Cubic critical, medium range

Percentage Accurate: 31.5% → 95.1%

Time: 18.8s

Alternatives: 12

Speedup: 23.2×

• Could not converge on a ground truth

  1. reg0 = (bf 4289125927543952)
  2. reg1 = (bf #e3692064761512988.5)
  3. reg2 = (bf #e1.162657755207949851389167599662208034468e-16)

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: 31.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -1.0546875 (/ (* (pow a 3.0) (pow c 4.0)) (pow b 6.0)))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
    (+ (* c -0.5) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0))))))
  b))

C

double code(double a, double b, double c) {
	return ((-1.0546875 * ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 6.0))) + ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0)))))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-1.0546875d0) * (((a ** 3.0d0) * (c ** 4.0d0)) / (b ** 6.0d0))) + (((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 4.0d0))) + ((c * (-0.5d0)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 2.0d0)))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((-1.0546875 * ((Math.pow(a, 3.0) * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + ((-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 4.0))) + ((c * -0.5) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 2.0)))))) / b;
}

Python

def code(a, b, c):
	return ((-1.0546875 * ((math.pow(a, 3.0) * math.pow(c, 4.0)) / math.pow(b, 6.0))) + ((-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 4.0))) + ((c * -0.5) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 2.0)))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-1.0546875 * Float64(Float64((a ^ 3.0) * (c ^ 4.0)) / (b ^ 6.0))) + Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-1.0546875 * (((a ^ 3.0) * (c ^ 4.0)) / (b ^ 6.0))) + ((-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + ((c * -0.5) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 2.0)))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-1.0546875 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Simplified 29.0%

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

5. Taylor expanded in b around inf 96.0%

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

   1. +-commutative 96.0%

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

   2. *-un-lft-identity 96.0%

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

   3. fma-define 96.0%

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

7. Applied egg-rr 96.0%

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

   1. fma-undefine 96.0%

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

9. Simplified 96.0%

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

10. Taylor expanded in b around inf 96.0%

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

11. Final simplification 96.0%

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

Alternative 2: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5 + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* c -0.5)
   (*
    a
    (+
     (* -0.375 (/ (pow c 2.0) (pow b 2.0)))
     (*
      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 ((c * -0.5) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 2.0))) + (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 = ((c * (-0.5d0)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 2.0d0))) + (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 ((c * -0.5) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 2.0))) + (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 ((c * -0.5) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 2.0))) + (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(c * -0.5) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 2.0))) + 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 = ((c * -0.5) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 2.0))) + (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[(c * -0.5), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $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 28.9%

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

3. Simplified 29.0%

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

5. Taylor expanded in b around inf 96.0%

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

   1. +-commutative 96.0%

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

   2. *-un-lft-identity 96.0%

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

   3. fma-define 96.0%

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

7. Applied egg-rr 96.0%

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

   1. fma-undefine 96.0%

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

9. Simplified 96.0%

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

10. Taylor expanded in a around 0 96.0%

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

11. Final simplification 96.0%

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

Alternative 3: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{-1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{4}} + -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -1.0546875 (/ (* (pow a 3.0) (pow c 4.0)) (pow b 6.0)))
   (*
    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 ((-1.0546875 * ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 6.0))) + (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 = (((-1.0546875d0) * (((a ** 3.0d0) * (c ** 4.0d0)) / (b ** 6.0d0))) + (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 ((-1.0546875 * ((Math.pow(a, 3.0) * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + (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 ((-1.0546875 * ((math.pow(a, 3.0) * math.pow(c, 4.0)) / math.pow(b, 6.0))) + (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(Float64(-1.0546875 * Float64(Float64((a ^ 3.0) * (c ^ 4.0)) / (b ^ 6.0))) + 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 = ((-1.0546875 * (((a ^ 3.0) * (c ^ 4.0)) / (b ^ 6.0))) + (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[(N[(-1.0546875 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Simplified 29.0%

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

5. Taylor expanded in b around inf 96.0%

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

   1. +-commutative 96.0%

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

   2. *-un-lft-identity 96.0%

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

   3. fma-define 96.0%

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

7. Applied egg-rr 96.0%

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

   1. fma-undefine 96.0%

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

9. Simplified 96.0%

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

10. Taylor expanded in b around inf 96.0%

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

11. Taylor expanded in c around 0 95.9%

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

12. Final simplification 95.9%

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

Alternative 4: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -0.5625 \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (*
     c
     (+
      (* -0.375 (/ a (pow b 2.0)))
      (*
       c
       (+
        (* -1.0546875 (/ (* (pow a 3.0) c) (pow b 6.0)))
        (* -0.5625 (/ (pow a 2.0) (pow b 4.0)))))))
    0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * ((c * ((-0.375 * (a / pow(b, 2.0))) + (c * ((-1.0546875 * ((pow(a, 3.0) * c) / pow(b, 6.0))) + (-0.5625 * (pow(a, 2.0) / pow(b, 4.0))))))) - 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 * (((-0.375d0) * (a / (b ** 2.0d0))) + (c * (((-1.0546875d0) * (((a ** 3.0d0) * c) / (b ** 6.0d0))) + ((-0.5625d0) * ((a ** 2.0d0) / (b ** 4.0d0))))))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((c * ((-0.375 * (a / Math.pow(b, 2.0))) + (c * ((-1.0546875 * ((Math.pow(a, 3.0) * c) / Math.pow(b, 6.0))) + (-0.5625 * (Math.pow(a, 2.0) / Math.pow(b, 4.0))))))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((c * ((-0.375 * (a / math.pow(b, 2.0))) + (c * ((-1.0546875 * ((math.pow(a, 3.0) * c) / math.pow(b, 6.0))) + (-0.5625 * (math.pow(a, 2.0) / math.pow(b, 4.0))))))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 2.0))) + Float64(c * Float64(Float64(-1.0546875 * Float64(Float64((a ^ 3.0) * c) / (b ^ 6.0))) + Float64(-0.5625 * Float64((a ^ 2.0) / (b ^ 4.0))))))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((c * ((-0.375 * (a / (b ^ 2.0))) + (c * ((-1.0546875 * (((a ^ 3.0) * c) / (b ^ 6.0))) + (-0.5625 * ((a ^ 2.0) / (b ^ 4.0))))))) - 0.5)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-1.0546875 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

3. Simplified 29.0%

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

5. Taylor expanded in b around inf 96.0%

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

   1. +-commutative 96.0%

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

   2. *-un-lft-identity 96.0%

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

   3. fma-define 96.0%

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

7. Applied egg-rr 96.0%

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

   1. fma-undefine 96.0%

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

9. Simplified 96.0%

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

10. Taylor expanded in c around 0 95.8%

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

Alternative 5: 93.7% 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 28.9%

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

3. Simplified 29.0%

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

5. Taylor expanded in a around 0 94.7%

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

Alternative 6: 93.6% 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 28.9%

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

3. Simplified 29.0%

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

5. Taylor expanded in b around inf 96.0%

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

   1. +-commutative 96.0%

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

   2. *-un-lft-identity 96.0%

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

   3. fma-define 96.0%

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

7. Applied egg-rr 96.0%

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

   1. fma-undefine 96.0%

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

9. Simplified 96.0%

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

10. Taylor expanded in c around 0 94.6%

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

11. Final simplification 94.6%

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

Alternative 7: 90.6% 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 28.9%

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

3. Simplified 29.0%

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

5. Taylor expanded in a around 0 92.1%

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

Alternative 8: 90.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.9%

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

3. Simplified 29.0%

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

5. Taylor expanded in c around 0 91.5%

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

6. Taylor expanded in b around inf 92.1%

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

   1. +-commutative 92.1%

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

   2. fma-define 92.1%

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

   3. associate-*r/ 92.1%

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

   4. unpow2 92.1%

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

   5. unpow2 92.1%

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

   6. times-frac 92.1%

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

   7. unpow1 92.1%

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

   8. pow-plus 92.1%

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

   9. metadata-eval 92.1%

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

8. Simplified 92.1%

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

9. Final simplification 92.1%

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

Alternative 9: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.9%

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

   1. log1p-expm1-u 18.3%

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

   2. log1p-undefine 15.7%

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

4. Applied egg-rr 15.7%

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

   1. log1p-define 18.3%

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

   2. log1p-expm1-u 28.9%

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

   3. clear-num 28.9%

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

   4. inv-pow 28.9%

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

   5. *-commutative 28.9%

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

   6. neg-mul-1 28.9%

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

   7. fma-define 28.9%

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

   8. pow2 28.9%

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

   9. associate-*l* 28.9%

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

6. Applied egg-rr 28.9%

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

   1. unpow-1 28.9%

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

   2. unpow2 28.9%

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

   3. fmm-def 29.0%

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

   4. *-commutative 29.0%

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

   5. distribute-rgt-neg-in 29.0%

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

   6. *-commutative 29.0%

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

   7. metadata-eval 29.0%

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

8. Simplified 29.0%

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

9. Taylor expanded in b around inf 92.0%

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

    1. *-commutative 92.0%

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

    2. associate-*r/ 92.0%

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

    3. metadata-eval 92.0%

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

11. Simplified 92.0%

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

Alternative 10: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((-0.375 * ((a * c) / 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) * ((a * c) / (b ** 3.0d0))) - (0.5d0 / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.9%

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

3. Simplified 29.0%

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

5. Taylor expanded in c around 0 91.7%

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

   1. associate-*r/ 91.7%

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

   2. metadata-eval 91.7%

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

7. Simplified 91.7%

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

Alternative 11: 81.2% 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 28.9%

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

3. Simplified 29.0%

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

5. Taylor expanded in b around inf 82.9%

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

Alternative 12: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.9%

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

   1. log1p-expm1-u 18.3%

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

   2. log1p-undefine 15.7%

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

4. Applied egg-rr 15.7%

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

   1. log1p-define 18.3%

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

   2. log1p-expm1-u 28.9%

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

   3. clear-num 28.9%

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

   4. inv-pow 28.9%

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

   5. *-commutative 28.9%

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

   6. neg-mul-1 28.9%

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

   7. fma-define 28.9%

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

   8. pow2 28.9%

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

   9. associate-*l* 28.9%

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

6. Applied egg-rr 28.9%

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

   1. unpow-1 28.9%

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

   2. unpow2 28.9%

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

   3. fmm-def 29.0%

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

   4. *-commutative 29.0%

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

   5. distribute-rgt-neg-in 29.0%

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

   6. *-commutative 29.0%

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

   7. metadata-eval 29.0%

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

8. Simplified 29.0%

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

9. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
10. 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%

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

11. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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)))