Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.7%

Time: 14.8s

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: 17.8% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, a \cdot \left(-5 \cdot \left({c}^{4} \cdot \frac{{a}^{2}}{{b}^{6}}\right) - {\left(\frac{c}{b}\right)}^{2}\right) - c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -2.0
   (* (pow a 2.0) (/ (pow c 3.0) (pow b 4.0)))
   (-
    (*
     a
     (-
      (* -5.0 (* (pow c 4.0) (/ (pow a 2.0) (pow b 6.0))))
      (pow (/ c b) 2.0)))
    c))
  b))

C

double code(double a, double b, double c) {
	return fma(-2.0, (pow(a, 2.0) * (pow(c, 3.0) / pow(b, 4.0))), ((a * ((-5.0 * (pow(c, 4.0) * (pow(a, 2.0) / pow(b, 6.0)))) - pow((c / b), 2.0))) - c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-2.0, Float64((a ^ 2.0) * Float64((c ^ 3.0) / (b ^ 4.0))), Float64(Float64(a * Float64(Float64(-5.0 * Float64((c ^ 4.0) * Float64((a ^ 2.0) / (b ^ 6.0)))) - (Float64(c / b) ^ 2.0))) - c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(N[(-5.0 * N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 95.8%

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

7. Simplified 95.8%

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

8. Taylor expanded in a around 0 95.8%

   \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \color{blue}{a \cdot \left(-5 \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} - \frac{{c}^{2}}{{b}^{2}}\right)} - c\right)}{b} \]
9. Step-by-step derivation

   1. *-commutative 95.8%

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

   2. associate-/l* 95.8%

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

   3. unpow2 95.8%

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

   4. unpow2 95.8%

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

   5. times-frac 95.8%

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

   6. unpow2 95.8%

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

10. Simplified 95.8%

    \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \color{blue}{a \cdot \left(-5 \cdot \left({c}^{4} \cdot \frac{{a}^{2}}{{b}^{6}}\right) - {\left(\frac{c}{b}\right)}^{2}\right)} - c\right)}{b} \]
11. Add Preprocessing

Alternative 2: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \frac{-0.25 \cdot \left(a \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right)}{b}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (fma
      -2.0
      (/ (pow c 3.0) (pow b 5.0))
      (/ (* -0.25 (* a (* (/ (pow c 4.0) (pow b 6.0)) 20.0))) b)))
    (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * fma(-2.0, (pow(c, 3.0) / pow(b, 5.0)), ((-0.25 * (a * ((pow(c, 4.0) / pow(b, 6.0)) * 20.0))) / b))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
}

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * fma(-2.0, Float64((c ^ 3.0) / (b ^ 5.0)), Float64(Float64(-0.25 * Float64(a * Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 20.0))) / b))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in a around 0 95.8%

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

   1. +-commutative 95.8%

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

   2. mul-1-neg 95.8%

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

   3. unsub-neg 95.8%

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

7. Simplified 95.8%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \frac{-0.25 \cdot \left(a \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right)}{b}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]
8. Add Preprocessing

Alternative 3: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-1 + c \cdot \left(c \cdot \left({a}^{3} \cdot \left(-5 \cdot \frac{c}{{b}^{6}} - \frac{2}{a \cdot {b}^{4}}\right)\right) - \frac{a}{{b}^{2}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (+
    -1.0
    (*
     c
     (-
      (*
       c
       (*
        (pow a 3.0)
        (- (* -5.0 (/ c (pow b 6.0))) (/ 2.0 (* a (pow b 4.0))))))
      (/ a (pow b 2.0))))))
  b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((c * (pow(a, 3.0) * ((-5.0 * (c / pow(b, 6.0))) - (2.0 / (a * pow(b, 4.0)))))) - (a / 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 * ((-1.0d0) + (c * ((c * ((a ** 3.0d0) * (((-5.0d0) * (c / (b ** 6.0d0))) - (2.0d0 / (a * (b ** 4.0d0)))))) - (a / (b ** 2.0d0)))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((c * (Math.pow(a, 3.0) * ((-5.0 * (c / Math.pow(b, 6.0))) - (2.0 / (a * Math.pow(b, 4.0)))))) - (a / Math.pow(b, 2.0)))))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 + (c * ((c * (math.pow(a, 3.0) * ((-5.0 * (c / math.pow(b, 6.0))) - (2.0 / (a * math.pow(b, 4.0)))))) - (a / math.pow(b, 2.0)))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 + Float64(c * Float64(Float64(c * Float64((a ^ 3.0) * Float64(Float64(-5.0 * Float64(c / (b ^ 6.0))) - Float64(2.0 / Float64(a * (b ^ 4.0)))))) - Float64(a / (b ^ 2.0)))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (-1.0 + (c * ((c * ((a ^ 3.0) * ((-5.0 * (c / (b ^ 6.0))) - (2.0 / (a * (b ^ 4.0)))))) - (a / (b ^ 2.0)))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-1.0 + N[(c * N[(N[(c * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(-5.0 * N[(c / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(a * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 95.8%

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

7. Simplified 95.8%

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

8. Taylor expanded in a around 0 95.8%

   \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \color{blue}{a \cdot \left(-5 \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} - \frac{{c}^{2}}{{b}^{2}}\right)} - c\right)}{b} \]
9. Step-by-step derivation

   1. *-commutative 95.8%

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

   2. associate-/l* 95.8%

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

   3. unpow2 95.8%

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

   4. unpow2 95.8%

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

   5. times-frac 95.8%

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

   6. unpow2 95.8%

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

10. Simplified 95.8%

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

11. Taylor expanded in c around 0 95.7%

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

12. Taylor expanded in a around inf 95.7%

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

    1. associate-*r/ 95.7%

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

    2. metadata-eval 95.7%

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

14. Simplified 95.7%

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

15. Final simplification 95.7%

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

Alternative 4: 96.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 95.8%

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

7. Simplified 95.8%

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

8. Taylor expanded in a around 0 95.8%

   \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \color{blue}{a \cdot \left(-5 \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} - \frac{{c}^{2}}{{b}^{2}}\right)} - c\right)}{b} \]
9. Step-by-step derivation

   1. *-commutative 95.8%

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

   2. associate-/l* 95.8%

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

   3. unpow2 95.8%

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

   4. unpow2 95.8%

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

   5. times-frac 95.8%

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

   6. unpow2 95.8%

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

10. Simplified 95.8%

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

11. Taylor expanded in a around 0 94.8%

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

    1. +-commutative 94.8%

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

    2. mul-1-neg 94.8%

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

    3. unsub-neg 94.8%

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

    4. mul-1-neg 94.8%

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

    5. unsub-neg 94.8%

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

    6. associate-/l* 94.8%

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

13. Simplified 94.8%

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

Alternative 5: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-1 + c \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{4}} + \frac{-1}{{b}^{2}}\right)\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (+
    -1.0
    (* c (* a (+ (* -2.0 (/ (* a c) (pow b 4.0))) (/ -1.0 (pow b 2.0)))))))
  b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 + (c * (a * ((-2.0 * ((a * c) / pow(b, 4.0))) + (-1.0 / 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 * ((-1.0d0) + (c * (a * (((-2.0d0) * ((a * c) / (b ** 4.0d0))) + ((-1.0d0) / (b ** 2.0d0))))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 + (c * (a * ((-2.0 * ((a * c) / Math.pow(b, 4.0))) + (-1.0 / Math.pow(b, 2.0))))))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 + (c * (a * ((-2.0 * ((a * c) / math.pow(b, 4.0))) + (-1.0 / math.pow(b, 2.0))))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 + Float64(c * Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * c) / (b ^ 4.0))) + Float64(-1.0 / (b ^ 2.0))))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (-1.0 + (c * (a * ((-2.0 * ((a * c) / (b ^ 4.0))) + (-1.0 / (b ^ 2.0))))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-1.0 + N[(c * N[(a * N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 95.8%

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

7. Simplified 95.8%

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

8. Taylor expanded in a around 0 95.8%

   \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \color{blue}{a \cdot \left(-5 \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} - \frac{{c}^{2}}{{b}^{2}}\right)} - c\right)}{b} \]
9. Step-by-step derivation

   1. *-commutative 95.8%

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

   2. associate-/l* 95.8%

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

   3. unpow2 95.8%

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

   4. unpow2 95.8%

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

   5. times-frac 95.8%

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

   6. unpow2 95.8%

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

10. Simplified 95.8%

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

11. Taylor expanded in c around 0 95.7%

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

12. Taylor expanded in a around 0 94.8%

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

13. Final simplification 94.8%

    \[\leadsto \frac{c \cdot \left(-1 + c \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{4}} + \frac{-1}{{b}^{2}}\right)\right)\right)}{b} \]
14. Add Preprocessing

Alternative 6: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in a around 0 93.1%

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

   1. mul-1-neg 93.1%

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

   2. unsub-neg 93.1%

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

   3. mul-1-neg 93.1%

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

   4. distribute-neg-frac2 93.1%

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

   5. associate-/l* 93.1%

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

7. Simplified 93.1%

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

Alternative 7: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 95.8%

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

7. Simplified 95.8%

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

8. Taylor expanded in a around 0 95.8%

   \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \color{blue}{a \cdot \left(-5 \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} - \frac{{c}^{2}}{{b}^{2}}\right)} - c\right)}{b} \]
9. Step-by-step derivation

   1. *-commutative 95.8%

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

   2. associate-/l* 95.8%

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

   3. unpow2 95.8%

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

   4. unpow2 95.8%

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

   5. times-frac 95.8%

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

   6. unpow2 95.8%

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

10. Simplified 95.8%

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

11. Taylor expanded in b around inf 93.1%

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

    1. neg-mul-1 93.1%

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

    2. mul-1-neg 93.1%

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

    3. unsub-neg 93.1%

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

    4. associate-/l* 93.1%

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

    5. unpow2 93.1%

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

    6. unpow2 93.1%

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

    7. times-frac 93.1%

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

    8. unpow2 93.1%

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

13. Simplified 93.1%

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

14. Final simplification 93.1%

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

Alternative 8: 90.4% 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 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in a around 0 88.4%

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

   1. associate-*r/ 88.4%

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

   2. mul-1-neg 88.4%

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

7. Simplified 88.4%

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

8. Final simplification 88.4%

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

Alternative 9: 3.3% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. frac-2neg 20.3%

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

   2. div-inv 20.3%

      \[\leadsto \color{blue}{\left(-\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   3. sub-neg 20.3%

      \[\leadsto \left(-\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + \left(-b\right)\right)}\right) \cdot \frac{1}{-a \cdot 2} \]

   4. distribute-neg-in 20.3%

      \[\leadsto \color{blue}{\left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) + \left(-\left(-b\right)\right)\right)} \cdot \frac{1}{-a \cdot 2} \]

   5. pow2 20.3%

      \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, \color{blue}{{b}^{2}}\right)}\right) + \left(-\left(-b\right)\right)\right) \cdot \frac{1}{-a \cdot 2} \]

   6. add-sqr-sqrt 0.0%

      \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \left(-\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

   7. sqrt-unprod 1.6%

      \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \left(-\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

   8. sqr-neg 1.6%

      \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \left(-\sqrt{\color{blue}{b \cdot b}}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

   9. sqrt-prod 1.6%

      \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \left(-\color{blue}{\sqrt{b} \cdot \sqrt{b}}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

   10. add-sqr-sqrt 1.6%

       \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \left(-\color{blue}{b}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

   11. add-sqr-sqrt 0.0%

       \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}\right) \cdot \frac{1}{-a \cdot 2} \]

   12. sqrt-unprod 20.3%

       \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}\right) \cdot \frac{1}{-a \cdot 2} \]

   13. sqr-neg 20.3%

       \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \sqrt{\color{blue}{b \cdot b}}\right) \cdot \frac{1}{-a \cdot 2} \]

   14. sqrt-prod 20.5%

       \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \color{blue}{\sqrt{b} \cdot \sqrt{b}}\right) \cdot \frac{1}{-a \cdot 2} \]

   15. add-sqr-sqrt 20.3%

       \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \color{blue}{b}\right) \cdot \frac{1}{-a \cdot 2} \]

   16. distribute-rgt-neg-in 20.3%

       \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + b\right) \cdot \frac{1}{\color{blue}{a \cdot \left(-2\right)}} \]

   17. metadata-eval 20.3%

       \[\leadsto \left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + b\right) \cdot \frac{1}{a \cdot \color{blue}{-2}} \]

6. Applied egg-rr 20.3%

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

   1. expm1-log1p-u 20.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + b\right)\right)} \cdot \frac{1}{a \cdot -2} \]

   2. expm1-undefine 16.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(-\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + b\right)} - 1\right)} \cdot \frac{1}{a \cdot -2} \]

   3. neg-mul-1 16.4%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{-1 \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}} + b\right)} - 1\right) \cdot \frac{1}{a \cdot -2} \]

   4. fma-define 16.4%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1, \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}, b\right)}\right)} - 1\right) \cdot \frac{1}{a \cdot -2} \]

8. Applied egg-rr 16.4%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}, b\right)\right)} - 1\right)} \cdot \frac{1}{a \cdot -2} \]
9. Step-by-step derivation

   1. expm1-define 20.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-1, \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}, b\right)\right)\right)} \cdot \frac{1}{a \cdot -2} \]

10. Simplified 20.3%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-1, \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}, b\right)\right)\right)} \cdot \frac{1}{a \cdot -2} \]

11. Taylor expanded in a around 0 3.3%

    \[\leadsto \color{blue}{-0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
12. Step-by-step derivation

    1. associate-*r/ 3.3%

       \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + -1 \cdot b\right)}{a}} \]

    2. distribute-rgt1-in 3.3%

       \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

    3. metadata-eval 3.3%

       \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.3%

       \[\leadsto \frac{-0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.3%

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

13. Simplified 3.3%

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

Reproduce

herbie shell --seed 2024144 
(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)))