Cubic critical, wide range

Percentage Accurate: 18.0% → 97.6%

Time: 13.7s

Alternatives: 6

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.0% 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: 97.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (pow((a * c), 4.0) / (pow(b, 7.0) / (-1.0546875 / 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 = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + (((a * c) ** 4.0d0) / ((b ** 7.0d0) / ((-1.0546875d0) / a)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.0%

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

2. Taylor expanded in b around inf 98.2%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]
3. Step-by-step derivation

   1. clear-num 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\frac{1}{\frac{a \cdot {b}^{7}}{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}}}\right)\right) \]

   2. un-div-inv 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{\frac{a \cdot {b}^{7}}{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}}}\right)\right) \]

   3. *-commutative 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666}{\frac{\color{blue}{{b}^{7} \cdot a}}{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}}\right)\right) \]

   4. associate-/l* 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666}{\color{blue}{\frac{{b}^{7}}{\frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a}}}}\right)\right) \]

4. Applied egg-rr 98.2%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{\frac{{b}^{7}}{\frac{\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot 1.265625\right)}{a}}}}\right)\right) \]
5. Step-by-step derivation

   1. associate-/l* 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666}{\color{blue}{\frac{{b}^{7} \cdot a}{\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot 1.265625\right)}}}\right)\right) \]

   2. *-commutative 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666}{\frac{\color{blue}{a \cdot {b}^{7}}}{\mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot 1.265625\right)}}\right)\right) \]

   3. associate-/r/ 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, {\left(a \cdot c\right)}^{4} \cdot 1.265625\right)}\right)\right) \]

   4. fma-udef 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \color{blue}{\left(5.0625 \cdot {\left(a \cdot c\right)}^{4} + {\left(a \cdot c\right)}^{4} \cdot 1.265625\right)}\right)\right) \]

   5. *-commutative 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \left(5.0625 \cdot {\left(a \cdot c\right)}^{4} + \color{blue}{1.265625 \cdot {\left(a \cdot c\right)}^{4}}\right)\right)\right) \]

   6. distribute-rgt-out 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot \left(5.0625 + 1.265625\right)\right)}\right)\right) \]

   7. metadata-eval 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \left({\left(a \cdot c\right)}^{4} \cdot \color{blue}{6.328125}\right)\right)\right) \]

6. Simplified 98.2%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}\right)\right) \]
7. Step-by-step derivation

   1. associate-*l/ 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{7}}}\right)\right) \]

   2. associate-/l* 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{\frac{a \cdot {b}^{7}}{{\left(a \cdot c\right)}^{4} \cdot 6.328125}}}\right)\right) \]

8. Applied egg-rr 98.2%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{\frac{a \cdot {b}^{7}}{{\left(a \cdot c\right)}^{4} \cdot 6.328125}}}\right)\right) \]
9. Step-by-step derivation

   1. associate-/r/ 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}\right)\right) \]

   2. associate-/r* 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{\frac{-0.16666666666666666}{a}}{{b}^{7}}} \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)\right)\right) \]

   3. associate-*l/ 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{\frac{-0.16666666666666666}{a} \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{{b}^{7}}}\right)\right) \]

   4. *-commutative 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right) \cdot \frac{-0.16666666666666666}{a}}}{{b}^{7}}\right)\right) \]

   5. associate-/l* 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{\frac{{b}^{7}}{\frac{-0.16666666666666666}{a}}}}\right)\right) \]

   6. associate-/l* 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{\frac{\frac{{b}^{7}}{\frac{-0.16666666666666666}{a}}}{6.328125}}}\right)\right) \]

   7. associate-/l/ 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{{\left(a \cdot c\right)}^{4}}{\color{blue}{\frac{{b}^{7}}{6.328125 \cdot \frac{-0.16666666666666666}{a}}}}\right)\right) \]

   8. associate-*r/ 98.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{\color{blue}{\frac{6.328125 \cdot -0.16666666666666666}{a}}}}\right)\right) \]

   9. metadata-eval 98.2%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Alternative 2: 96.8% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+ (* -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 (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-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 = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-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 (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))));
}

Python

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

Wolfram

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

Derivation

1. Initial program 18.0%

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

2. Taylor expanded in b around inf 97.3%

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

3. Final simplification 97.3%

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

Alternative 3: 95.3% 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 18.0%

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

2. Taylor expanded in b around inf 95.7%

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

3. Final simplification 95.7%

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

Alternative 4: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1.125 \cdot \frac{a \cdot \left(c \cdot \left(a \cdot c\right)\right)}{{b}^{3}} + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)}{a \cdot 3} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return ((-1.125 * ((a * (c * (a * c))) / pow(b, 3.0))) + (-1.5 * (a * (c / b)))) / (a * 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 = (((-1.125d0) * ((a * (c * (a * c))) / (b ** 3.0d0))) + ((-1.5d0) * (a * (c / b)))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return ((-1.125 * ((a * (c * (a * c))) / Math.pow(b, 3.0))) + (-1.5 * (a * (c / b)))) / (a * 3.0);
}

Python

def code(a, b, c):
	return ((-1.125 * ((a * (c * (a * c))) / math.pow(b, 3.0))) + (-1.5 * (a * (c / b)))) / (a * 3.0)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((-1.125 * ((a * (c * (a * c))) / (b ^ 3.0))) + (-1.5 * (a * (c / b)))) / (a * 3.0);
end

Wolfram

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

Derivation

1. Initial program 18.0%

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

2. Taylor expanded in b around inf 95.1%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
3. Step-by-step derivation

   1. fma-def 95.1%

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

   2. associate-/l* 95.1%

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

   3. associate-/l* 95.1%

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

4. Simplified 95.1%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \frac{{a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}\right)}}{3 \cdot a} \]
5. Step-by-step derivation

   1. fma-udef 95.1%

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

   2. +-commutative 95.1%

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

   3. associate-/r/ 95.1%

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

   4. associate-*l/ 95.1%

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

   5. pow-prod-down 95.1%

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

   6. div-inv 95.0%

      \[\leadsto \frac{-1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} + -1.5 \cdot \color{blue}{\left(a \cdot \frac{1}{\frac{b}{c}}\right)}}{3 \cdot a} \]

   7. clear-num 95.2%

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

6. Applied egg-rr 95.2%

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

   1. unpow2 95.2%

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

   2. *-commutative 95.2%

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

   3. associate-*r* 95.2%

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

8. Applied egg-rr 95.2%

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

9. Final simplification 95.2%

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

Alternative 5: 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 18.0%

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

2. Taylor expanded in b around inf 89.9%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
3. Step-by-step derivation

   1. associate-*r/ 89.9%

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

   2. associate-*r* 90.1%

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

4. Simplified 90.1%

   \[\leadsto \frac{\color{blue}{\frac{\left(-1.5 \cdot a\right) \cdot c}{b}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. div-inv 90.0%

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

   2. div-inv 89.9%

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

   3. associate-*l* 89.9%

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

   4. *-commutative 89.9%

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

   5. *-commutative 89.9%

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

   6. metadata-eval 89.9%

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

   7. associate-/r* 89.7%

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

   8. metadata-eval 89.7%

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

   9. metadata-eval 89.7%

      \[\leadsto \left(c \cdot \left(a \cdot -1.5\right)\right) \cdot \left(\frac{1}{b} \cdot \frac{\color{blue}{0.3333333333333333}}{a}\right) \]

6. Applied egg-rr 89.7%

   \[\leadsto \color{blue}{\left(c \cdot \left(a \cdot -1.5\right)\right) \cdot \left(\frac{1}{b} \cdot \frac{0.3333333333333333}{a}\right)} \]
7. Step-by-step derivation

   1. associate-*r* 89.8%

      \[\leadsto \color{blue}{\left(\left(c \cdot \left(a \cdot -1.5\right)\right) \cdot \frac{1}{b}\right) \cdot \frac{0.3333333333333333}{a}} \]

   2. associate-*r/ 89.8%

      \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot -1.5\right)\right) \cdot 1}{b}} \cdot \frac{0.3333333333333333}{a} \]

   3. associate-/l* 89.8%

      \[\leadsto \color{blue}{\frac{c \cdot \left(a \cdot -1.5\right)}{\frac{b}{1}}} \cdot \frac{0.3333333333333333}{a} \]

   4. *-commutative 89.8%

      \[\leadsto \frac{\color{blue}{\left(a \cdot -1.5\right) \cdot c}}{\frac{b}{1}} \cdot \frac{0.3333333333333333}{a} \]

   5. /-rgt-identity 89.8%

      \[\leadsto \frac{\left(a \cdot -1.5\right) \cdot c}{\color{blue}{b}} \cdot \frac{0.3333333333333333}{a} \]

   6. associate-*r/ 90.0%

      \[\leadsto \color{blue}{\left(\left(a \cdot -1.5\right) \cdot \frac{c}{b}\right)} \cdot \frac{0.3333333333333333}{a} \]

   7. *-commutative 90.0%

      \[\leadsto \left(\color{blue}{\left(-1.5 \cdot a\right)} \cdot \frac{c}{b}\right) \cdot \frac{0.3333333333333333}{a} \]

   8. associate-*r* 89.9%

      \[\leadsto \color{blue}{\left(-1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)} \cdot \frac{0.3333333333333333}{a} \]

   9. associate-*l* 90.0%

      \[\leadsto \color{blue}{-1.5 \cdot \left(\left(a \cdot \frac{c}{b}\right) \cdot \frac{0.3333333333333333}{a}\right)} \]

   10. associate-*r/ 89.9%

       \[\leadsto -1.5 \cdot \left(\color{blue}{\frac{a \cdot c}{b}} \cdot \frac{0.3333333333333333}{a}\right) \]

   11. *-commutative 89.9%

       \[\leadsto -1.5 \cdot \left(\frac{\color{blue}{c \cdot a}}{b} \cdot \frac{0.3333333333333333}{a}\right) \]

   12. associate-/l* 89.9%

       \[\leadsto -1.5 \cdot \left(\color{blue}{\frac{c}{\frac{b}{a}}} \cdot \frac{0.3333333333333333}{a}\right) \]

8. Simplified 89.9%

   \[\leadsto \color{blue}{-1.5 \cdot \left(\frac{c}{\frac{b}{a}} \cdot \frac{0.3333333333333333}{a}\right)} \]

9. Taylor expanded in c around 0 90.5%

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

11. Simplified 90.2%

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

12. Final simplification 90.2%

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

Alternative 6: 90.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.0%

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

2. Taylor expanded in b around inf 90.5%

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

   1. associate-*r/ 90.5%

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

   2. *-commutative 90.5%

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

4. Simplified 90.5%

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

5. Final simplification 90.5%

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

Reproduce

herbie shell --seed 2023326 
(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)))