Quadratic roots, wide range

Percentage Accurate: 17.7% → 97.5%

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: 17.7% 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: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \mathsf{fma}\left(-1, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \frac{20 \cdot {a}^{3}}{{b}^{7}}\right)\right)\right) + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (fma
     -1.0
     (/ a (pow b 3.0))
     (*
      c
      (fma
       -2.0
       (/ (pow a 2.0) (pow b 5.0))
       (* -0.25 (* c (/ (* 20.0 (pow a 3.0)) (pow b 7.0))))))))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((c * fma(-1.0, (a / pow(b, 3.0)), (c * fma(-2.0, (pow(a, 2.0) / pow(b, 5.0)), (-0.25 * (c * ((20.0 * pow(a, 3.0)) / pow(b, 7.0)))))))) + (-1.0 / b));
}

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * fma(-1.0, Float64(a / (b ^ 3.0)), Float64(c * fma(-2.0, Float64((a ^ 2.0) / (b ^ 5.0)), Float64(-0.25 * Float64(c * Float64(Float64(20.0 * (a ^ 3.0)) / (b ^ 7.0)))))))) + Float64(-1.0 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(-1.0 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(c * N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(c * N[(N[(20.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.5%

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

   1. *-commutative 15.5%

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

3. Simplified 15.5%

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

5. Taylor expanded in c around 0 98.1%

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

7. Simplified 98.1%

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

8. Taylor expanded in a around 0 98.1%

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

   1. associate-*r/ 98.1%

      \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-1, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \color{blue}{\frac{20 \cdot {a}^{3}}{{b}^{7}}}\right)\right)\right) - \frac{1}{b}\right) \]

10. Simplified 98.1%

    \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-1, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \color{blue}{\frac{20 \cdot {a}^{3}}{{b}^{7}}}\right)\right)\right) - \frac{1}{b}\right) \]

11. Final simplification 98.1%

    \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-1, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \frac{20 \cdot {a}^{3}}{{b}^{7}}\right)\right)\right) + \frac{-1}{b}\right) \]
12. Add Preprocessing

Alternative 2: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-2 \cdot \frac{a}{b} + {a}^{2} \cdot \left(-2 \cdot \frac{c}{{b}^{3}} + a \cdot \left(-10 \cdot \frac{a \cdot {c}^{3}}{{b}^{7}} + -4 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (+
    (* -2.0 (/ a b))
    (*
     (pow a 2.0)
     (+
      (* -2.0 (/ c (pow b 3.0)))
      (*
       a
       (+
        (* -10.0 (/ (* a (pow c 3.0)) (pow b 7.0)))
        (* -4.0 (/ (pow c 2.0) (pow b 5.0)))))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (c * ((-2.0 * (a / b)) + (pow(a, 2.0) * ((-2.0 * (c / pow(b, 3.0))) + (a * ((-10.0 * ((a * pow(c, 3.0)) / pow(b, 7.0))) + (-4.0 * (pow(c, 2.0) / pow(b, 5.0))))))))) / (a * 2.0);
}

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 * (((-2.0d0) * (a / b)) + ((a ** 2.0d0) * (((-2.0d0) * (c / (b ** 3.0d0))) + (a * (((-10.0d0) * ((a * (c ** 3.0d0)) / (b ** 7.0d0))) + ((-4.0d0) * ((c ** 2.0d0) / (b ** 5.0d0))))))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((-2.0 * (a / b)) + (Math.pow(a, 2.0) * ((-2.0 * (c / Math.pow(b, 3.0))) + (a * ((-10.0 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 7.0))) + (-4.0 * (Math.pow(c, 2.0) / Math.pow(b, 5.0))))))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (c * ((-2.0 * (a / b)) + (math.pow(a, 2.0) * ((-2.0 * (c / math.pow(b, 3.0))) + (a * ((-10.0 * ((a * math.pow(c, 3.0)) / math.pow(b, 7.0))) + (-4.0 * (math.pow(c, 2.0) / math.pow(b, 5.0))))))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(-2.0 * Float64(a / b)) + Float64((a ^ 2.0) * Float64(Float64(-2.0 * Float64(c / (b ^ 3.0))) + Float64(a * Float64(Float64(-10.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 7.0))) + Float64(-4.0 * Float64((c ^ 2.0) / (b ^ 5.0))))))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((-2.0 * (a / b)) + ((a ^ 2.0) * ((-2.0 * (c / (b ^ 3.0))) + (a * ((-10.0 * ((a * (c ^ 3.0)) / (b ^ 7.0))) + (-4.0 * ((c ^ 2.0) / (b ^ 5.0))))))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(-2.0 * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[(-2.0 * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-10.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.5%

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

   1. *-commutative 15.5%

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

3. Simplified 15.5%

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

5. Taylor expanded in c around 0 97.9%

   \[\leadsto \frac{\color{blue}{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-4 \cdot \frac{{a}^{3}}{{b}^{5}} + -0.5 \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)}}{a \cdot 2} \]

6. Taylor expanded in a around 0 97.9%

   \[\leadsto \frac{c \cdot \left(-2 \cdot \frac{a}{b} + \color{blue}{{a}^{2} \cdot \left(-2 \cdot \frac{c}{{b}^{3}} + a \cdot \left(-10 \cdot \frac{a \cdot {c}^{3}}{{b}^{7}} + -4 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)}\right)}{a \cdot 2} \]
7. Add Preprocessing

Alternative 3: 97.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.5%

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

   1. *-commutative 15.5%

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

3. Simplified 15.5%

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

5. Taylor expanded in a around 0 97.8%

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

6. Taylor expanded in c around inf 97.8%

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

   1. mul-1-neg 97.8%

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

   2. unsub-neg 97.8%

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

   3. associate-/r* 97.8%

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

8. Simplified 97.8%

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

9. Final simplification 97.8%

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

Alternative 4: 96.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (- (* -2.0 (/ (* c (pow a 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 * ((-2.0 * ((c * pow(a, 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 * (((-2.0d0) * ((c * (a ** 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 * ((-2.0 * ((c * Math.pow(a, 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 * ((-2.0 * ((c * math.pow(a, 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(-2.0 * Float64(Float64(c * (a ^ 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 * ((-2.0 * ((c * (a ^ 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[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $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 15.5%

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

   1. *-commutative 15.5%

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

3. Simplified 15.5%

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

5. Taylor expanded in c around 0 97.5%

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

6. Final simplification 97.5%

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

Alternative 5: 95.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.5%

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

   1. *-commutative 15.5%

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

3. Simplified 15.5%

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

5. Taylor expanded in c around 0 96.2%

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

   1. associate-*r/ 96.2%

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

   2. neg-mul-1 96.2%

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

   3. distribute-rgt-neg-in 96.2%

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

7. Simplified 96.2%

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

8. Taylor expanded in a around inf 96.1%

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

   1. distribute-lft-out 96.1%

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

   2. associate-/r* 96.0%

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

10. Simplified 96.0%

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

11. Taylor expanded in b around inf 96.0%

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

    1. unpow2 96.0%

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

    2. unpow2 96.0%

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

    3. times-frac 96.0%

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

    4. sqr-neg 96.0%

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

    5. mul-1-neg 96.0%

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

    6. associate-*r/ 96.0%

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

    7. *-commutative 96.0%

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

    8. associate-*r/ 96.0%

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

    9. mul-1-neg 96.0%

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

    10. associate-*r/ 96.0%

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

    11. *-commutative 96.0%

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

    12. associate-*r/ 96.0%

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

    13. unpow2 96.0%

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

    14. associate-*r/ 96.0%

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

    15. *-commutative 96.0%

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

    16. mul-1-neg 96.0%

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

    17. distribute-frac-neg 96.0%

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

    18. distribute-frac-neg2 96.0%

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

13. Simplified 96.0%

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

14. Taylor expanded in b around inf 96.6%

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

    1. distribute-lft-out 96.6%

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

    2. associate-*r/ 96.6%

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

    3. mul-1-neg 96.6%

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

    4. distribute-neg-frac2 96.6%

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

    5. +-commutative 96.6%

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

    6. associate-/l* 96.6%

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

    7. fma-define 96.6%

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

    8. unpow2 96.6%

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

    9. unpow2 96.6%

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

    10. times-frac 96.6%

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

    11. unpow2 96.6%

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

16. Simplified 96.6%

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

Alternative 6: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.5%

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

   1. *-commutative 15.5%

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

3. Simplified 15.5%

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

5. Taylor expanded in c around 0 96.2%

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

   1. associate-*r/ 96.2%

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

   2. neg-mul-1 96.2%

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

   3. distribute-rgt-neg-in 96.2%

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

7. Simplified 96.2%

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

   1. div-inv 96.2%

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

   2. pow-flip 96.2%

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

   3. metadata-eval 96.2%

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

9. Applied egg-rr 96.2%

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

    1. associate-*l* 96.2%

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

11. Simplified 96.2%

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

12. Final simplification 96.2%

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

Alternative 7: 95.0% 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(c / a) + Float64(Float64(c / b) * Float64(c / b))) / Float64(-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 15.5%

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

   1. *-commutative 15.5%

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

3. Simplified 15.5%

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

5. Taylor expanded in c around 0 96.2%

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

   1. associate-*r/ 96.2%

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

   2. neg-mul-1 96.2%

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

   3. distribute-rgt-neg-in 96.2%

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

7. Simplified 96.2%

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

8. Taylor expanded in a around inf 96.1%

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

   1. distribute-lft-out 96.1%

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

   2. associate-/r* 96.0%

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

10. Simplified 96.0%

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

11. Taylor expanded in b around inf 96.0%

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

    1. unpow2 96.0%

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

    2. unpow2 96.0%

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

    3. times-frac 96.0%

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

    4. sqr-neg 96.0%

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

    5. mul-1-neg 96.0%

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

    6. associate-*r/ 96.0%

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

    7. *-commutative 96.0%

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

    8. associate-*r/ 96.0%

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

    9. mul-1-neg 96.0%

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

    10. associate-*r/ 96.0%

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

    11. *-commutative 96.0%

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

    12. associate-*r/ 96.0%

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

    13. unpow2 96.0%

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

    14. associate-*r/ 96.0%

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

    15. *-commutative 96.0%

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

    16. mul-1-neg 96.0%

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

    17. distribute-frac-neg 96.0%

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

    18. distribute-frac-neg2 96.0%

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

13. Simplified 96.0%

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

    1. unpow2 96.0%

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

15. Applied egg-rr 96.0%

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

16. Final simplification 96.0%

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

Alternative 8: 90.6% 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 15.5%

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

   1. *-commutative 15.5%

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

3. Simplified 15.5%

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

5. Taylor expanded in b around inf 92.1%

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

   1. associate-*r/ 92.1%

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

   2. mul-1-neg 92.1%

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

7. Simplified 92.1%

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

8. Final simplification 92.1%

   \[\leadsto \frac{c}{-b} \]
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 15.5%

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

   1. *-commutative 15.5%

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

3. Simplified 15.5%

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

5. Taylor expanded in c around 0 96.2%

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

   1. associate-*r/ 96.2%

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

   2. neg-mul-1 96.2%

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

   3. distribute-rgt-neg-in 96.2%

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

7. Simplified 96.2%

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

8. Taylor expanded in a around 0 91.8%

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

   1. expm1-log1p-u 80.0%

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

   2. expm1-undefine 18.6%

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

10. Applied egg-rr 18.6%

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

    1. sub-neg 18.6%

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

    2. metadata-eval 18.6%

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

    3. +-commutative 18.6%

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

    4. log1p-undefine 18.6%

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

    5. rem-exp-log 30.5%

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

    6. associate-*r/ 30.5%

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

    7. *-commutative 30.5%

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

    8. associate-*r/ 30.5%

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

    9. mul-1-neg 30.5%

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

    10. unsub-neg 30.5%

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

12. Simplified 30.5%

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

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 2024103 
(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)))