Cubic critical, wide range

Percentage Accurate: 18.3% → 98.3%

Time: 22.1s

Alternatives: 9

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: 18.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, {c}^{4} \cdot \left(-1.0546875 \cdot \frac{a}{{b}^{7}} + c \cdot \left(-4.9833984375 \cdot \frac{c \cdot {a}^{3}}{{b}^{11}} + -2.21484375 \cdot \frac{{a}^{2}}{{b}^{9}}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (*
   a
   (fma
    -0.375
    (/ (pow c 2.0) (pow b 3.0))
    (*
     a
     (fma
      -0.5625
      (/ (pow c 3.0) (pow b 5.0))
      (*
       (pow c 4.0)
       (+
        (* -1.0546875 (/ a (pow b 7.0)))
        (*
         c
         (+
          (* -4.9833984375 (/ (* c (pow a 3.0)) (pow b 11.0)))
          (* -2.21484375 (/ (pow a 2.0) (pow b 9.0)))))))))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (a * fma(-0.375, (pow(c, 2.0) / pow(b, 3.0)), (a * fma(-0.5625, (pow(c, 3.0) / pow(b, 5.0)), (pow(c, 4.0) * ((-1.0546875 * (a / pow(b, 7.0))) + (c * ((-4.9833984375 * ((c * pow(a, 3.0)) / pow(b, 11.0))) + (-2.21484375 * (pow(a, 2.0) / pow(b, 9.0))))))))))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(a * fma(-0.375, Float64((c ^ 2.0) / (b ^ 3.0)), Float64(a * fma(-0.5625, Float64((c ^ 3.0) / (b ^ 5.0)), Float64((c ^ 4.0) * Float64(Float64(-1.0546875 * Float64(a / (b ^ 7.0))) + Float64(c * Float64(Float64(-4.9833984375 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 11.0))) + Float64(-2.21484375 * Float64((a ^ 2.0) / (b ^ 9.0))))))))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-1.0546875 * N[(a / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-4.9833984375 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 11.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.21484375 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in a around 0 97.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + a \cdot \left(-0.16666666666666666 \cdot \frac{1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}}{b} + a \cdot \left(-0.16666666666666666 \cdot \frac{a \cdot \left(1.125 \cdot \frac{{c}^{2} \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{4}} + \left(1.5 \cdot \frac{c \cdot \left(1.5 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}} + 2.84765625 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + -0.16666666666666666 \cdot \frac{1.5 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}}{b}\right)\right)\right)\right)} \]

5. Simplified 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \left(-0.16666666666666666 \cdot \left(\frac{a \cdot \mathsf{fma}\left(1.125, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{4}}, \mathsf{fma}\left(1.5, c \cdot \frac{\mathsf{fma}\left(1.5, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{2}}, 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}}, \frac{2.84765625 \cdot {c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(1.5, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{2}}, 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right)\right)\right)} \]

6. Taylor expanded in c around 0 97.8%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, \color{blue}{{c}^{4} \cdot \left(-1.0546875 \cdot \frac{a}{{b}^{7}} + c \cdot \left(-4.9833984375 \cdot \frac{{a}^{3} \cdot c}{{b}^{11}} + -2.21484375 \cdot \frac{{a}^{2}}{{b}^{9}}\right)\right)}\right)\right)\right) \]

7. Final simplification 97.8%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, {c}^{4} \cdot \left(-1.0546875 \cdot \frac{a}{{b}^{7}} + c \cdot \left(-4.9833984375 \cdot \frac{c \cdot {a}^{3}}{{b}^{11}} + -2.21484375 \cdot \frac{{a}^{2}}{{b}^{9}}\right)\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, {c}^{4} \cdot \left(-1.0546875 \cdot \frac{a}{{b}^{7}} + -2.21484375 \cdot \frac{c \cdot {a}^{2}}{{b}^{9}}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (*
   a
   (fma
    -0.375
    (/ (pow c 2.0) (pow b 3.0))
    (*
     a
     (fma
      -0.5625
      (/ (pow c 3.0) (pow b 5.0))
      (*
       (pow c 4.0)
       (+
        (* -1.0546875 (/ a (pow b 7.0)))
        (* -2.21484375 (/ (* c (pow a 2.0)) (pow b 9.0)))))))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (a * fma(-0.375, (pow(c, 2.0) / pow(b, 3.0)), (a * fma(-0.5625, (pow(c, 3.0) / pow(b, 5.0)), (pow(c, 4.0) * ((-1.0546875 * (a / pow(b, 7.0))) + (-2.21484375 * ((c * pow(a, 2.0)) / pow(b, 9.0))))))))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(a * fma(-0.375, Float64((c ^ 2.0) / (b ^ 3.0)), Float64(a * fma(-0.5625, Float64((c ^ 3.0) / (b ^ 5.0)), Float64((c ^ 4.0) * Float64(Float64(-1.0546875 * Float64(a / (b ^ 7.0))) + Float64(-2.21484375 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 9.0))))))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-1.0546875 * N[(a / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.21484375 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in a around 0 97.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + a \cdot \left(-0.16666666666666666 \cdot \frac{1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}}{b} + a \cdot \left(-0.16666666666666666 \cdot \frac{a \cdot \left(1.125 \cdot \frac{{c}^{2} \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{4}} + \left(1.5 \cdot \frac{c \cdot \left(1.5 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}} + 2.84765625 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + -0.16666666666666666 \cdot \frac{1.5 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}}{b}\right)\right)\right)\right)} \]

5. Simplified 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \left(-0.16666666666666666 \cdot \left(\frac{a \cdot \mathsf{fma}\left(1.125, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{4}}, \mathsf{fma}\left(1.5, c \cdot \frac{\mathsf{fma}\left(1.5, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{2}}, 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}}, \frac{2.84765625 \cdot {c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(1.5, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{2}}, 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right)\right)\right)} \]

6. Taylor expanded in c around 0 97.5%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, \color{blue}{{c}^{4} \cdot \left(-2.21484375 \cdot \frac{{a}^{2} \cdot c}{{b}^{9}} + -1.0546875 \cdot \frac{a}{{b}^{7}}\right)}\right)\right)\right) \]

7. Final simplification 97.5%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, {c}^{4} \cdot \left(-1.0546875 \cdot \frac{a}{{b}^{7}} + -2.21484375 \cdot \frac{c \cdot {a}^{2}}{{b}^{9}}\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 3: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{a}^{4}}{{b}^{6}}\\ c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot t\_0 + t\_0 \cdot 5.0625\right)}{b \cdot a}\right)\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (pow a 4.0) (pow b 6.0))))
   (*
    c
    (+
     (*
      c
      (+
       (* -0.375 (/ a (pow b 3.0)))
       (*
        c
        (+
         (* -0.5625 (/ (pow a 2.0) (pow b 5.0)))
         (*
          -0.16666666666666666
          (/ (* c (+ (* 1.265625 t_0) (* t_0 5.0625))) (* b a)))))))
     (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(a, 4.0) / pow(b, 6.0);
	return c * ((c * ((-0.375 * (a / pow(b, 3.0))) + (c * ((-0.5625 * (pow(a, 2.0) / pow(b, 5.0))) + (-0.16666666666666666 * ((c * ((1.265625 * t_0) + (t_0 * 5.0625))) / (b * a))))))) + (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
    real(8) :: t_0
    t_0 = (a ** 4.0d0) / (b ** 6.0d0)
    code = c * ((c * (((-0.375d0) * (a / (b ** 3.0d0))) + (c * (((-0.5625d0) * ((a ** 2.0d0) / (b ** 5.0d0))) + ((-0.16666666666666666d0) * ((c * ((1.265625d0 * t_0) + (t_0 * 5.0625d0))) / (b * a))))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(a, 4.0) / Math.pow(b, 6.0);
	return c * ((c * ((-0.375 * (a / Math.pow(b, 3.0))) + (c * ((-0.5625 * (Math.pow(a, 2.0) / Math.pow(b, 5.0))) + (-0.16666666666666666 * ((c * ((1.265625 * t_0) + (t_0 * 5.0625))) / (b * a))))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	t_0 = math.pow(a, 4.0) / math.pow(b, 6.0)
	return c * ((c * ((-0.375 * (a / math.pow(b, 3.0))) + (c * ((-0.5625 * (math.pow(a, 2.0) / math.pow(b, 5.0))) + (-0.16666666666666666 * ((c * ((1.265625 * t_0) + (t_0 * 5.0625))) / (b * a))))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	t_0 = Float64((a ^ 4.0) / (b ^ 6.0))
	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(-0.16666666666666666 * Float64(Float64(c * Float64(Float64(1.265625 * t_0) + Float64(t_0 * 5.0625))) / Float64(b * a))))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (a ^ 4.0) / (b ^ 6.0);
	tmp = c * ((c * ((-0.375 * (a / (b ^ 3.0))) + (c * ((-0.5625 * ((a ^ 2.0) / (b ^ 5.0))) + (-0.16666666666666666 * ((c * ((1.265625 * t_0) + (t_0 * 5.0625))) / (b * a))))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]}, 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[(-0.16666666666666666 * N[(N[(c * N[(N[(1.265625 * t$95$0), $MachinePrecision] + N[(t$95$0 * 5.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in c around 0 96.6%

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

4. Final simplification 96.6%

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

Alternative 4: 96.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (-0.375 * (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(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in a around 0 96.1%

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

4. Final simplification 96.1%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right) \]
5. 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 20.5%

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

3. Taylor expanded in c around 0 95.8%

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

4. Final simplification 95.8%

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

Alternative 6: 95.1% 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 20.5%

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

3. Taylor expanded in a around 0 94.4%

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

4. Final simplification 94.4%

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

Alternative 7: 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 20.5%

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

3. Taylor expanded in c around 0 94.0%

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

   1. associate-*r/ 94.0%

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

   2. associate-*r/ 94.0%

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

   3. metadata-eval 94.0%

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

5. Simplified 94.0%

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

6. Final simplification 94.0%

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

Alternative 8: 89.8% 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 20.5%

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

3. Taylor expanded in a around 0 94.4%

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

4. Taylor expanded in c around 0 88.7%

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

   1. associate-*r/ 88.7%

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

   2. *-commutative 88.7%

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

   3. associate-/l* 88.4%

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

6. Simplified 88.4%

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

7. Final simplification 88.4%

   \[\leadsto c \cdot \frac{-0.5}{b} \]
8. 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 20.5%

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

3. Taylor expanded in b around inf 88.7%

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

   1. associate-*r/ 88.7%

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

   2. *-commutative 88.7%

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

5. Simplified 88.7%

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

6. Final simplification 88.7%

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

Reproduce

herbie shell --seed 2024055 
(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)))