Quadratic roots, wide range

Percentage Accurate: 18.0% → 97.6%

Time: 13.4s

Alternatives: 8

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: 18.0% 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: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{fma}\left(-4, \frac{{a}^{2}}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-2, \frac{c}{b} + \frac{a}{{b}^{3}} \cdot {c}^{2}, \frac{-0.5}{a} \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  0.5
  (fma
   -4.0
   (/ (pow a 2.0) (/ (pow b 5.0) (pow c 3.0)))
   (fma
    -2.0
    (+ (/ c b) (* (/ a (pow b 3.0)) (pow c 2.0)))
    (* (/ -0.5 a) (/ (pow (* a c) 4.0) (/ (pow b 7.0) 20.0)))))))

C

double code(double a, double b, double c) {
	return 0.5 * fma(-4.0, (pow(a, 2.0) / (pow(b, 5.0) / pow(c, 3.0))), fma(-2.0, ((c / b) + ((a / pow(b, 3.0)) * pow(c, 2.0))), ((-0.5 / a) * (pow((a * c), 4.0) / (pow(b, 7.0) / 20.0)))));
}

Julia

function code(a, b, c)
	return Float64(0.5 * fma(-4.0, Float64((a ^ 2.0) / Float64((b ^ 5.0) / (c ^ 3.0))), fma(-2.0, Float64(Float64(c / b) + Float64(Float64(a / (b ^ 3.0)) * (c ^ 2.0))), Float64(Float64(-0.5 / a) * Float64((Float64(a * c) ^ 4.0) / Float64((b ^ 7.0) / 20.0))))))
end

Wolfram

code[a_, b_, c_] := N[(0.5 * N[(-4.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(c / b), $MachinePrecision] + N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / a), $MachinePrecision] * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.7%

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

3. Simplified 16.7%

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

   1. div-sub 16.5%

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

   2. sub-neg 16.5%

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

   3. *-un-lft-identity 16.5%

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

   4. *-commutative 16.5%

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

   5. times-frac 16.5%

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

   6. metadata-eval 16.5%

      \[\leadsto \color{blue}{0.5} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} + \left(-\frac{b}{a \cdot 2}\right) \]

   7. pow2 16.5%

      \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, \color{blue}{{b}^{2}}\right)}}{a} + \left(-\frac{b}{a \cdot 2}\right) \]

   8. *-un-lft-identity 16.5%

      \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} + \left(-\frac{\color{blue}{1 \cdot b}}{a \cdot 2}\right) \]

   9. *-commutative 16.5%

      \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} + \left(-\frac{1 \cdot b}{\color{blue}{2 \cdot a}}\right) \]

   10. times-frac 16.5%

       \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} + \left(-\color{blue}{\frac{1}{2} \cdot \frac{b}{a}}\right) \]

   11. metadata-eval 16.5%

       \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} + \left(-\color{blue}{0.5} \cdot \frac{b}{a}\right) \]

6. Applied egg-rr 16.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} + \left(-0.5 \cdot \frac{b}{a}\right)} \]
7. Step-by-step derivation

   1. sub-neg 16.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} - 0.5 \cdot \frac{b}{a}} \]

   2. distribute-lft-out-- 16.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} - \frac{b}{a}\right)} \]

8. Simplified 16.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} - \frac{b}{a}\right)} \]

9. Taylor expanded in b around inf 98.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(-4 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{c}{b} + \left(-2 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)\right)} \]

11. Simplified 98.1%

    \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(-4, \frac{{a}^{2}}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-2, \frac{c}{b} + \frac{a}{{b}^{3}} \cdot {c}^{2}, \frac{-0.5}{a} \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)} \]

12. Final simplification 98.1%

    \[\leadsto 0.5 \cdot \mathsf{fma}\left(-4, \frac{{a}^{2}}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-2, \frac{c}{b} + \frac{a}{{b}^{3}} \cdot {c}^{2}, \frac{-0.5}{a} \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right) \]
13. Add Preprocessing

Alternative 2: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (* -0.25 (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0))))
    (/ (* a (pow c 2.0)) (pow b 3.0)))
   (/ c b))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0)))) - ((a * 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 = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * ((((a * c) ** 4.0d0) / a) * (20.0d0 / (b ** 7.0d0)))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))) - (c / b))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (((-0.25 * ((Math.pow((a * c), 4.0) / a) * (20.0 / Math.pow(b, 7.0)))) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) - (c / b));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (((-0.25 * ((math.pow((a * c), 4.0) / a) * (20.0 / math.pow(b, 7.0)))) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) - (c / b))

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0)))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + (((-0.25 * ((((a * c) ^ 4.0) / a) * (20.0 / (b ^ 7.0)))) - ((a * (c ^ 2.0)) / (b ^ 3.0))) - (c / b));
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 98.1%

   \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

6. Taylor expanded in c around 0 98.1%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
7. Step-by-step derivation

   1. distribute-rgt-out 98.1%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

   2. associate-*r* 98.1%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}\right)\right) \]

   3. *-commutative 98.1%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}\right)\right) \]

   4. times-frac 98.1%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{4 + 16}{{b}^{7}}\right)}\right)\right) \]

8. Simplified 98.1%

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

9. Final simplification 98.1%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \]
10. Add Preprocessing

Alternative 3: 96.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 97.4%

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

   1. associate-+r+ 97.4%

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

   2. mul-1-neg 97.4%

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

   3. unsub-neg 97.4%

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

   4. mul-1-neg 97.4%

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

   5. unsub-neg 97.4%

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

   6. associate-*r/ 97.4%

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

   7. *-commutative 97.4%

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

   8. associate-/l* 97.4%

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

7. Simplified 97.4%

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

8. Final simplification 97.4%

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

Alternative 4: 96.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 97.0%

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

   1. associate-/l* 96.9%

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

   2. associate-/r/ 97.0%

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

7. Applied egg-rr 97.0%

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

   1. pow-prod-down 97.0%

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

   2. *-un-lft-identity 97.0%

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

   3. cube-mult 97.0%

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

   4. times-frac 97.0%

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

   5. unpow2 97.0%

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

   6. frac-times 97.0%

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

   7. associate-*l/ 97.0%

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

   8. associate-/r/ 97.0%

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

   9. associate-*l/ 97.0%

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

   10. associate-/r/ 97.0%

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

   11. pow2 97.0%

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

9. Applied egg-rr 97.0%

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

10. Final simplification 97.0%

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

Alternative 5: 96.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 97.0%

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

   1. expm1-log1p-u 97.0%

      \[\leadsto \frac{-4 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({a}^{3} \cdot {c}^{3}\right)\right)}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

   2. expm1-udef 96.5%

      \[\leadsto \frac{-4 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({a}^{3} \cdot {c}^{3}\right)} - 1}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

   3. pow-prod-down 96.5%

      \[\leadsto \frac{-4 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left(a \cdot c\right)}^{3}}\right)} - 1}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

7. Applied egg-rr 96.5%

   \[\leadsto \frac{-4 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{3}\right)} - 1}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]
8. Step-by-step derivation

   1. expm1-def 97.0%

      \[\leadsto \frac{-4 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot c\right)}^{3}\right)\right)}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

   2. expm1-log1p 97.0%

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

9. Simplified 97.0%

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

    1. pow-prod-down 97.0%

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

    2. *-un-lft-identity 97.0%

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

    3. cube-mult 97.0%

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

    4. times-frac 97.0%

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

    5. unpow2 97.0%

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

    6. frac-times 97.0%

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

    7. associate-*l/ 97.0%

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

    8. associate-/r/ 97.0%

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

    9. associate-*l/ 97.0%

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

    10. associate-/r/ 97.0%

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

    11. pow2 97.0%

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

11. Applied egg-rr 97.0%

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

12. Final simplification 97.0%

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

Alternative 6: 95.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 96.2%

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

   1. mul-1-neg 96.2%

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

   2. unsub-neg 96.2%

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

   3. mul-1-neg 96.2%

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

   4. distribute-neg-frac 96.2%

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

   5. associate-/l* 96.2%

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

7. Simplified 96.2%

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

8. Final simplification 96.2%

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

Alternative 7: 95.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 95.8%

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

   1. distribute-lft-out 95.8%

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

   2. associate-/l* 95.7%

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

   3. associate-/l* 95.7%

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

7. Simplified 95.7%

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

   1. *-un-lft-identity 95.7%

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

   2. associate-/l* 95.7%

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

   3. +-commutative 95.7%

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

   4. associate-/l* 95.7%

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

   5. div-inv 95.7%

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

   6. fma-def 95.7%

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

   7. pow-prod-down 95.7%

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

   8. pow-flip 95.7%

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

   9. metadata-eval 95.7%

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

   10. associate-/r/ 95.7%

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

9. Applied egg-rr 95.7%

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

10. Taylor expanded in a around 0 95.9%

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

11. Final simplification 95.9%

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

Alternative 8: 90.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(-c) / 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 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 91.2%

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

   1. mul-1-neg 91.2%

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

   2. distribute-neg-frac 91.2%

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

7. Simplified 91.2%

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

8. Final simplification 91.2%

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

Reproduce

herbie shell --seed 2024035 
(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)))