Cubic critical, wide range

Percentage Accurate: 18.0% → 97.6%

Time: 13.1s

Alternatives: 5

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, 0.1× speedup

Math

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

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 3.0) (/ (/ (pow b 7.0) (pow c 4.0)) -1.0546875))))))

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, 3.0) / ((pow(b, 7.0) / pow(c, 4.0)) / -1.0546875))));
}

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 ** 3.0d0) / (((b ** 7.0d0) / (c ** 4.0d0)) / (-1.0546875d0)))))
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, 3.0) / ((Math.pow(b, 7.0) / Math.pow(c, 4.0)) / -1.0546875))));
}

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, 3.0) / ((math.pow(b, 7.0) / math.pow(c, 4.0)) / -1.0546875))))

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((a ^ 3.0) / Float64(Float64((b ^ 7.0) / (c ^ 4.0)) / -1.0546875)))))
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 ^ 3.0) / (((b ^ 7.0) / (c ^ 4.0)) / -1.0546875))));
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[a, 3.0], $MachinePrecision] / N[(N[(N[Power[b, 7.0], $MachinePrecision] / N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / -1.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.8%

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

2. Taylor expanded in b around inf 98.1%

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

3. Taylor expanded in a around 0 98.1%

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

   1. associate-/l* 98.1%

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

   2. distribute-rgt-out 98.1%

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

   3. metadata-eval 98.1%

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

5. Simplified 98.1%

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

   1. associate-*r/ 98.1%

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

   2. div-inv 98.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot {a}^{3}}{\color{blue}{{b}^{7} \cdot \frac{1}{{c}^{4} \cdot 6.328125}}}\right)\right) \]

   3. *-commutative 98.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot {a}^{3}}{{b}^{7} \cdot \frac{1}{\color{blue}{6.328125 \cdot {c}^{4}}}}\right)\right) \]

   4. associate-/r* 98.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot {a}^{3}}{{b}^{7} \cdot \color{blue}{\frac{\frac{1}{6.328125}}{{c}^{4}}}}\right)\right) \]

   5. metadata-eval 98.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot {a}^{3}}{{b}^{7} \cdot \frac{\color{blue}{0.1580246913580247}}{{c}^{4}}}\right)\right) \]

7. Applied egg-rr 98.1%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666 \cdot {a}^{3}}{{b}^{7} \cdot \frac{0.1580246913580247}{{c}^{4}}}}\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\color{blue}{{a}^{3} \cdot -0.16666666666666666}}{{b}^{7} \cdot \frac{0.1580246913580247}{{c}^{4}}}\right)\right) \]

   2. associate-/l* 98.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7} \cdot \frac{0.1580246913580247}{{c}^{4}}}{-0.16666666666666666}}}\right)\right) \]

   3. associate-*r/ 98.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{{a}^{3}}{\frac{\color{blue}{\frac{{b}^{7} \cdot 0.1580246913580247}{{c}^{4}}}}{-0.16666666666666666}}\right)\right) \]

   4. associate-*l/ 98.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{{a}^{3}}{\frac{\color{blue}{\frac{{b}^{7}}{{c}^{4}} \cdot 0.1580246913580247}}{-0.16666666666666666}}\right)\right) \]

   5. associate-/l* 98.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{{a}^{3}}{\color{blue}{\frac{\frac{{b}^{7}}{{c}^{4}}}{\frac{-0.16666666666666666}{0.1580246913580247}}}}\right)\right) \]

   6. metadata-eval 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 2: 96.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* a (* a (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 * ((a * (a * 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 * (a * (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 * ((a * (a * 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 * ((a * (a * 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 * Float64(a * (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 * (a * (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[(a * N[(a * N[Power[c, 3.0], $MachinePrecision]), $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 16.8%

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

2. Taylor expanded in b around inf 97.5%

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

   1. add-sqr-sqrt 97.5%

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

   2. sqr-pow 97.5%

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

   3. times-frac 97.5%

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

   4. sqrt-prod 97.5%

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

   5. unpow2 97.5%

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

   6. sqrt-prod 97.5%

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

   7. add-sqr-sqrt 97.5%

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

   8. sqrt-pow1 97.5%

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

   9. metadata-eval 97.5%

      \[\leadsto -0.5625 \cdot \left(\frac{a \cdot {c}^{\color{blue}{1.5}}}{{b}^{\left(\frac{5}{2}\right)}} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{3}}}{{b}^{\left(\frac{5}{2}\right)}}\right) + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   10. metadata-eval 97.5%

       \[\leadsto -0.5625 \cdot \left(\frac{a \cdot {c}^{1.5}}{{b}^{\color{blue}{2.5}}} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{3}}}{{b}^{\left(\frac{5}{2}\right)}}\right) + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   11. sqrt-prod 97.5%

       \[\leadsto -0.5625 \cdot \left(\frac{a \cdot {c}^{1.5}}{{b}^{2.5}} \cdot \frac{\color{blue}{\sqrt{{a}^{2}} \cdot \sqrt{{c}^{3}}}}{{b}^{\left(\frac{5}{2}\right)}}\right) + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   12. unpow2 97.5%

       \[\leadsto -0.5625 \cdot \left(\frac{a \cdot {c}^{1.5}}{{b}^{2.5}} \cdot \frac{\sqrt{\color{blue}{a \cdot a}} \cdot \sqrt{{c}^{3}}}{{b}^{\left(\frac{5}{2}\right)}}\right) + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   13. sqrt-prod 97.5%

       \[\leadsto -0.5625 \cdot \left(\frac{a \cdot {c}^{1.5}}{{b}^{2.5}} \cdot \frac{\color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)} \cdot \sqrt{{c}^{3}}}{{b}^{\left(\frac{5}{2}\right)}}\right) + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   14. add-sqr-sqrt 97.5%

       \[\leadsto -0.5625 \cdot \left(\frac{a \cdot {c}^{1.5}}{{b}^{2.5}} \cdot \frac{\color{blue}{a} \cdot \sqrt{{c}^{3}}}{{b}^{\left(\frac{5}{2}\right)}}\right) + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   15. sqrt-pow1 97.5%

       \[\leadsto -0.5625 \cdot \left(\frac{a \cdot {c}^{1.5}}{{b}^{2.5}} \cdot \frac{a \cdot \color{blue}{{c}^{\left(\frac{3}{2}\right)}}}{{b}^{\left(\frac{5}{2}\right)}}\right) + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   16. metadata-eval 97.5%

       \[\leadsto -0.5625 \cdot \left(\frac{a \cdot {c}^{1.5}}{{b}^{2.5}} \cdot \frac{a \cdot {c}^{\color{blue}{1.5}}}{{b}^{\left(\frac{5}{2}\right)}}\right) + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   17. metadata-eval 97.5%

       \[\leadsto -0.5625 \cdot \left(\frac{a \cdot {c}^{1.5}}{{b}^{2.5}} \cdot \frac{a \cdot {c}^{1.5}}{{b}^{\color{blue}{2.5}}}\right) + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

4. Applied egg-rr 97.5%

   \[\leadsto -0.5625 \cdot \color{blue}{\left(\frac{a \cdot {c}^{1.5}}{{b}^{2.5}} \cdot \frac{a \cdot {c}^{1.5}}{{b}^{2.5}}\right)} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]
5. Step-by-step derivation

   1. associate-*r/ 97.5%

      \[\leadsto -0.5625 \cdot \color{blue}{\frac{\frac{a \cdot {c}^{1.5}}{{b}^{2.5}} \cdot \left(a \cdot {c}^{1.5}\right)}{{b}^{2.5}}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   2. associate-*l/ 97.5%

      \[\leadsto -0.5625 \cdot \frac{\color{blue}{\frac{\left(a \cdot {c}^{1.5}\right) \cdot \left(a \cdot {c}^{1.5}\right)}{{b}^{2.5}}}}{{b}^{2.5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   3. associate-/r* 97.5%

      \[\leadsto -0.5625 \cdot \color{blue}{\frac{\left(a \cdot {c}^{1.5}\right) \cdot \left(a \cdot {c}^{1.5}\right)}{{b}^{2.5} \cdot {b}^{2.5}}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   4. associate-*l* 97.5%

      \[\leadsto -0.5625 \cdot \frac{\color{blue}{a \cdot \left({c}^{1.5} \cdot \left(a \cdot {c}^{1.5}\right)\right)}}{{b}^{2.5} \cdot {b}^{2.5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   5. *-commutative 97.5%

      \[\leadsto -0.5625 \cdot \frac{a \cdot \left({c}^{1.5} \cdot \color{blue}{\left({c}^{1.5} \cdot a\right)}\right)}{{b}^{2.5} \cdot {b}^{2.5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   6. associate-*l* 97.5%

      \[\leadsto -0.5625 \cdot \frac{a \cdot \color{blue}{\left(\left({c}^{1.5} \cdot {c}^{1.5}\right) \cdot a\right)}}{{b}^{2.5} \cdot {b}^{2.5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   7. metadata-eval 97.5%

      \[\leadsto -0.5625 \cdot \frac{a \cdot \left(\left({c}^{\color{blue}{\left(\frac{3}{2}\right)}} \cdot {c}^{1.5}\right) \cdot a\right)}{{b}^{2.5} \cdot {b}^{2.5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   8. metadata-eval 97.5%

      \[\leadsto -0.5625 \cdot \frac{a \cdot \left(\left({c}^{\left(\frac{3}{2}\right)} \cdot {c}^{\color{blue}{\left(\frac{3}{2}\right)}}\right) \cdot a\right)}{{b}^{2.5} \cdot {b}^{2.5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   9. sqr-pow 97.5%

      \[\leadsto -0.5625 \cdot \frac{a \cdot \left(\color{blue}{{c}^{3}} \cdot a\right)}{{b}^{2.5} \cdot {b}^{2.5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   10. metadata-eval 97.5%

       \[\leadsto -0.5625 \cdot \frac{a \cdot \left({c}^{3} \cdot a\right)}{{b}^{\color{blue}{\left(\frac{5}{2}\right)}} \cdot {b}^{2.5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   11. metadata-eval 97.5%

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

   12. sqr-pow 97.5%

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

6. Simplified 97.5%

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

7. Final simplification 97.5%

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

Alternative 3: 95.2% 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 16.8%

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

2. Taylor expanded in b around inf 96.2%

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

3. Final simplification 96.2%

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

Alternative 4: 94.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((-1.5 * ((a * c) / b)) + (-1.125 * ((1.0 / b) * pow((c * (a / b)), 2.0)))) / (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 = (((-1.5d0) * ((a * c) / b)) + ((-1.125d0) * ((1.0d0 / b) * ((c * (a / b)) ** 2.0d0)))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return ((-1.5 * ((a * c) / b)) + (-1.125 * ((1.0 / b) * Math.pow((c * (a / b)), 2.0)))) / (a * 3.0);
}

Python

def code(a, b, c):
	return ((-1.5 * ((a * c) / b)) + (-1.125 * ((1.0 / b) * math.pow((c * (a / b)), 2.0)))) / (a * 3.0)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((-1.5 * ((a * c) / b)) + (-1.125 * ((1.0 / b) * ((c * (a / b)) ^ 2.0)))) / (a * 3.0);
end

Wolfram

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

Derivation

1. Initial program 16.8%

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

2. Taylor expanded in b around inf 95.4%

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

   1. *-un-lft-identity 95.4%

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

   2. metadata-eval 95.4%

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

   3. cube-mult 95.4%

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

   4. times-frac 95.4%

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

   5. metadata-eval 95.4%

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

   6. unpow2 95.4%

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

   7. unpow2 95.4%

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

   8. swap-sqr 95.4%

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

   9. frac-times 95.4%

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

   10. pow1 95.4%

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

   11. metadata-eval 95.4%

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

   12. pow1 95.4%

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

   13. metadata-eval 95.4%

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

   14. pow-sqr 95.4%

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

   15. associate-/l* 95.4%

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

   16. associate-/r/ 95.4%

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

   17. metadata-eval 95.4%

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

   18. metadata-eval 95.4%

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

4. Applied egg-rr 95.4%

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

5. Final simplification 95.4%

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

Alternative 5: 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 16.8%

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

2. Taylor expanded in b around inf 91.3%

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

   1. associate-*r/ 91.3%

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

   2. *-commutative 91.3%

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

4. Simplified 91.3%

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

5. Final simplification 91.3%

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

Reproduce

herbie shell --seed 2023336 
(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)))