Cubic critical, wide range

Percentage Accurate: 17.9% → 97.7%

Time: 39.0s

Alternatives: 13

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.9% 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: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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 * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * (((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + (((a * (-1.0546875d0)) * (c ** 4.0d0)) / (b ** 7.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * ((-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (((a * -1.0546875) * Math.pow(c, 4.0)) / Math.pow(b, 7.0))))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * ((-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (((a * -1.0546875) * math.pow(c, 4.0)) / math.pow(b, 7.0))))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((-0.5625 * ((c ^ 3.0) / (b ^ 5.0))) + (((a * -1.0546875) * (c ^ 4.0)) / (b ^ 7.0))))));
end

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in a around 0 96.8%

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

4. Taylor expanded in c around 0 96.8%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
5. Step-by-step derivation

   1. associate-*r/ 96.8%

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

   2. associate-*r* 96.8%

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

6. Simplified 96.8%

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

7. Final simplification 96.8%

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

Alternative 2: 97.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around inf 18.2%

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

   1. add-cube-cbrt 18.4%

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

   2. fma-define 18.7%

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

   3. pow2 18.7%

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

   4. cancel-sign-sub-inv 18.7%

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

   5. metadata-eval 18.7%

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

5. Applied egg-rr 18.7%

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

   1. clear-num 18.7%

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

   2. inv-pow 18.7%

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

7. Applied egg-rr 18.2%

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

   1. unpow-1 18.2%

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

   2. *-commutative 18.2%

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

   3. *-lft-identity 18.2%

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

   4. times-frac 18.2%

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

   5. metadata-eval 18.2%

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

   6. fma-undefine 18.2%

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

   7. neg-mul-1 18.2%

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

   8. +-commutative 18.2%

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

   9. unsub-neg 18.2%

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

9. Simplified 18.2%

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

10. Taylor expanded in a around 0 96.7%

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

12. Simplified 96.7%

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

13. Taylor expanded in c around 0 96.7%

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

14. Final simplification 96.7%

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

Alternative 3: 97.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around 0 96.6%

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

5. Simplified 96.6%

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

6. Taylor expanded in c around 0 96.6%

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

7. Taylor expanded in a around 0 96.6%

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

8. Taylor expanded in b around 0 96.6%

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

Alternative 4: 96.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = c / (b ** 3.0d0)
    code = 1.0d0 / (((-2.0d0) * (b / c)) + (a * (((-3.0d0) * (a * ((t_0 * (-0.75d0)) + (t_0 * 0.375d0)))) + (1.5d0 * (1.0d0 / b)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around inf 18.2%

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

   1. add-cube-cbrt 18.4%

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

   2. fma-define 18.7%

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

   3. pow2 18.7%

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

   4. cancel-sign-sub-inv 18.7%

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

   5. metadata-eval 18.7%

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

5. Applied egg-rr 18.7%

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

   1. clear-num 18.7%

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

   2. inv-pow 18.7%

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

7. Applied egg-rr 18.2%

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

   1. unpow-1 18.2%

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

   2. *-commutative 18.2%

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

   3. *-lft-identity 18.2%

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

   4. times-frac 18.2%

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

   5. metadata-eval 18.2%

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

   6. fma-undefine 18.2%

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

   7. neg-mul-1 18.2%

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

   8. +-commutative 18.2%

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

   9. unsub-neg 18.2%

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

9. Simplified 18.2%

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

10. Taylor expanded in a around 0 95.8%

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

11. Final simplification 95.8%

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

Alternative 5: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{a \cdot \frac{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \left(\frac{1.5}{b} - a \cdot \left(\frac{c}{{b}^{3}} \cdot -1.125\right)\right)\right)}{a}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (*
   a
   (/
    (fma -2.0 (/ b c) (* a (- (/ 1.5 b) (* a (* (/ c (pow b 3.0)) -1.125)))))
    a))))

C

double code(double a, double b, double c) {
	return 1.0 / (a * (fma(-2.0, (b / c), (a * ((1.5 / b) - (a * ((c / pow(b, 3.0)) * -1.125))))) / a));
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(a * Float64(fma(-2.0, Float64(b / c), Float64(a * Float64(Float64(1.5 / b) - Float64(a * Float64(Float64(c / (b ^ 3.0)) * -1.125))))) / a)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(a * N[(N[(-2.0 * N[(b / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] - N[(a * N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -1.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in a around inf 18.3%

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

   1. add-cube-cbrt 18.3%

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

   2. pow3 18.3%

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

   3. neg-mul-1 18.3%

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

   4. fma-define 18.3%

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

   5. cancel-sign-sub-inv 18.3%

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

   6. metadata-eval 18.3%

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

5. Applied egg-rr 18.3%

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

   1. rem-cube-cbrt 18.3%

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

   2. clear-num 18.3%

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

   3. inv-pow 18.3%

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

   4. *-commutative 18.3%

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

   5. +-commutative 18.3%

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

   6. *-commutative 18.3%

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

   7. fma-define 18.3%

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

7. Applied egg-rr 18.3%

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

   1. unpow-1 18.3%

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

   2. associate-/l* 18.3%

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

   3. fma-define 18.3%

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

   4. neg-mul-1 18.3%

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

   5. +-commutative 18.3%

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

   6. unsub-neg 18.3%

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

9. Simplified 18.3%

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

10. Taylor expanded in a around 0 95.8%

    \[\leadsto \frac{1}{a \cdot \color{blue}{\frac{-2 \cdot \frac{b}{c} + a \cdot \left(-1 \cdot \left(a \cdot \left(-2.25 \cdot \frac{c}{{b}^{3}} + 1.125 \cdot \frac{c}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{1}{b}\right)}{a}}} \]
11. Step-by-step derivation

12. Simplified 95.8%

    \[\leadsto \frac{1}{a \cdot \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \left(\frac{1.5}{b} - a \cdot \left(\frac{c}{{b}^{3}} \cdot -1.125\right)\right)\right)}{a}}} \]
13. Add Preprocessing

Alternative 6: 96.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around 0 96.6%

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

5. Simplified 96.6%

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

6. Taylor expanded in c around 0 96.6%

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

7. Taylor expanded in a around 0 95.6%

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

   1. associate-*r/ 95.6%

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

   2. associate-*r/ 95.6%

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

   3. metadata-eval 95.6%

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

9. Simplified 95.6%

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

10. Final simplification 95.6%

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

Alternative 7: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{3 \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \left(\frac{0.5}{b} - a \cdot \left(-0.375 \cdot \frac{c}{{b}^{3}}\right)\right)\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (*
   3.0
   (fma
    -0.6666666666666666
    (/ b c)
    (* a (- (/ 0.5 b) (* a (* -0.375 (/ c (pow b 3.0))))))))))

C

double code(double a, double b, double c) {
	return 1.0 / (3.0 * fma(-0.6666666666666666, (b / c), (a * ((0.5 / b) - (a * (-0.375 * (c / pow(b, 3.0))))))));
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(3.0 * fma(-0.6666666666666666, Float64(b / c), Float64(a * Float64(Float64(0.5 / b) - Float64(a * Float64(-0.375 * Float64(c / (b ^ 3.0)))))))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(3.0 * N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(a * N[(N[(0.5 / b), $MachinePrecision] - N[(a * N[(-0.375 * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around inf 18.2%

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

   1. add-cube-cbrt 18.4%

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

   2. fma-define 18.7%

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

   3. pow2 18.7%

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

   4. cancel-sign-sub-inv 18.7%

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

   5. metadata-eval 18.7%

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

5. Applied egg-rr 18.7%

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

   1. clear-num 18.7%

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

   2. inv-pow 18.7%

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

7. Applied egg-rr 18.2%

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

   1. unpow-1 18.2%

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

   2. *-commutative 18.2%

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

   3. *-lft-identity 18.2%

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

   4. times-frac 18.2%

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

   5. metadata-eval 18.2%

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

   6. fma-undefine 18.2%

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

   7. neg-mul-1 18.2%

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

   8. +-commutative 18.2%

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

   9. unsub-neg 18.2%

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

9. Simplified 18.2%

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

10. Taylor expanded in a around 0 95.5%

    \[\leadsto \frac{1}{3 \cdot \color{blue}{\left(-0.6666666666666666 \cdot \frac{b}{c} + a \cdot \left(-1 \cdot \left(a \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right) + 0.5 \cdot \frac{1}{b}\right)\right)}} \]
11. Step-by-step derivation

    1. fma-define 95.5%

       \[\leadsto \frac{1}{3 \cdot \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \left(-1 \cdot \left(a \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right) + 0.5 \cdot \frac{1}{b}\right)\right)}} \]

    2. +-commutative 95.5%

       \[\leadsto \frac{1}{3 \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \color{blue}{\left(0.5 \cdot \frac{1}{b} + -1 \cdot \left(a \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right)\right)}\right)} \]

    3. mul-1-neg 95.5%

       \[\leadsto \frac{1}{3 \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \left(0.5 \cdot \frac{1}{b} + \color{blue}{\left(-a \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right)}\right)\right)} \]

    4. unsub-neg 95.5%

       \[\leadsto \frac{1}{3 \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \color{blue}{\left(0.5 \cdot \frac{1}{b} - a \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right)}\right)} \]

    5. associate-*r/ 95.5%

       \[\leadsto \frac{1}{3 \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{b}} - a \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right)\right)} \]

    6. metadata-eval 95.5%

       \[\leadsto \frac{1}{3 \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \left(\frac{\color{blue}{0.5}}{b} - a \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right)\right)} \]

    7. distribute-rgt-out 95.5%

       \[\leadsto \frac{1}{3 \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \left(\frac{0.5}{b} - a \cdot \color{blue}{\left(\frac{c}{{b}^{3}} \cdot \left(-0.75 + 0.375\right)\right)}\right)\right)} \]

    8. metadata-eval 95.5%

       \[\leadsto \frac{1}{3 \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \left(\frac{0.5}{b} - a \cdot \left(\frac{c}{{b}^{3}} \cdot \color{blue}{-0.375}\right)\right)\right)} \]

12. Simplified 95.5%

    \[\leadsto \frac{1}{3 \cdot \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \left(\frac{0.5}{b} - a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.375\right)\right)\right)}} \]

13. Final simplification 95.5%

    \[\leadsto \frac{1}{3 \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \left(\frac{0.5}{b} - a \cdot \left(-0.375 \cdot \frac{c}{{b}^{3}}\right)\right)\right)} \]
14. Add Preprocessing

Alternative 8: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around inf 18.2%

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

   1. add-cube-cbrt 18.4%

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

   2. fma-define 18.7%

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

   3. pow2 18.7%

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

   4. cancel-sign-sub-inv 18.7%

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

   5. metadata-eval 18.7%

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

5. Applied egg-rr 18.7%

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

   1. clear-num 18.7%

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

   2. inv-pow 18.7%

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

7. Applied egg-rr 18.2%

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

   1. unpow-1 18.2%

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

   2. *-commutative 18.2%

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

   3. *-lft-identity 18.2%

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

   4. times-frac 18.2%

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

   5. metadata-eval 18.2%

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

   6. fma-undefine 18.2%

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

   7. neg-mul-1 18.2%

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

   8. +-commutative 18.2%

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

   9. unsub-neg 18.2%

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

9. Simplified 18.2%

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

10. Taylor expanded in c around 0 94.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]
11. Step-by-step derivation

    1. *-un-lft-identity 94.1%

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

    2. associate-/r/ 94.1%

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

    3. +-commutative 94.1%

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

    4. fma-define 94.1%

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

    5. associate-/l* 94.1%

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

    6. *-commutative 94.1%

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

12. Applied egg-rr 94.1%

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

    1. *-lft-identity 94.1%

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

    2. associate-*l/ 94.4%

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

    3. *-lft-identity 94.4%

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

    4. associate-*r/ 94.4%

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

    5. *-commutative 94.4%

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

14. Simplified 94.4%

    \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, b \cdot -2\right)}} \]
15. Add Preprocessing

Alternative 9: 95.2% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (((b * (-2.0d0)) + (1.5d0 * ((c * a) / b))) / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around inf 18.2%

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

   1. add-cube-cbrt 18.4%

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

   2. fma-define 18.7%

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

   3. pow2 18.7%

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

   4. cancel-sign-sub-inv 18.7%

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

   5. metadata-eval 18.7%

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

5. Applied egg-rr 18.7%

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

   1. clear-num 18.7%

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

   2. inv-pow 18.7%

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

7. Applied egg-rr 18.2%

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

   1. unpow-1 18.2%

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

   2. *-commutative 18.2%

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

   3. *-lft-identity 18.2%

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

   4. times-frac 18.2%

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

   5. metadata-eval 18.2%

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

   6. fma-undefine 18.2%

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

   7. neg-mul-1 18.2%

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

   8. +-commutative 18.2%

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

   9. unsub-neg 18.2%

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

9. Simplified 18.2%

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

10. Taylor expanded in c around 0 94.1%

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

11. Final simplification 94.1%

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

Alternative 10: 95.2% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around inf 18.2%

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

   1. add-cube-cbrt 18.4%

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

   2. fma-define 18.7%

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

   3. pow2 18.7%

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

   4. cancel-sign-sub-inv 18.7%

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

   5. metadata-eval 18.7%

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

5. Applied egg-rr 18.7%

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

   1. clear-num 18.7%

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

   2. inv-pow 18.7%

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

7. Applied egg-rr 18.2%

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

   1. unpow-1 18.2%

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

   2. *-commutative 18.2%

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

   3. *-lft-identity 18.2%

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

   4. times-frac 18.2%

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

   5. metadata-eval 18.2%

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

   6. fma-undefine 18.2%

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

   7. neg-mul-1 18.2%

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

   8. +-commutative 18.2%

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

   9. unsub-neg 18.2%

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

9. Simplified 18.2%

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

10. Taylor expanded in a around 0 94.1%

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

Alternative 11: 90.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in b around inf 89.8%

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

   1. associate-*r/ 89.8%

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

   2. *-commutative 89.8%

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

5. Simplified 89.8%

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

6. Final simplification 89.8%

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

Alternative 12: 90.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around inf 18.2%

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

4. Taylor expanded in c around 0 93.9%

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

   1. associate-*r/ 93.9%

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

   2. metadata-eval 93.9%

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

6. Simplified 93.9%

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

7. Taylor expanded in a around 0 89.5%

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

Alternative 13: 3.3% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around inf 18.2%

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

   1. add-cube-cbrt 18.2%

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

   2. pow3 18.2%

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

   3. neg-mul-1 18.2%

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

   4. fma-define 18.2%

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

   5. cancel-sign-sub-inv 18.2%

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

   6. metadata-eval 18.2%

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

   7. *-commutative 18.2%

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

5. Applied egg-rr 18.2%

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

6. Taylor expanded in c around 0 3.3%

   \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{0.3333333333333333}\right)}^{3} \cdot \left(b + -1 \cdot b\right)}{a}} \]
7. Step-by-step derivation

   1. rem-cube-cbrt 3.3%

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

   2. distribute-rgt1-in 3.3%

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

   3. metadata-eval 3.3%

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

   4. mul0-lft 3.3%

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

   5. metadata-eval 3.3%

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

8. Simplified 3.3%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))