Cubic critical, wide range

Percentage Accurate: 18.2% → 98.0%

Time: 26.5s

Alternatives: 10

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: 18.2% 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: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left({c}^{2} \cdot \left(c \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} + c \cdot \left(-2.21484375 \cdot \frac{c \cdot {a}^{3}}{{b}^{9}} + -1.0546875 \cdot \frac{{a}^{2}}{{b}^{7}}\right)\right) - \frac{0.375}{{b}^{3}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (*
   a
   (*
    (pow c 2.0)
    (-
     (*
      c
      (+
       (* -0.5625 (/ a (pow b 5.0)))
       (*
        c
        (+
         (* -2.21484375 (/ (* c (pow a 3.0)) (pow b 9.0)))
         (* -1.0546875 (/ (pow a 2.0) (pow b 7.0)))))))
     (/ 0.375 (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (a * (pow(c, 2.0) * ((c * ((-0.5625 * (a / pow(b, 5.0))) + (c * ((-2.21484375 * ((c * pow(a, 3.0)) / pow(b, 9.0))) + (-1.0546875 * (pow(a, 2.0) / pow(b, 7.0))))))) - (0.375 / pow(b, 3.0))))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(a * Float64((c ^ 2.0) * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(a / (b ^ 5.0))) + Float64(c * Float64(Float64(-2.21484375 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 9.0))) + Float64(-1.0546875 * Float64((a ^ 2.0) / (b ^ 7.0))))))) - Float64(0.375 / (b ^ 3.0))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(c * N[(N[(-0.5625 * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-2.21484375 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in a around 0 98.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + a \cdot \left(-0.16666666666666666 \cdot \frac{a \cdot \left(1.5 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}}{b}\right)\right)\right)} \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left(-0.16666666666666666 \cdot \left(\frac{a \cdot \mathsf{fma}\left(1.5, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{2}}, 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b} + \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}\right)\right)\right)\right)\right)} \]

6. Taylor expanded in c around inf 98.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \color{blue}{\left({c}^{5} \cdot \left(-2.21484375 \cdot \frac{a}{{b}^{9}} - 1.0546875 \cdot \frac{1}{{b}^{7} \cdot c}\right)\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\color{blue}{\frac{-2.21484375 \cdot a}{{b}^{9}}} - 1.0546875 \cdot \frac{1}{{b}^{7} \cdot c}\right)\right)\right)\right)\right) \]

   2. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \color{blue}{\frac{1.0546875 \cdot 1}{{b}^{7} \cdot c}}\right)\right)\right)\right)\right) \]

   3. metadata-eval 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \frac{\color{blue}{1.0546875}}{{b}^{7} \cdot c}\right)\right)\right)\right)\right) \]

   4. *-commutative 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \frac{1.0546875}{\color{blue}{c \cdot {b}^{7}}}\right)\right)\right)\right)\right) \]

8. Simplified 98.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \color{blue}{\left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \frac{1.0546875}{c \cdot {b}^{7}}\right)\right)}\right)\right)\right) \]

9. Taylor expanded in c around 0 98.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \color{blue}{\left({c}^{2} \cdot \left(c \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} + c \cdot \left(-2.21484375 \cdot \frac{{a}^{3} \cdot c}{{b}^{9}} + -1.0546875 \cdot \frac{{a}^{2}}{{b}^{7}}\right)\right) - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right)}\right) \]

10. Taylor expanded in b around 0 98.1%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left({c}^{2} \cdot \left(c \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} + c \cdot \left(-2.21484375 \cdot \frac{{a}^{3} \cdot c}{{b}^{9}} + -1.0546875 \cdot \frac{{a}^{2}}{{b}^{7}}\right)\right) - \color{blue}{\frac{0.375}{{b}^{3}}}\right)\right)\right) \]

11. Final simplification 98.1%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left({c}^{2} \cdot \left(c \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} + c \cdot \left(-2.21484375 \cdot \frac{c \cdot {a}^{3}}{{b}^{9}} + -1.0546875 \cdot \frac{{a}^{2}}{{b}^{7}}\right)\right) - \frac{0.375}{{b}^{3}}\right)\right)\right) \]
12. Add Preprocessing

Alternative 2: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left({c}^{2} \cdot \left(c \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} + -1.0546875 \cdot \frac{c \cdot {a}^{2}}{{b}^{7}}\right) + 0.375 \cdot \frac{-1}{{b}^{3}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (*
   a
   (*
    (pow c 2.0)
    (+
     (*
      c
      (+
       (* -0.5625 (/ a (pow b 5.0)))
       (* -1.0546875 (/ (* c (pow a 2.0)) (pow b 7.0)))))
     (* 0.375 (/ -1.0 (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (a * (pow(c, 2.0) * ((c * ((-0.5625 * (a / pow(b, 5.0))) + (-1.0546875 * ((c * pow(a, 2.0)) / pow(b, 7.0))))) + (0.375 * (-1.0 / pow(b, 3.0)))))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(a * Float64((c ^ 2.0) * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(a / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 7.0))))) + Float64(0.375 * Float64(-1.0 / (b ^ 3.0)))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(c * N[(N[(-0.5625 * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.375 * N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in a around 0 98.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + a \cdot \left(-0.16666666666666666 \cdot \frac{a \cdot \left(1.5 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}}{b}\right)\right)\right)} \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left(-0.16666666666666666 \cdot \left(\frac{a \cdot \mathsf{fma}\left(1.5, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{2}}, 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b} + \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}\right)\right)\right)\right)\right)} \]

6. Taylor expanded in c around inf 98.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \color{blue}{\left({c}^{5} \cdot \left(-2.21484375 \cdot \frac{a}{{b}^{9}} - 1.0546875 \cdot \frac{1}{{b}^{7} \cdot c}\right)\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\color{blue}{\frac{-2.21484375 \cdot a}{{b}^{9}}} - 1.0546875 \cdot \frac{1}{{b}^{7} \cdot c}\right)\right)\right)\right)\right) \]

   2. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \color{blue}{\frac{1.0546875 \cdot 1}{{b}^{7} \cdot c}}\right)\right)\right)\right)\right) \]

   3. metadata-eval 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \frac{\color{blue}{1.0546875}}{{b}^{7} \cdot c}\right)\right)\right)\right)\right) \]

   4. *-commutative 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \frac{1.0546875}{\color{blue}{c \cdot {b}^{7}}}\right)\right)\right)\right)\right) \]

8. Simplified 98.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \color{blue}{\left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \frac{1.0546875}{c \cdot {b}^{7}}\right)\right)}\right)\right)\right) \]

9. Taylor expanded in c around 0 97.6%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \color{blue}{\left({c}^{2} \cdot \left(c \cdot \left(-1.0546875 \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + -0.5625 \cdot \frac{a}{{b}^{5}}\right) - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right)}\right) \]

10. Final simplification 97.6%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left({c}^{2} \cdot \left(c \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} + -1.0546875 \cdot \frac{c \cdot {a}^{2}}{{b}^{7}}\right) + 0.375 \cdot \frac{-1}{{b}^{3}}\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \left({a}^{3} \cdot -1.0546875\right) \cdot \frac{c}{{b}^{7}}\right)\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.375 (/ a (pow b 3.0)))
     (*
      c
      (+
       (* -0.5625 (/ (pow a 2.0) (pow b 5.0)))
       (* (* (pow a 3.0) -1.0546875) (/ c (pow b 7.0)))))))
   (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-0.375 * (a / pow(b, 3.0))) + (c * ((-0.5625 * (pow(a, 2.0) / pow(b, 5.0))) + ((pow(a, 3.0) * -1.0546875) * (c / pow(b, 7.0))))))) + (0.5 * (-1.0 / 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 * ((c * (((-0.375d0) * (a / (b ** 3.0d0))) + (c * (((-0.5625d0) * ((a ** 2.0d0) / (b ** 5.0d0))) + (((a ** 3.0d0) * (-1.0546875d0)) * (c / (b ** 7.0d0))))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-0.375 * (a / Math.pow(b, 3.0))) + (c * ((-0.5625 * (Math.pow(a, 2.0) / Math.pow(b, 5.0))) + ((Math.pow(a, 3.0) * -1.0546875) * (c / Math.pow(b, 7.0))))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((c * ((-0.375 * (a / math.pow(b, 3.0))) + (c * ((-0.5625 * (math.pow(a, 2.0) / math.pow(b, 5.0))) + ((math.pow(a, 3.0) * -1.0546875) * (c / math.pow(b, 7.0))))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 3.0))) + Float64(c * Float64(Float64(-0.5625 * Float64((a ^ 2.0) / (b ^ 5.0))) + Float64(Float64((a ^ 3.0) * -1.0546875) * Float64(c / (b ^ 7.0))))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-0.375 * (a / (b ^ 3.0))) + (c * ((-0.5625 * ((a ^ 2.0) / (b ^ 5.0))) + (((a ^ 3.0) * -1.0546875) * (c / (b ^ 7.0))))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[a, 3.0], $MachinePrecision] * -1.0546875), $MachinePrecision] * N[(c / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in c around 0 97.2%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]

4. Taylor expanded in a around 0 97.2%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]
5. Step-by-step derivation

   1. associate-/l* 97.2%

      \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -1.0546875 \cdot \color{blue}{\left({a}^{3} \cdot \frac{c}{{b}^{7}}\right)}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

   2. associate-*r* 97.2%

      \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \color{blue}{\left(-1.0546875 \cdot {a}^{3}\right) \cdot \frac{c}{{b}^{7}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

6. Simplified 97.2%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \color{blue}{\left(-1.0546875 \cdot {a}^{3}\right) \cdot \frac{c}{{b}^{7}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

7. Final simplification 97.2%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \left({a}^{3} \cdot -1.0546875\right) \cdot \frac{c}{{b}^{7}}\right)\right) + 0.5 \cdot \frac{-1}{b}\right) \]
8. Add Preprocessing

Alternative 4: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \left({c}^{2} \cdot \left(-0.5625 \cdot \frac{c \cdot a}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (*
   a
   (*
    (pow c 2.0)
    (- (* -0.5625 (/ (* c a) (pow b 5.0))) (/ 0.375 (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (a * (pow(c, 2.0) * ((-0.5625 * ((c * a) / pow(b, 5.0))) - (0.375 / pow(b, 3.0))))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(a * Float64((c ^ 2.0) * Float64(Float64(-0.5625 * Float64(Float64(c * a) / (b ^ 5.0))) - Float64(0.375 / (b ^ 3.0))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(-0.5625 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in a around 0 98.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + a \cdot \left(-0.16666666666666666 \cdot \frac{a \cdot \left(1.5 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}}{b}\right)\right)\right)} \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left(-0.16666666666666666 \cdot \left(\frac{a \cdot \mathsf{fma}\left(1.5, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{2}}, 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b} + \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}\right)\right)\right)\right)\right)} \]

6. Taylor expanded in c around inf 98.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \color{blue}{\left({c}^{5} \cdot \left(-2.21484375 \cdot \frac{a}{{b}^{9}} - 1.0546875 \cdot \frac{1}{{b}^{7} \cdot c}\right)\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\color{blue}{\frac{-2.21484375 \cdot a}{{b}^{9}}} - 1.0546875 \cdot \frac{1}{{b}^{7} \cdot c}\right)\right)\right)\right)\right) \]

   2. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \color{blue}{\frac{1.0546875 \cdot 1}{{b}^{7} \cdot c}}\right)\right)\right)\right)\right) \]

   3. metadata-eval 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \frac{\color{blue}{1.0546875}}{{b}^{7} \cdot c}\right)\right)\right)\right)\right) \]

   4. *-commutative 98.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \frac{1.0546875}{\color{blue}{c \cdot {b}^{7}}}\right)\right)\right)\right)\right) \]

8. Simplified 98.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \color{blue}{\left({c}^{5} \cdot \left(\frac{-2.21484375 \cdot a}{{b}^{9}} - \frac{1.0546875}{c \cdot {b}^{7}}\right)\right)}\right)\right)\right) \]

9. Taylor expanded in c around 0 96.8%

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

    1. associate-*r/ 96.8%

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

    2. metadata-eval 96.8%

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

11. Simplified 96.8%

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

12. Final simplification 96.8%

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

Alternative 5: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + -0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}}\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.375 (/ a (pow b 3.0)))
     (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))))
   (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-0.375 * (a / pow(b, 3.0))) + (-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))))) + (0.5 * (-1.0 / 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 * ((c * (((-0.375d0) * (a / (b ** 3.0d0))) + ((-0.5625d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-0.375 * (a / Math.pow(b, 3.0))) + (-0.5625 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((c * ((-0.375 * (a / math.pow(b, 3.0))) + (-0.5625 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-0.375 * (a / (b ^ 3.0))) + (-0.5625 * ((c * (a ^ 2.0)) / (b ^ 5.0))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in c around 0 96.5%

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

4. Final simplification 96.5%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + -0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}}\right) + 0.5 \cdot \frac{-1}{b}\right) \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

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

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in a around 0 95.5%

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

4. Final simplification 95.5%

   \[\leadsto -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]
5. Add Preprocessing

Alternative 7: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in b around inf 95.5%

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

4. Final simplification 95.5%

   \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \]
5. Add Preprocessing

Alternative 8: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in c around 0 95.1%

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

   1. associate-*r/ 95.1%

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

   2. associate-*r/ 95.1%

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

   3. metadata-eval 95.1%

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

5. Simplified 95.1%

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

6. Final simplification 95.1%

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

Alternative 9: 90.1% accurate, 23.2× speedup

Math

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

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(Float64(-0.5 * c) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in b around inf 91.1%

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

   1. associate-*r/ 91.1%

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

   2. *-commutative 91.1%

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

5. Simplified 91.1%

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

6. Final simplification 91.1%

   \[\leadsto \frac{-0.5 \cdot c}{b} \]
7. Add Preprocessing

Alternative 10: 3.3% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.2%

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

   1. add-sqr-sqrt 17.2%

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

   2. difference-of-squares 17.2%

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

   3. associate-*l* 17.2%

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

   4. associate-*l* 17.2%

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

4. Applied egg-rr 17.2%

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

   1. *-commutative 17.2%

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

   2. *-commutative 17.2%

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

6. Simplified 17.2%

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

7. Taylor expanded in b around inf 3.3%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.3%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)}{a}} \]

   2. distribute-lft1-in 3.3%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}}{a} \]

   3. metadata-eval 3.3%

      \[\leadsto \frac{0.16666666666666666 \cdot \left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}{a} \]

   4. mul0-lft 3.3%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.3%

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

9. Simplified 3.3%

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

10. Final simplification 3.3%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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)))