Quadratic roots, wide range

Percentage Accurate: 17.9% → 97.7%

Time: 28.5s

Alternatives: 9

Speedup: 29.0×

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

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Initial Program: 17.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Alternative 1: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 96.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

6. Taylor expanded in c around 0 96.6%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \color{blue}{\left(20 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}}{b}\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ 96.6%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \color{blue}{\frac{20 \cdot {c}^{4}}{{b}^{6}}}}{b}\right)\right) \]

8. Simplified 96.6%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \color{blue}{\frac{20 \cdot {c}^{4}}{{b}^{6}}}}{b}\right)\right) \]

9. Taylor expanded in c around -inf 96.6%

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

10. Final simplification 96.6%

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

Alternative 2: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left({a}^{2} \cdot -2\right) \cdot \frac{{c}^{3}}{{b}^{4}} - \mathsf{fma}\left(a, {\left(\frac{-c}{b}\right)}^{2}, c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (* (* (pow a 2.0) -2.0) (/ (pow c 3.0) (pow b 4.0)))
   (fma a (pow (/ (- c) b) 2.0) c))
  b))

C

double code(double a, double b, double c) {
	return (((pow(a, 2.0) * -2.0) * (pow(c, 3.0) / pow(b, 4.0))) - fma(a, pow((-c / b), 2.0), c)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64((a ^ 2.0) * -2.0) * Float64((c ^ 3.0) / (b ^ 4.0))) - fma(a, (Float64(Float64(-c) / b) ^ 2.0), c)) / b)
end

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 95.3%

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

   1. distribute-lft-in 95.3%

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

   2. *-commutative 95.3%

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

   3. associate-/l* 95.3%

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

   4. mul-1-neg 95.3%

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

   5. div-inv 95.3%

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

   6. pow-flip 95.3%

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

   7. metadata-eval 95.3%

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

7. Applied egg-rr 95.3%

   \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(\left({a}^{2} \cdot \frac{c}{{b}^{5}}\right) \cdot -2\right) + c \cdot \left(-a \cdot {b}^{-3}\right)\right)} - \frac{1}{b}\right) \]
8. Step-by-step derivation

   1. distribute-lft-out 95.3%

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

   2. unsub-neg 95.3%

      \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(\left({a}^{2} \cdot \frac{c}{{b}^{5}}\right) \cdot -2 - a \cdot {b}^{-3}\right)} - \frac{1}{b}\right) \]

   3. associate-*l* 95.3%

      \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{{a}^{2} \cdot \left(\frac{c}{{b}^{5}} \cdot -2\right)} - a \cdot {b}^{-3}\right) - \frac{1}{b}\right) \]

   4. associate-*l/ 95.3%

      \[\leadsto c \cdot \left(c \cdot \left({a}^{2} \cdot \color{blue}{\frac{c \cdot -2}{{b}^{5}}} - a \cdot {b}^{-3}\right) - \frac{1}{b}\right) \]

9. Simplified 95.3%

   \[\leadsto c \cdot \left(\color{blue}{c \cdot \left({a}^{2} \cdot \frac{c \cdot -2}{{b}^{5}} - a \cdot {b}^{-3}\right)} - \frac{1}{b}\right) \]

10. Taylor expanded in b around inf 95.6%

    \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
11. Step-by-step derivation

12. Simplified 95.6%

    \[\leadsto \color{blue}{\frac{\left(-2 \cdot {a}^{2}\right) \cdot \frac{{c}^{3}}{{b}^{4}} - \mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{b}} \]

13. Final simplification 95.6%

    \[\leadsto \frac{\left({a}^{2} \cdot -2\right) \cdot \frac{{c}^{3}}{{b}^{4}} - \mathsf{fma}\left(a, {\left(\frac{-c}{b}\right)}^{2}, c\right)}{b} \]
14. Add Preprocessing

Alternative 3: 96.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((c * ((pow(a, 2.0) * ((c * -2.0) / pow(b, 5.0))) - (a * pow(b, -3.0)))) + (-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
    code = c * ((c * (((a ** 2.0d0) * ((c * (-2.0d0)) / (b ** 5.0d0))) - (a * (b ** (-3.0d0))))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((Math.pow(a, 2.0) * ((c * -2.0) / Math.pow(b, 5.0))) - (a * Math.pow(b, -3.0)))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * ((math.pow(a, 2.0) * ((c * -2.0) / math.pow(b, 5.0))) - (a * math.pow(b, -3.0)))) + (-1.0 / b))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * (((a ^ 2.0) * ((c * -2.0) / (b ^ 5.0))) - (a * (b ^ -3.0)))) + (-1.0 / b));
end

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 95.3%

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

   1. distribute-lft-in 95.3%

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

   2. *-commutative 95.3%

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

   3. associate-/l* 95.3%

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

   4. mul-1-neg 95.3%

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

   5. div-inv 95.3%

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

   6. pow-flip 95.3%

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

   7. metadata-eval 95.3%

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

7. Applied egg-rr 95.3%

   \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(\left({a}^{2} \cdot \frac{c}{{b}^{5}}\right) \cdot -2\right) + c \cdot \left(-a \cdot {b}^{-3}\right)\right)} - \frac{1}{b}\right) \]
8. Step-by-step derivation

   1. distribute-lft-out 95.3%

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

   2. unsub-neg 95.3%

      \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(\left({a}^{2} \cdot \frac{c}{{b}^{5}}\right) \cdot -2 - a \cdot {b}^{-3}\right)} - \frac{1}{b}\right) \]

   3. associate-*l* 95.3%

      \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{{a}^{2} \cdot \left(\frac{c}{{b}^{5}} \cdot -2\right)} - a \cdot {b}^{-3}\right) - \frac{1}{b}\right) \]

   4. associate-*l/ 95.3%

      \[\leadsto c \cdot \left(c \cdot \left({a}^{2} \cdot \color{blue}{\frac{c \cdot -2}{{b}^{5}}} - a \cdot {b}^{-3}\right) - \frac{1}{b}\right) \]

9. Simplified 95.3%

   \[\leadsto c \cdot \left(\color{blue}{c \cdot \left({a}^{2} \cdot \frac{c \cdot -2}{{b}^{5}} - a \cdot {b}^{-3}\right)} - \frac{1}{b}\right) \]

10. Final simplification 95.3%

    \[\leadsto c \cdot \left(c \cdot \left({a}^{2} \cdot \frac{c \cdot -2}{{b}^{5}} - a \cdot {b}^{-3}\right) + \frac{-1}{b}\right) \]
11. Add Preprocessing

Alternative 4: 95.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 94.0%

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

   1. mul-1-neg 94.0%

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

   2. unsub-neg 94.0%

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

   3. mul-1-neg 94.0%

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

   4. associate-/l* 94.0%

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

7. Simplified 94.0%

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

8. Final simplification 94.0%

   \[\leadsto \frac{c + a \cdot \frac{{c}^{2}}{{b}^{2}}}{-b} \]
9. Add Preprocessing

Alternative 5: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 93.7%

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

   1. associate-*r/ 93.7%

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

   2. neg-mul-1 93.7%

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

   3. distribute-rgt-neg-in 93.7%

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

7. Simplified 93.7%

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

8. Taylor expanded in a around inf 93.6%

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

   1. mul-1-neg 93.6%

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

   2. unsub-neg 93.6%

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

   3. associate-*r/ 93.6%

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

   4. neg-mul-1 93.6%

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

   5. associate-/r* 93.6%

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

10. Simplified 93.6%

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

11. Taylor expanded in c around 0 93.5%

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

    1. sub-neg 93.5%

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

    2. +-commutative 93.5%

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

    3. mul-1-neg 93.5%

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

    4. unsub-neg 93.5%

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

    5. neg-sub0 93.5%

       \[\leadsto a \cdot \left(c \cdot \left(\color{blue}{\left(0 - \frac{1}{a \cdot b}\right)} - \frac{c}{{b}^{3}}\right)\right) \]

    6. div0 93.5%

       \[\leadsto a \cdot \left(c \cdot \left(\left(\color{blue}{\frac{0}{a}} - \frac{1}{a \cdot b}\right) - \frac{c}{{b}^{3}}\right)\right) \]

    7. *-commutative 93.5%

       \[\leadsto a \cdot \left(c \cdot \left(\left(\frac{0}{a} - \frac{1}{\color{blue}{b \cdot a}}\right) - \frac{c}{{b}^{3}}\right)\right) \]

    8. associate-/r* 93.4%

       \[\leadsto a \cdot \left(c \cdot \left(\left(\frac{0}{a} - \color{blue}{\frac{\frac{1}{b}}{a}}\right) - \frac{c}{{b}^{3}}\right)\right) \]

    9. div-sub 93.4%

       \[\leadsto a \cdot \left(c \cdot \left(\color{blue}{\frac{0 - \frac{1}{b}}{a}} - \frac{c}{{b}^{3}}\right)\right) \]

    10. neg-sub0 93.4%

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

    11. distribute-neg-frac 93.4%

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

    12. metadata-eval 93.4%

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

13. Simplified 93.4%

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

14. Taylor expanded in a around inf 93.6%

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

    1. mul-1-neg 93.6%

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

    2. unsub-neg 93.6%

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

    3. associate-/r* 93.6%

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

    4. associate-*r/ 93.6%

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

    5. unpow3 93.6%

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

    6. unpow2 93.6%

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

    7. associate-/r* 93.6%

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

    8. div-sub 93.6%

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

    9. unsub-neg 93.6%

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

    10. mul-1-neg 93.6%

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

    11. distribute-lft-out 93.6%

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

    12. associate-*r/ 93.6%

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

    13. mul-1-neg 93.6%

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

    14. distribute-neg-frac2 93.6%

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

16. Simplified 93.6%

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

    1. associate-*r/ 93.9%

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

    2. +-commutative 93.9%

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

18. Applied egg-rr 93.9%

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

Alternative 6: 95.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (* (pow b -3.0) (* c a)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 93.7%

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

   1. associate-*r/ 93.7%

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

   2. neg-mul-1 93.7%

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

   3. distribute-rgt-neg-in 93.7%

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

7. Simplified 93.7%

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

   1. pow1 93.7%

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

   2. div-inv 93.7%

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

   3. fma-neg 93.7%

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

   4. distribute-rgt-neg-out 93.7%

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

   5. pow-flip 93.7%

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

   6. metadata-eval 93.7%

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

9. Applied egg-rr 93.7%

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

    1. unpow1 93.7%

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

    2. fma-define 93.7%

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

    3. +-commutative 93.7%

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

    4. cancel-sign-sub-inv 93.7%

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

    5. distribute-neg-frac 93.7%

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

    6. metadata-eval 93.7%

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

    7. *-commutative 93.7%

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

11. Simplified 93.7%

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

12. Final simplification 93.7%

    \[\leadsto c \cdot \left(\frac{-1}{b} - {b}^{-3} \cdot \left(c \cdot a\right)\right) \]
13. Add Preprocessing

Alternative 7: 94.9% accurate, 7.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return a * (((-c / a) - ((c / b) * (c / b))) / 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 = a * (((-c / a) - ((c / b) * (c / b))) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 93.7%

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

   1. associate-*r/ 93.7%

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

   2. neg-mul-1 93.7%

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

   3. distribute-rgt-neg-in 93.7%

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

7. Simplified 93.7%

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

8. Taylor expanded in a around inf 93.6%

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

   1. mul-1-neg 93.6%

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

   2. unsub-neg 93.6%

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

   3. associate-*r/ 93.6%

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

   4. neg-mul-1 93.6%

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

   5. associate-/r* 93.6%

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

10. Simplified 93.6%

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

11. Taylor expanded in c around 0 93.5%

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

    1. sub-neg 93.5%

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

    2. +-commutative 93.5%

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

    3. mul-1-neg 93.5%

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

    4. unsub-neg 93.5%

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

    5. neg-sub0 93.5%

       \[\leadsto a \cdot \left(c \cdot \left(\color{blue}{\left(0 - \frac{1}{a \cdot b}\right)} - \frac{c}{{b}^{3}}\right)\right) \]

    6. div0 93.5%

       \[\leadsto a \cdot \left(c \cdot \left(\left(\color{blue}{\frac{0}{a}} - \frac{1}{a \cdot b}\right) - \frac{c}{{b}^{3}}\right)\right) \]

    7. *-commutative 93.5%

       \[\leadsto a \cdot \left(c \cdot \left(\left(\frac{0}{a} - \frac{1}{\color{blue}{b \cdot a}}\right) - \frac{c}{{b}^{3}}\right)\right) \]

    8. associate-/r* 93.4%

       \[\leadsto a \cdot \left(c \cdot \left(\left(\frac{0}{a} - \color{blue}{\frac{\frac{1}{b}}{a}}\right) - \frac{c}{{b}^{3}}\right)\right) \]

    9. div-sub 93.4%

       \[\leadsto a \cdot \left(c \cdot \left(\color{blue}{\frac{0 - \frac{1}{b}}{a}} - \frac{c}{{b}^{3}}\right)\right) \]

    10. neg-sub0 93.4%

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

    11. distribute-neg-frac 93.4%

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

    12. metadata-eval 93.4%

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

13. Simplified 93.4%

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

14. Taylor expanded in a around inf 93.6%

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

    1. mul-1-neg 93.6%

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

    2. unsub-neg 93.6%

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

    3. associate-/r* 93.6%

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

    4. associate-*r/ 93.6%

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

    5. unpow3 93.6%

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

    6. unpow2 93.6%

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

    7. associate-/r* 93.6%

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

    8. div-sub 93.6%

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

    9. unsub-neg 93.6%

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

    10. mul-1-neg 93.6%

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

    11. distribute-lft-out 93.6%

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

    12. associate-*r/ 93.6%

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

    13. mul-1-neg 93.6%

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

    14. distribute-neg-frac2 93.6%

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

16. Simplified 93.6%

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

    1. unpow2 93.6%

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

18. Applied egg-rr 93.6%

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

19. Final simplification 93.6%

    \[\leadsto a \cdot \frac{\frac{-c}{a} - \frac{c}{b} \cdot \frac{c}{b}}{b} \]
20. Add Preprocessing

Alternative 8: 90.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 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 = c / -b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 89.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ 89.6%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   2. mul-1-neg 89.6%

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

7. Simplified 89.6%

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

8. Final simplification 89.6%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Alternative 9: 3.3% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 93.7%

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

   1. associate-*r/ 93.7%

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

   2. neg-mul-1 93.7%

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

   3. distribute-rgt-neg-in 93.7%

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

7. Simplified 93.7%

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

8. Taylor expanded in a around 0 89.4%

   \[\leadsto c \cdot \color{blue}{\frac{-1}{b}} \]
9. Step-by-step derivation

   1. expm1-log1p-u 78.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)\right)} \]

   2. expm1-undefine 17.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)} - 1} \]

   3. associate-*r/ 17.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c \cdot -1}{b}}\right)} - 1 \]

10. Applied egg-rr 17.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c \cdot -1}{b}\right)} - 1} \]
11. Step-by-step derivation

    1. sub-neg 17.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c \cdot -1}{b}\right)} + \left(-1\right)} \]

    2. metadata-eval 17.6%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{c \cdot -1}{b}\right)} + \color{blue}{-1} \]

    3. +-commutative 17.6%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{c \cdot -1}{b}\right)}} \]

    4. log1p-undefine 17.6%

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

    5. rem-exp-log 28.1%

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

    6. *-commutative 28.1%

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

    7. associate-*r/ 28.1%

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

    8. mul-1-neg 28.1%

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

    9. unsub-neg 28.1%

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

12. Simplified 28.1%

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

13. Taylor expanded in c around 0 3.3%

    \[\leadsto -1 + \color{blue}{1} \]

14. Final simplification 3.3%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))