7%

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

   11. neg-mul-1 20.7%

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

   12. distribute-rgt-neg-in 20.7%

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

   13. times-frac 20.7%

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

   14. metadata-eval 20.7%

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

   15. neg-mul-1 20.7%

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

3. Simplified 20.8%

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

4. Taylor expanded in b around inf 96.8%

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

   1. fma-def 96.8%

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

   2. unpow2 96.8%

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

   3. fma-def 96.8%

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

6. Simplified 96.8%

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

   1. pow1 96.8%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \color{blue}{{\left({c}^{4} \cdot {a}^{4}\right)}^{1}}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]

   2. pow-prod-down 96.8%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot {\color{blue}{\left({\left(c \cdot a\right)}^{4}\right)}}^{1}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]

8. Applied egg-rr 96.8%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \color{blue}{{\left({\left(c \cdot a\right)}^{4}\right)}^{1}}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]
9. Step-by-step derivation

   1. unpow1 96.8%

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

10. Simplified 96.8%

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

    1. unpow2 96.8%

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

    2. associate-*l* 96.8%

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

    3. associate-*l* 96.8%

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

12. Applied egg-rr 96.8%

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

13. Final simplification 96.8%

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

Alternative 2: 96.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5625
  (* (* a a) (/ (pow c 3.0) (pow b 5.0)))
  (fma c (/ -0.5 b) (* a (* -0.375 (/ (* c c) (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(c, (-0.5 / b), (a * (-0.375 * ((c * c) / 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(c, Float64(-0.5 / b), Float64(a * Float64(-0.375 * Float64(Float64(c * c) / (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[(c * N[(-0.5 / b), $MachinePrecision] + N[(a * N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.7%

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

   1. /-rgt-identity 20.7%

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

   2. metadata-eval 20.7%

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

   3. associate-/l* 20.7%

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

   4. associate-*r/ 20.7%

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

   5. *-commutative 20.7%

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

   6. associate-*l/ 20.7%

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

   7. associate-*r/ 20.7%

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

   8. metadata-eval 20.7%

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

   9. metadata-eval 20.7%

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

   10. times-frac 20.7%

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

   11. neg-mul-1 20.7%

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

   12. distribute-rgt-neg-in 20.7%

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

   13. times-frac 20.7%

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

   14. metadata-eval 20.7%

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

   15. neg-mul-1 20.7%

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

3. Simplified 20.8%

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

4. Taylor expanded in b around inf 95.3%

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

   1. fma-def 95.3%

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

   2. associate-/l* 95.3%

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

   3. unpow2 95.3%

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

   4. fma-def 95.3%

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

   5. associate-/l* 95.3%

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

   6. unpow2 95.3%

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

   7. associate-*r/ 95.2%

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

6. Simplified 95.2%

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

7. Taylor expanded in c around 0 95.8%

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

   1. fma-def 95.8%

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

   2. unpow2 95.8%

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

   3. associate-*l/ 95.8%

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

   4. *-commutative 95.8%

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

   5. associate-*r/ 95.8%

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

   6. associate-*l/ 95.4%

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

   7. *-commutative 95.4%

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

   8. fma-def 95.4%

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

   9. *-commutative 95.4%

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

   10. unpow2 95.4%

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

   11. associate-*l/ 95.4%

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

   12. *-commutative 95.4%

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

   13. associate-*l* 95.4%

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

9. Simplified 95.4%

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

10. Final simplification 95.4%

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

Alternative 3: 96.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.7%

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

   1. /-rgt-identity 20.7%

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

   2. metadata-eval 20.7%

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

   3. associate-/l* 20.7%

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

   4. associate-*r/ 20.7%

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

   5. *-commutative 20.7%

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

   6. associate-*l/ 20.7%

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

   7. associate-*r/ 20.7%

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

   8. metadata-eval 20.7%

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

   9. metadata-eval 20.7%

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

   10. times-frac 20.7%

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

   11. neg-mul-1 20.7%

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

   12. distribute-rgt-neg-in 20.7%

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

   13. times-frac 20.7%

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

   14. metadata-eval 20.7%

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

   15. neg-mul-1 20.7%

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

3. Simplified 20.8%

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

4. Taylor expanded in b around inf 95.8%

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

   1. fma-def 95.8%

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

   2. associate-/l* 95.8%

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

   3. unpow2 95.8%

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

   4. +-commutative 95.8%

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

   5. fma-def 95.8%

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

   6. associate-/l* 95.8%

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

   7. unpow2 95.8%

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

   8. associate-*r/ 95.8%

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

6. Simplified 95.8%

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

7. Final simplification 95.8%

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

Alternative 4: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -4e-23)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -4e-23) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)) <= (-4d-23)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -4e-23) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -4e-23:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -4e-23)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -4e-23)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -4e-23], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.99999999999999984e-23

   1. Initial program 65.4%

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

      1. neg-sub0 65.4%

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

      2. associate-+l- 65.4%

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

      3. sub0-neg 65.4%

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

      4. neg-mul-1 65.4%

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

      5. associate-*r/ 65.4%

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

      6. metadata-eval 65.4%

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

      7. metadata-eval 65.4%

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

      8. times-frac 65.4%

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

      9. *-commutative 65.4%

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

      10. times-frac 65.3%

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

      11. associate-*l/ 65.4%

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

   3. Simplified 65.4%

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

   if -3.99999999999999984e-23 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 3.9%

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

      1. /-rgt-identity 3.9%

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

      2. metadata-eval 3.9%

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

      3. associate-/l* 3.9%

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

      4. associate-*r/ 3.9%

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

      5. *-commutative 3.9%

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

      6. associate-*l/ 3.9%

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

      7. associate-*r/ 3.9%

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

      8. metadata-eval 3.9%

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

      9. metadata-eval 3.9%

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

      10. times-frac 3.9%

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

      11. neg-mul-1 3.9%

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

      12. distribute-rgt-neg-in 3.9%

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

      13. times-frac 3.9%

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

      14. metadata-eval 3.9%

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

      15. neg-mul-1 3.9%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in b around inf 99.8%

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

      1. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 95.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.7%

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

   1. /-rgt-identity 20.7%

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

   2. metadata-eval 20.7%

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

   3. associate-/l* 20.7%

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

   4. associate-*r/ 20.7%

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

   5. *-commutative 20.7%

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

   6. associate-*l/ 20.7%

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

   7. associate-*r/ 20.7%

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

   8. metadata-eval 20.7%

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

   9. metadata-eval 20.7%

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

   10. times-frac 20.7%

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

   11. neg-mul-1 20.7%

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

   12. distribute-rgt-neg-in 20.7%

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

   13. times-frac 20.7%

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

   14. metadata-eval 20.7%

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

   15. neg-mul-1 20.7%

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

3. Simplified 20.8%

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

4. Taylor expanded in b around inf 93.8%

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

   1. +-commutative 93.8%

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

   2. fma-def 93.8%

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

   3. associate-/l* 93.8%

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

   4. unpow2 93.8%

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

   5. associate-*r/ 93.8%

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

6. Simplified 93.8%

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

7. Final simplification 93.8%

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

Alternative 6: 89.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.7%

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

   1. /-rgt-identity 20.7%

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

   2. metadata-eval 20.7%

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

   3. associate-/l* 20.7%

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

   4. associate-*r/ 20.7%

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

   5. *-commutative 20.7%

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

   6. associate-*l/ 20.7%

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

   7. associate-*r/ 20.7%

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

   8. metadata-eval 20.7%

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

   9. metadata-eval 20.7%

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

   10. times-frac 20.7%

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

   11. neg-mul-1 20.7%

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

   12. distribute-rgt-neg-in 20.7%

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

   13. times-frac 20.7%

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

   14. metadata-eval 20.7%

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

   15. neg-mul-1 20.7%

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

3. Simplified 20.8%

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

4. Taylor expanded in b around inf 88.0%

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

   1. associate-*r/ 87.8%

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

6. Simplified 87.8%

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

7. Taylor expanded in c around 0 88.4%

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

   1. associate-*r/ 88.4%

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

   2. associate-/l* 88.1%

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

   3. associate-/r/ 88.0%

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

9. Simplified 88.0%

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

10. Final simplification 88.0%

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

Alternative 7: 89.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.7%

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

   1. /-rgt-identity 20.7%

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

   2. metadata-eval 20.7%

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

   3. associate-/l* 20.7%

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

   4. associate-*r/ 20.7%

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

   5. *-commutative 20.7%

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

   6. associate-*l/ 20.7%

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

   7. associate-*r/ 20.7%

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

   8. metadata-eval 20.7%

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

   9. metadata-eval 20.7%

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

   10. times-frac 20.7%

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

   11. neg-mul-1 20.7%

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

   12. distribute-rgt-neg-in 20.7%

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

   13. times-frac 20.7%

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

   14. metadata-eval 20.7%

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

   15. neg-mul-1 20.7%

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

3. Simplified 20.8%

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

4. Taylor expanded in b around inf 88.0%

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

   1. associate-*r/ 87.8%

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

6. Simplified 87.8%

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

7. Taylor expanded in c around 0 88.4%

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

   1. associate-*r/ 88.4%

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

   2. associate-/l* 88.1%

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

   3. associate-/r/ 88.0%

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

9. Simplified 88.0%

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

10. Taylor expanded in b around 0 88.4%

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

    1. associate-*r/ 88.4%

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

    2. associate-/l* 88.1%

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

12. Simplified 88.1%

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

13. Final simplification 88.1%

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

Alternative 8: 90.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.7%

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

   1. /-rgt-identity 20.7%

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

   2. metadata-eval 20.7%

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

   3. associate-/l* 20.7%

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

   4. associate-*r/ 20.7%

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

   5. *-commutative 20.7%

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

   6. associate-*l/ 20.7%

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

   7. associate-*r/ 20.7%

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

   8. metadata-eval 20.7%

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

   9. metadata-eval 20.7%

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

   10. times-frac 20.7%

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

   11. neg-mul-1 20.7%

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

   12. distribute-rgt-neg-in 20.7%

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

   13. times-frac 20.7%

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

   14. metadata-eval 20.7%

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

   15. neg-mul-1 20.7%

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

3. Simplified 20.8%

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

4. Taylor expanded in b around inf 88.4%

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

   1. associate-*r/ 88.4%

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

6. Simplified 88.4%

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

7. Final simplification 88.4%

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

Reproduce

herbie shell --seed 2023242 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))