Cubic critical, medium range

Percentage Accurate: 31.8% → 95.4%

Time: 17.5s

Alternatives: 7

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.8% 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: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, -3 \cdot \left(a \cdot \left(-0.5625 \cdot \frac{{c}^{2}}{{b}^{5}}\right) + \frac{c}{{b}^{3}} \cdot -0.375\right), \frac{1.5}{b}\right)\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma
   -2.0
   (/ b c)
   (*
    a
    (fma
     a
     (*
      -3.0
      (+
       (* a (* -0.5625 (/ (pow c 2.0) (pow b 5.0))))
       (* (/ c (pow b 3.0)) -0.375)))
     (/ 1.5 b))))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(-2.0, (b / c), (a * fma(a, (-3.0 * ((a * (-0.5625 * (pow(c, 2.0) / pow(b, 5.0)))) + ((c / pow(b, 3.0)) * -0.375))), (1.5 / b))));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(-2.0, Float64(b / c), Float64(a * fma(a, Float64(-3.0 * Float64(Float64(a * Float64(-0.5625 * Float64((c ^ 2.0) / (b ^ 5.0)))) + Float64(Float64(c / (b ^ 3.0)) * -0.375))), Float64(1.5 / b)))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(a * N[(a * N[(-3.0 * N[(N[(a * N[(-0.5625 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

   1. /-rgt-identity 33.8%

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

   2. metadata-eval 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in a around inf 33.7%

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

   1. clear-num 33.7%

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

   2. inv-pow 33.7%

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

   3. *-commutative 33.7%

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

   4. *-commutative 33.7%

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

   5. fma-define 33.7%

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

7. Applied egg-rr 33.7%

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

   1. unpow-1 33.7%

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

9. Simplified 33.7%

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

10. Taylor expanded in a around 0 94.4%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-3 \cdot \left(a \cdot \left(-0.75 \cdot \frac{c \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(-0.2222222222222222 \cdot \frac{b \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + 0.5625 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) + -3 \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{1}{b}\right)}} \]

12. Simplified 94.4%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, -3 \cdot \left(a \cdot \mathsf{fma}\left(-0.75, c \cdot \frac{\frac{c}{{b}^{3}} \cdot -0.375}{{b}^{2}}, \mathsf{fma}\left(-0.2222222222222222, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{c}^{2}}, 0.5625 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right) + \frac{c}{{b}^{3}} \cdot -0.375\right), \frac{1.5}{b}\right)\right)}} \]

13. Taylor expanded in c around 0 94.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, -3 \cdot \left(a \cdot \color{blue}{\left(-0.5625 \cdot \frac{{c}^{2}}{{b}^{5}}\right)} + \frac{c}{{b}^{3}} \cdot -0.375\right), \frac{1.5}{b}\right)\right)} \]
14. Add Preprocessing

Alternative 2: 93.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(-0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (/
   (fma
    -2.0
    b
    (*
     c
     (fma -3.0 (* c (* -0.375 (/ (pow a 2.0) (pow b 3.0)))) (/ (* a 1.5) b))))
   c)))

C

double code(double a, double b, double c) {
	return 1.0 / (fma(-2.0, b, (c * fma(-3.0, (c * (-0.375 * (pow(a, 2.0) / pow(b, 3.0)))), ((a * 1.5) / b)))) / c);
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(fma(-2.0, b, Float64(c * fma(-3.0, Float64(c * Float64(-0.375 * Float64((a ^ 2.0) / (b ^ 3.0)))), Float64(Float64(a * 1.5) / b)))) / c))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * b + N[(c * N[(-3.0 * N[(c * N[(-0.375 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

   1. /-rgt-identity 33.8%

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

   2. metadata-eval 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in a around inf 33.7%

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

   1. clear-num 33.7%

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

   2. inv-pow 33.7%

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

   3. *-commutative 33.7%

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

   4. *-commutative 33.7%

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

   5. fma-define 33.7%

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

7. Applied egg-rr 33.7%

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

   1. unpow-1 33.7%

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

9. Simplified 33.7%

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

10. Taylor expanded in c around 0 92.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + c \cdot \left(-3 \cdot \left(c \cdot \left(-0.75 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{a}{b}\right)}{c}}} \]
11. Step-by-step derivation

    1. fma-define 92.6%

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-2, b, c \cdot \left(-3 \cdot \left(c \cdot \left(-0.75 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{a}{b}\right)\right)}}{c}} \]

    2. fma-define 92.6%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \color{blue}{\mathsf{fma}\left(-3, c \cdot \left(-0.75 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right), 1.5 \cdot \frac{a}{b}\right)}\right)}{c}} \]

    3. distribute-rgt-out 92.6%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \color{blue}{\left(\frac{{a}^{2}}{{b}^{3}} \cdot \left(-0.75 + 0.375\right)\right)}, 1.5 \cdot \frac{a}{b}\right)\right)}{c}} \]

    4. metadata-eval 92.6%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot \color{blue}{-0.375}\right), 1.5 \cdot \frac{a}{b}\right)\right)}{c}} \]

    5. associate-*r/ 92.6%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot -0.375\right), \color{blue}{\frac{1.5 \cdot a}{b}}\right)\right)}{c}} \]

    6. *-commutative 92.6%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot -0.375\right), \frac{\color{blue}{a \cdot 1.5}}{b}\right)\right)}{c}} \]

12. Simplified 92.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot -0.375\right), \frac{a \cdot 1.5}{b}\right)\right)}{c}}} \]

13. Final simplification 92.6%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(-0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}{c}} \]
14. Add Preprocessing

Alternative 3: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}}\\ \frac{1}{-2 \cdot \frac{b}{c} + a \cdot \left(-3 \cdot \left(a \cdot \left(t\_0 \cdot -0.75 + t\_0 \cdot 0.375\right)\right) + 1.5 \cdot \frac{1}{b}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 3.0))))
   (/
    1.0
    (+
     (* -2.0 (/ b c))
     (*
      a
      (+ (* -3.0 (* a (+ (* t_0 -0.75) (* t_0 0.375)))) (* 1.5 (/ 1.0 b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 3.0);
	return 1.0 / ((-2.0 * (b / c)) + (a * ((-3.0 * (a * ((t_0 * -0.75) + (t_0 * 0.375)))) + (1.5 * (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 = c / (b ** 3.0d0)
    code = 1.0d0 / (((-2.0d0) * (b / c)) + (a * (((-3.0d0) * (a * ((t_0 * (-0.75d0)) + (t_0 * 0.375d0)))) + (1.5d0 * (1.0d0 / b)))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c / Math.pow(b, 3.0);
	return 1.0 / ((-2.0 * (b / c)) + (a * ((-3.0 * (a * ((t_0 * -0.75) + (t_0 * 0.375)))) + (1.5 * (1.0 / b)))));
}

Python

def code(a, b, c):
	t_0 = c / math.pow(b, 3.0)
	return 1.0 / ((-2.0 * (b / c)) + (a * ((-3.0 * (a * ((t_0 * -0.75) + (t_0 * 0.375)))) + (1.5 * (1.0 / b)))))

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 3.0))
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(a * Float64(Float64(-3.0 * Float64(a * Float64(Float64(t_0 * -0.75) + Float64(t_0 * 0.375)))) + Float64(1.5 * Float64(1.0 / b))))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c / (b ^ 3.0);
	tmp = 1.0 / ((-2.0 * (b / c)) + (a * ((-3.0 * (a * ((t_0 * -0.75) + (t_0 * 0.375)))) + (1.5 * (1.0 / b)))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-3.0 * N[(a * N[(N[(t$95$0 * -0.75), $MachinePrecision] + N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.8%

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

   1. /-rgt-identity 33.8%

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

   2. metadata-eval 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in a around inf 33.7%

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

   1. clear-num 33.7%

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

   2. inv-pow 33.7%

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

   3. *-commutative 33.7%

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

   4. *-commutative 33.7%

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

   5. fma-define 33.7%

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

7. Applied egg-rr 33.7%

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

   1. unpow-1 33.7%

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

9. Simplified 33.7%

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

10. Taylor expanded in a around 0 92.6%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + a \cdot \left(-3 \cdot \left(a \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{1}{b}\right)}} \]

11. Final simplification 92.6%

    \[\leadsto \frac{1}{-2 \cdot \frac{b}{c} + a \cdot \left(-3 \cdot \left(a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.75 + \frac{c}{{b}^{3}} \cdot 0.375\right)\right) + 1.5 \cdot \frac{1}{b}\right)} \]
12. Add Preprocessing

Alternative 4: 90.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-2 \cdot b + 1.5 \cdot \frac{c \cdot a}{b}}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (+ (* -2.0 b) (* 1.5 (/ (* c a) b))) c)))

C

double code(double a, double b, double c) {
	return 1.0 / (((-2.0 * b) + (1.5 * ((c * a) / 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 = 1.0d0 / ((((-2.0d0) * b) + (1.5d0 * ((c * a) / b))) / c)
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (((-2.0 * b) + (1.5 * ((c * a) / b))) / c);
}

Python

def code(a, b, c):
	return 1.0 / (((-2.0 * b) + (1.5 * ((c * a) / b))) / c)

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(Float64(-2.0 * b) + Float64(1.5 * Float64(Float64(c * a) / b))) / c))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (((-2.0 * b) + (1.5 * ((c * a) / b))) / c);
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(-2.0 * b), $MachinePrecision] + N[(1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

   1. /-rgt-identity 33.8%

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

   2. metadata-eval 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in a around inf 33.7%

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

   1. clear-num 33.7%

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

   2. inv-pow 33.7%

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

   3. *-commutative 33.7%

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

   4. *-commutative 33.7%

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

   5. fma-define 33.7%

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

7. Applied egg-rr 33.7%

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

   1. unpow-1 33.7%

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

9. Simplified 33.7%

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

10. Taylor expanded in c around 0 89.5%

    \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]

11. Final simplification 89.5%

    \[\leadsto \frac{1}{\frac{-2 \cdot b + 1.5 \cdot \frac{c \cdot a}{b}}{c}} \]
12. Add Preprocessing

Alternative 5: 90.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
}

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

   1. /-rgt-identity 33.8%

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

   2. metadata-eval 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in a around inf 33.7%

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

   1. clear-num 33.7%

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

   2. inv-pow 33.7%

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

   3. *-commutative 33.7%

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

   4. *-commutative 33.7%

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

   5. fma-define 33.7%

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

7. Applied egg-rr 33.7%

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

   1. unpow-1 33.7%

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

9. Simplified 33.7%

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

10. Taylor expanded in a around 0 89.5%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
11. Add Preprocessing

Alternative 6: 81.0% 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 33.8%

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

3. Taylor expanded in b around inf 79.6%

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

   1. associate-*r/ 79.6%

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

   2. *-commutative 79.6%

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

5. Simplified 79.6%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
6. Add Preprocessing

Alternative 7: 80.8% 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 33.8%

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

3. Taylor expanded in b around inf 79.6%

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

   1. associate-*r/ 79.6%

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

   2. *-commutative 79.6%

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

5. Simplified 79.6%

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

6. Taylor expanded in c around 0 79.6%

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

   1. associate-*r/ 79.6%

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

   2. *-commutative 79.6%

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

   3. associate-*r/ 79.4%

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

8. Simplified 79.4%

   \[\leadsto \color{blue}{c \cdot \frac{-0.5}{b}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))