Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.8%

Time: 14.9s

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: 97.8% 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 17.6%

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

   1. *-commutative 17.6%

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

3. Simplified 17.6%

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

5. Taylor expanded in a around 0 98.0%

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

6. Taylor expanded in c around 0 98.0%

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

7. Taylor expanded in c around -inf 98.0%

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

8. Final simplification 98.0%

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

Alternative 2: 97.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

   1. *-commutative 17.6%

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

3. Simplified 17.6%

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

5. Taylor expanded in c around 0 96.9%

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

6. Taylor expanded in c around -inf 96.7%

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

   1. mul-1-neg 96.7%

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

   2. *-commutative 96.7%

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

   3. distribute-rgt-neg-in 96.7%

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

8. Simplified 96.7%

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

9. Taylor expanded in b around -inf 97.3%

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

    1. mul-1-neg 97.3%

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

    2. distribute-neg-frac2 97.3%

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

    3. +-commutative 97.3%

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

    4. associate-/l* 97.3%

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

    5. fma-define 97.3%

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

    6. unpow2 97.3%

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

    7. unpow2 97.3%

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

    8. times-frac 97.3%

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

    9. unpow2 97.3%

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

    10. associate-/l* 97.3%

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

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

Alternative 3: 97.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

   1. *-commutative 17.6%

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

3. Simplified 17.6%

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

5. Taylor expanded in a around 0 98.0%

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

6. Taylor expanded in c around 0 98.0%

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

7. Taylor expanded in c around 0 97.3%

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

   1. neg-mul-1 97.3%

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

   2. unsub-neg 97.3%

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

   3. *-commutative 97.3%

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

   4. *-commutative 97.3%

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

   5. associate-/l* 97.3%

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

   6. associate-*r* 97.3%

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

   7. *-commutative 97.3%

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

   8. associate-*r/ 97.3%

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

   9. *-commutative 97.3%

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

   10. associate-/l* 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

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

Alternative 4: 96.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

   1. *-commutative 17.6%

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

3. Simplified 17.6%

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

5. Taylor expanded in c around 0 96.9%

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

6. Taylor expanded in a around 0 96.9%

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

   1. mul-1-neg 96.9%

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

   2. distribute-frac-neg 96.9%

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

   3. distribute-lft-in 96.9%

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

   4. distribute-frac-neg 96.9%

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

   5. distribute-rgt-neg-in 96.9%

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

   6. associate-/l* 96.9%

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

   7. unsub-neg 96.9%

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

8. Simplified 96.9%

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

9. Final simplification 96.9%

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

Alternative 5: 95.5% 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 17.6%

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

   1. *-commutative 17.6%

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

3. Simplified 17.6%

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

5. Taylor expanded in a around 0 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

   3. mul-1-neg 95.7%

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

   4. distribute-neg-frac2 95.7%

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

   5. associate-/l* 95.7%

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

7. Simplified 95.7%

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

8. Final simplification 95.7%

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

Alternative 6: 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 17.6%

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

   1. *-commutative 17.6%

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

3. Simplified 17.6%

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

5. Taylor expanded in c around 0 95.4%

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

   1. associate-*r/ 95.4%

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

   2. neg-mul-1 95.4%

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

   3. distribute-rgt-neg-in 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in c around inf 95.2%

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

   1. sub-neg 95.2%

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

   2. +-commutative 95.2%

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

   3. mul-1-neg 95.2%

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

   4. unsub-neg 95.2%

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

   5. associate-/r* 95.3%

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

   6. distribute-neg-frac 95.3%

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

   7. distribute-neg-frac 95.3%

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

   8. metadata-eval 95.3%

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

10. Simplified 95.3%

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

11. Taylor expanded in b around 0 95.2%

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

    1. *-commutative 95.2%

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

13. Simplified 95.2%

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

14. Taylor expanded in b around inf 95.7%

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

    1. distribute-lft-out 95.7%

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

    2. associate-*r/ 95.7%

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

    3. mul-1-neg 95.7%

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

    4. distribute-neg-frac2 95.7%

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

    5. +-commutative 95.7%

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

    6. associate-/l* 95.7%

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

    7. fma-define 95.7%

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

    8. unpow2 95.7%

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

    9. unpow2 95.7%

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

    10. times-frac 95.7%

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

    11. unpow2 95.7%

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

16. Simplified 95.7%

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

17. Final simplification 95.7%

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

Alternative 7: 95.1% accurate, 1.0× speedup

Math

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

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 17.6%

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

   1. *-commutative 17.6%

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

3. Simplified 17.6%

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

5. Taylor expanded in c around 0 95.4%

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

   1. associate-*r/ 95.4%

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

   2. neg-mul-1 95.4%

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

   3. distribute-rgt-neg-in 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in c around 0 95.4%

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

   1. sub-neg 95.4%

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

   2. associate-*r/ 95.4%

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

   3. mul-1-neg 95.4%

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

   4. distribute-rgt-neg-out 95.4%

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

   5. associate-*r/ 95.4%

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

   6. +-commutative 95.4%

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

   7. distribute-frac-neg 95.4%

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

   8. distribute-rgt-neg-in 95.4%

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

   9. associate-/l* 95.4%

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

   10. unsub-neg 95.4%

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

   11. distribute-neg-frac 95.4%

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

   12. metadata-eval 95.4%

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

   13. associate-/l* 95.4%

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

10. Simplified 95.4%

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

11. Final simplification 95.4%

    \[\leadsto c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \]
12. 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 17.6%

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

   1. *-commutative 17.6%

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

3. Simplified 17.6%

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

5. Taylor expanded in b around inf 90.6%

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

   1. associate-*r/ 90.6%

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

   2. mul-1-neg 90.6%

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

7. Simplified 90.6%

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

8. Final simplification 90.6%

   \[\leadsto \frac{c}{-b} \]
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 17.6%

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

   1. *-commutative 17.6%

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

3. Simplified 17.6%

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

5. Taylor expanded in c around 0 95.4%

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

   1. associate-*r/ 95.4%

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

   2. neg-mul-1 95.4%

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

   3. distribute-rgt-neg-in 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in a around 0 90.3%

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

   1. expm1-log1p-u 77.7%

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

   2. expm1-undefine 19.1%

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

10. Applied egg-rr 19.1%

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

    1. sub-neg 19.1%

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

    2. metadata-eval 19.1%

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

    3. +-commutative 19.1%

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

    4. log1p-undefine 19.1%

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

    5. rem-exp-log 31.7%

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

    6. associate-*r/ 31.7%

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

    7. *-commutative 31.7%

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

    8. associate-*r/ 31.7%

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

    9. mul-1-neg 31.7%

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

    10. unsub-neg 31.7%

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

12. Simplified 31.7%

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

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 2024059 
(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)))