Cubic critical, medium range

Percentage Accurate: 31.6% → 95.4%

Time: 13.7s

Alternatives: 9

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: 31.6% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right) + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -0.5 c)
   (*
    a
    (+
     (* -0.375 (* (/ c b) (/ c b)))
     (*
      a
      (+
       (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 6.0)))
       (* -0.5625 (/ (pow c 3.0) (pow b 4.0))))))))
  b))

C

double code(double a, double b, double c) {
	return ((-0.5 * c) + (a * ((-0.375 * ((c / b) * (c / b))) + (a * ((-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-0.5625 * (pow(c, 3.0) / pow(b, 4.0)))))))) / 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) + (a * (((-0.375d0) * ((c / b) * (c / b))) + (a * (((-1.0546875d0) * ((a * (c ** 4.0d0)) / (b ** 6.0d0))) + ((-0.5625d0) * ((c ** 3.0d0) / (b ** 4.0d0)))))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((-0.5 * c) + (a * ((-0.375 * ((c / b) * (c / b))) + (a * ((-1.0546875 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + (-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 4.0)))))))) / b;
}

Python

def code(a, b, c):
	return ((-0.5 * c) + (a * ((-0.375 * ((c / b) * (c / b))) + (a * ((-1.0546875 * ((a * math.pow(c, 4.0)) / math.pow(b, 6.0))) + (-0.5625 * (math.pow(c, 3.0) / math.pow(b, 4.0)))))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-0.5 * c) + Float64(a * Float64(Float64(-0.375 * Float64(Float64(c / b) * Float64(c / b))) + Float64(a * Float64(Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 6.0))) + Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 4.0)))))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-0.5 * c) + (a * ((-0.375 * ((c / b) * (c / b))) + (a * ((-1.0546875 * ((a * (c ^ 4.0)) / (b ^ 6.0))) + (-0.5625 * ((c ^ 3.0) / (b ^ 4.0)))))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.5 * c), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 27.8%

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

3. Simplified 27.9%

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

5. Taylor expanded in b around inf 96.8%

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

   1. +-commutative 96.8%

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

   2. *-un-lft-identity 96.8%

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

   3. fma-define 96.8%

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

7. Applied egg-rr 96.8%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right), -0.5 \cdot c\right)}}{b} \]
8. Step-by-step derivation

   1. fma-undefine 96.8%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(1 \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}}{b} \]

   2. *-lft-identity 96.8%

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

   3. unpow2 96.8%

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

   4. unpow2 96.8%

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

   5. times-frac 96.8%

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

   6. unpow1 96.8%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right), \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}{b} \]

   7. pow-plus 96.8%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}{b} \]

   8. metadata-eval 96.8%

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

   9. associate-/l* 96.8%

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

   10. associate-/l* 96.8%

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

   11. *-commutative 96.8%

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

9. Simplified 96.8%

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

10. Taylor expanded in a around 0 96.8%

    \[\leadsto \frac{\color{blue}{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}}{b} \]
11. Step-by-step derivation

    1. unpow2 96.8%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    2. pow2 96.8%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    3. frac-times 96.8%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

12. Applied egg-rr 96.8%

    \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]
13. Add Preprocessing

Alternative 2: 93.8% 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 27.8%

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

3. Simplified 27.9%

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

5. Taylor expanded in a around 0 95.2%

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

Alternative 3: 93.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{4}} + -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (*
     c
     (+
      (* -0.5625 (/ (* c (pow a 2.0)) (pow b 4.0)))
      (* -0.375 (/ a (pow b 2.0)))))
    0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 4.0))) + (-0.375 * (a / pow(b, 2.0))))) - 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 * ((c * (((-0.5625d0) * ((c * (a ** 2.0d0)) / (b ** 4.0d0))) + ((-0.375d0) * (a / (b ** 2.0d0))))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((c * ((-0.5625 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 4.0))) + (-0.375 * (a / Math.pow(b, 2.0))))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((c * ((-0.5625 * ((c * math.pow(a, 2.0)) / math.pow(b, 4.0))) + (-0.375 * (a / math.pow(b, 2.0))))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 4.0))) + Float64(-0.375 * Float64(a / (b ^ 2.0))))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((c * ((-0.5625 * ((c * (a ^ 2.0)) / (b ^ 4.0))) + (-0.375 * (a / (b ^ 2.0))))) - 0.5)) / b;
end

Wolfram

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

Derivation

1. Initial program 27.8%

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

3. Simplified 27.9%

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

5. Taylor expanded in b around inf 96.8%

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

   1. +-commutative 96.8%

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

   2. *-un-lft-identity 96.8%

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

   3. fma-define 96.8%

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

7. Applied egg-rr 96.8%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right), -0.5 \cdot c\right)}}{b} \]
8. Step-by-step derivation

   1. fma-undefine 96.8%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(1 \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}}{b} \]

   2. *-lft-identity 96.8%

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

   3. unpow2 96.8%

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

   4. unpow2 96.8%

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

   5. times-frac 96.8%

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

   6. unpow1 96.8%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right), \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}{b} \]

   7. pow-plus 96.8%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}{b} \]

   8. metadata-eval 96.8%

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

   9. associate-/l* 96.8%

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

   10. associate-/l* 96.8%

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

   11. *-commutative 96.8%

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

9. Simplified 96.8%

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

10. Taylor expanded in c around 0 95.2%

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

11. Final simplification 95.2%

    \[\leadsto \frac{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{4}} + -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b} \]
12. Add Preprocessing

Alternative 4: 90.7% 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 27.8%

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

3. Simplified 27.9%

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

5. Taylor expanded in a around 0 91.9%

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

Alternative 5: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.8%

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

   1. log1p-expm1-u 19.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   2. log1p-undefine 16.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

4. Applied egg-rr 16.8%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

5. Taylor expanded in b around inf 91.9%

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

   1. fma-define 91.9%

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

   2. associate-*r/ 91.9%

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

   3. unpow2 91.9%

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

   4. unpow2 91.9%

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

   5. times-frac 91.9%

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

   6. unpow1 91.9%

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

   7. pow-plus 91.9%

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

   8. metadata-eval 91.9%

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

7. Simplified 91.9%

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

   1. fma-undefine 91.9%

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

   2. *-commutative 91.9%

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

   3. associate-*r* 91.9%

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

9. Applied egg-rr 91.9%

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

10. Final simplification 91.9%

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

Alternative 6: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * ((-0.375 * ((c * a) / pow(b, 2.0))) - 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 ** 2.0d0))) - 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, 2.0))) - 0.5)) / b;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.8%

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

3. Simplified 27.9%

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

5. Taylor expanded in b around inf 96.8%

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

   1. +-commutative 96.8%

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

   2. *-un-lft-identity 96.8%

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

   3. fma-define 96.8%

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

7. Applied egg-rr 96.8%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right), -0.5 \cdot c\right)}}{b} \]
8. Step-by-step derivation

   1. fma-undefine 96.8%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(1 \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}}{b} \]

   2. *-lft-identity 96.8%

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

   3. unpow2 96.8%

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

   4. unpow2 96.8%

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

   5. times-frac 96.8%

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

   6. unpow1 96.8%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right), \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}{b} \]

   7. pow-plus 96.8%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}{b} \]

   8. metadata-eval 96.8%

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

   9. associate-/l* 96.8%

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

   10. associate-/l* 96.8%

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

   11. *-commutative 96.8%

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

9. Simplified 96.8%

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

10. Taylor expanded in c around 0 91.9%

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

11. Final simplification 91.9%

    \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5\right)}{b} \]
12. Add Preprocessing

Alternative 7: 90.4% accurate, 1.0× speedup

Math

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

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(-0.375 * Float64(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[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.8%

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

3. Simplified 27.9%

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

5. Taylor expanded in c around 0 91.6%

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

   1. associate-*r/ 91.6%

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

   2. metadata-eval 91.6%

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

7. Simplified 91.6%

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

8. Final simplification 91.6%

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

Alternative 8: 81.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.8%

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

3. Simplified 27.9%

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

5. Taylor expanded in b around inf 83.7%

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

Alternative 9: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 27.8%

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

   1. log1p-expm1-u 19.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   2. log1p-undefine 16.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

4. Applied egg-rr 16.8%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
5. Step-by-step derivation

   1. log1p-define 19.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   2. log1p-expm1-u 27.8%

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

   3. div-inv 27.8%

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

   4. neg-mul-1 27.8%

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

   5. fma-define 27.8%

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

   6. pow2 27.8%

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

   7. *-commutative 27.8%

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

   8. *-commutative 27.8%

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

   9. *-commutative 27.8%

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

6. Applied egg-rr 27.8%

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

7. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

9. Simplified 3.2%

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

10. Taylor expanded in a around 0 3.2%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(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)))