Quadratic roots, wide range

Percentage Accurate: 18.9% → 97.5%

Time: 15.3s

Alternatives: 11

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: 18.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}} - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (*
     -0.25
     (/
      (+
       (* 16.0 (* (pow a 4.0) (pow c 4.0)))
       (pow (* -2.0 (* (pow a 2.0) (pow c 2.0))) 2.0))
      (* a (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 * (((16.0 * (pow(a, 4.0) * pow(c, 4.0))) + pow((-2.0 * (pow(a, 2.0) * pow(c, 2.0))), 2.0)) / (a * 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) * (((16.0d0 * ((a ** 4.0d0) * (c ** 4.0d0))) + (((-2.0d0) * ((a ** 2.0d0) * (c ** 2.0d0))) ** 2.0d0)) / (a * (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 * (((16.0 * (Math.pow(a, 4.0) * Math.pow(c, 4.0))) + Math.pow((-2.0 * (Math.pow(a, 2.0) * Math.pow(c, 2.0))), 2.0)) / (a * 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 * (((16.0 * (math.pow(a, 4.0) * math.pow(c, 4.0))) + math.pow((-2.0 * (math.pow(a, 2.0) * math.pow(c, 2.0))), 2.0)) / (a * 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(16.0 * Float64((a ^ 4.0) * (c ^ 4.0))) + (Float64(-2.0 * Float64((a ^ 2.0) * (c ^ 2.0))) ^ 2.0)) / Float64(a * (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 * (((16.0 * ((a ^ 4.0) * (c ^ 4.0))) + ((-2.0 * ((a ^ 2.0) * (c ^ 2.0))) ^ 2.0)) / (a * (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[(16.0 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(a * 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 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 97.2%

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

6. Final simplification 97.2%

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

Alternative 2: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot \left(c \cdot -4\right)\right)}^{4}\\ \mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(0.0625, t\_0, t\_0 \cdot 0.015625\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.0625, \frac{\frac{16 \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{a}}{{b}^{3}}, \mathsf{fma}\left(0.25, \frac{c \cdot -4}{b} \cdot \frac{a}{a}, \frac{0.03125}{{b}^{5}} \cdot \frac{{\left(a \cdot c\right)}^{3} \cdot -64}{a}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a (* c -4.0)) 4.0)))
   (fma
    -0.25
    (/ (fma 0.0625 t_0 (* t_0 0.015625)) (* a (pow b 7.0)))
    (fma
     -0.0625
     (/ (/ (* 16.0 (* (* a c) (* a c))) a) (pow b 3.0))
     (fma
      0.25
      (* (/ (* c -4.0) b) (/ a a))
      (* (/ 0.03125 (pow b 5.0)) (/ (* (pow (* a c) 3.0) -64.0) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * (c * -4.0)), 4.0);
	return fma(-0.25, (fma(0.0625, t_0, (t_0 * 0.015625)) / (a * pow(b, 7.0))), fma(-0.0625, (((16.0 * ((a * c) * (a * c))) / a) / pow(b, 3.0)), fma(0.25, (((c * -4.0) / b) * (a / a)), ((0.03125 / pow(b, 5.0)) * ((pow((a * c), 3.0) * -64.0) / a)))));
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * -4.0)) ^ 4.0
	return fma(-0.25, Float64(fma(0.0625, t_0, Float64(t_0 * 0.015625)) / Float64(a * (b ^ 7.0))), fma(-0.0625, Float64(Float64(Float64(16.0 * Float64(Float64(a * c) * Float64(a * c))) / a) / (b ^ 3.0)), fma(0.25, Float64(Float64(Float64(c * -4.0) / b) * Float64(a / a)), Float64(Float64(0.03125 / (b ^ 5.0)) * Float64(Float64((Float64(a * c) ^ 3.0) * -64.0) / a)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision]}, N[(-0.25 * N[(N[(0.0625 * t$95$0 + N[(t$95$0 * 0.015625), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[(N[(N[(16.0 * N[(N[(a * c), $MachinePrecision] * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[(N[(N[(c * -4.0), $MachinePrecision] / b), $MachinePrecision] * N[(a / a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.03125 / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] * -64.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

   1. *-commutative 20.3%

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

   2. prod-diff 20.4%

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

   3. *-commutative 20.4%

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

   4. distribute-rgt-neg-in 20.4%

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

   5. distribute-lft-neg-in 20.4%

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

   6. metadata-eval 20.4%

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

   7. associate-*r* 20.4%

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

   8. distribute-lft-neg-in 20.4%

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

   9. metadata-eval 20.4%

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

   10. *-commutative 20.4%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 2.9%

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

   13. *-commutative 2.9%

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

   14. *-commutative 2.9%

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

   15. swap-sqr 2.9%

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

   16. metadata-eval 2.9%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right) + \mathsf{fma}\left(\sqrt{\color{blue}{16} \cdot \left(a \cdot a\right)}, c, \left(4 \cdot a\right) \cdot c\right)}}{a \cdot 2} \]

   17. metadata-eval 2.9%

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

   18. swap-sqr 2.9%

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

   19. sqrt-unprod 2.9%

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

   20. add-sqr-sqrt 2.9%

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

   21. *-commutative 2.9%

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

   22. add-sqr-sqrt 2.9%

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

   23. sqrt-unprod 2.9%

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

   24. swap-sqr 2.9%

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

6. Applied egg-rr 20.4%

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

   1. +-commutative 20.4%

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

   2. fma-undefine 20.4%

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

   3. associate-+l+ 20.4%

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

   4. *-commutative 20.4%

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

   5. associate-*l* 20.4%

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

   6. +-commutative 20.4%

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

   7. fma-undefine 20.3%

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

   8. unpow2 20.3%

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

   9. associate-+l+ 20.4%

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

   10. distribute-rgt-out 20.4%

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

   11. distribute-lft-out 20.4%

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

   12. metadata-eval 20.4%

       \[\leadsto \frac{\left(-b\right) + \sqrt{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot \color{blue}{-8}\right)\right)}}{a \cdot 2} \]

8. Simplified 20.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot -8\right)\right)}}}{a \cdot 2} \]

9. Taylor expanded in b around inf 96.8%

   \[\leadsto \color{blue}{-0.25 \cdot \frac{0.0625 \cdot {\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{4} + {\left(-0.125 \cdot {\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2}}{a \cdot {b}^{7}} + \left(-0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{3}} + \left(0.03125 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{a \cdot {b}^{5}} + 0.25 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot b}\right)\right)} \]

11. Simplified 97.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(0.0625, {\left(a \cdot \left(c \cdot -4\right)\right)}^{4}, {\left(a \cdot \left(c \cdot -4\right)\right)}^{4} \cdot 0.015625\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.0625, \frac{\frac{{\left(a \cdot c\right)}^{2} \cdot 16}{a}}{{b}^{3}}, \mathsf{fma}\left(0.25, \frac{c \cdot -4}{b} \cdot \frac{a}{a}, \frac{0.03125}{{b}^{5}} \cdot \frac{{\left(a \cdot c\right)}^{3} \cdot -64}{a}\right)\right)\right)} \]
12. Step-by-step derivation

    1. unpow2 93.3%

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

13. Applied egg-rr 97.2%

    \[\leadsto \mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(0.0625, {\left(a \cdot \left(c \cdot -4\right)\right)}^{4}, {\left(a \cdot \left(c \cdot -4\right)\right)}^{4} \cdot 0.015625\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.0625, \frac{\frac{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)} \cdot 16}{a}}{{b}^{3}}, \mathsf{fma}\left(0.25, \frac{c \cdot -4}{b} \cdot \frac{a}{a}, \frac{0.03125}{{b}^{5}} \cdot \frac{{\left(a \cdot c\right)}^{3} \cdot -64}{a}\right)\right)\right) \]

14. Final simplification 97.2%

    \[\leadsto \mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(0.0625, {\left(a \cdot \left(c \cdot -4\right)\right)}^{4}, {\left(a \cdot \left(c \cdot -4\right)\right)}^{4} \cdot 0.015625\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.0625, \frac{\frac{16 \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{a}}{{b}^{3}}, \mathsf{fma}\left(0.25, \frac{c \cdot -4}{b} \cdot \frac{a}{a}, \frac{0.03125}{{b}^{5}} \cdot \frac{{\left(a \cdot c\right)}^{3} \cdot -64}{a}\right)\right)\right) \]
15. Add Preprocessing

Alternative 3: 97.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot -8 + 4 \cdot \left(a \cdot c\right)\\ t_1 := {t\_0}^{2}\\ \frac{-0.5 \cdot \frac{0.0625 \cdot {t\_0}^{4} + {\left(-0.125 \cdot t\_1\right)}^{2}}{{b}^{7}} + \left(-0.125 \cdot \frac{t\_1}{{b}^{3}} + \left(0.0625 \cdot \frac{{t\_0}^{3}}{{b}^{5}} + 0.5 \cdot \frac{t\_0}{b}\right)\right)}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* (* a c) -8.0) (* 4.0 (* a c)))) (t_1 (pow t_0 2.0)))
   (/
    (+
     (*
      -0.5
      (/ (+ (* 0.0625 (pow t_0 4.0)) (pow (* -0.125 t_1) 2.0)) (pow b 7.0)))
     (+
      (* -0.125 (/ t_1 (pow b 3.0)))
      (+ (* 0.0625 (/ (pow t_0 3.0) (pow b 5.0))) (* 0.5 (/ t_0 b)))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = ((a * c) * -8.0) + (4.0 * (a * c));
	double t_1 = pow(t_0, 2.0);
	return ((-0.5 * (((0.0625 * pow(t_0, 4.0)) + pow((-0.125 * t_1), 2.0)) / pow(b, 7.0))) + ((-0.125 * (t_1 / pow(b, 3.0))) + ((0.0625 * (pow(t_0, 3.0) / pow(b, 5.0))) + (0.5 * (t_0 / 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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = ((a * c) * (-8.0d0)) + (4.0d0 * (a * c))
    t_1 = t_0 ** 2.0d0
    code = (((-0.5d0) * (((0.0625d0 * (t_0 ** 4.0d0)) + (((-0.125d0) * t_1) ** 2.0d0)) / (b ** 7.0d0))) + (((-0.125d0) * (t_1 / (b ** 3.0d0))) + ((0.0625d0 * ((t_0 ** 3.0d0) / (b ** 5.0d0))) + (0.5d0 * (t_0 / b))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = ((a * c) * -8.0) + (4.0 * (a * c));
	double t_1 = Math.pow(t_0, 2.0);
	return ((-0.5 * (((0.0625 * Math.pow(t_0, 4.0)) + Math.pow((-0.125 * t_1), 2.0)) / Math.pow(b, 7.0))) + ((-0.125 * (t_1 / Math.pow(b, 3.0))) + ((0.0625 * (Math.pow(t_0, 3.0) / Math.pow(b, 5.0))) + (0.5 * (t_0 / b))))) / (a * 2.0);
}

Python

def code(a, b, c):
	t_0 = ((a * c) * -8.0) + (4.0 * (a * c))
	t_1 = math.pow(t_0, 2.0)
	return ((-0.5 * (((0.0625 * math.pow(t_0, 4.0)) + math.pow((-0.125 * t_1), 2.0)) / math.pow(b, 7.0))) + ((-0.125 * (t_1 / math.pow(b, 3.0))) + ((0.0625 * (math.pow(t_0, 3.0) / math.pow(b, 5.0))) + (0.5 * (t_0 / b))))) / (a * 2.0)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(a * c) * -8.0) + Float64(4.0 * Float64(a * c)))
	t_1 = t_0 ^ 2.0
	return Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(0.0625 * (t_0 ^ 4.0)) + (Float64(-0.125 * t_1) ^ 2.0)) / (b ^ 7.0))) + Float64(Float64(-0.125 * Float64(t_1 / (b ^ 3.0))) + Float64(Float64(0.0625 * Float64((t_0 ^ 3.0) / (b ^ 5.0))) + Float64(0.5 * Float64(t_0 / b))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = ((a * c) * -8.0) + (4.0 * (a * c));
	t_1 = t_0 ^ 2.0;
	tmp = ((-0.5 * (((0.0625 * (t_0 ^ 4.0)) + ((-0.125 * t_1) ^ 2.0)) / (b ^ 7.0))) + ((-0.125 * (t_1 / (b ^ 3.0))) + ((0.0625 * ((t_0 ^ 3.0) / (b ^ 5.0))) + (0.5 * (t_0 / b))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] * -8.0), $MachinePrecision] + N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, N[(N[(N[(-0.5 * N[(N[(N[(0.0625 * N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(-0.125 * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.125 * N[(t$95$1 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0625 * N[(N[Power[t$95$0, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

   1. *-commutative 20.3%

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

   2. prod-diff 20.4%

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

   3. *-commutative 20.4%

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

   4. distribute-rgt-neg-in 20.4%

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

   5. distribute-lft-neg-in 20.4%

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

   6. metadata-eval 20.4%

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

   7. associate-*r* 20.4%

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

   8. distribute-lft-neg-in 20.4%

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

   9. metadata-eval 20.4%

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

   10. *-commutative 20.4%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 2.9%

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

   13. *-commutative 2.9%

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

   14. *-commutative 2.9%

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

   15. swap-sqr 2.9%

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

   16. metadata-eval 2.9%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right) + \mathsf{fma}\left(\sqrt{\color{blue}{16} \cdot \left(a \cdot a\right)}, c, \left(4 \cdot a\right) \cdot c\right)}}{a \cdot 2} \]

   17. metadata-eval 2.9%

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

   18. swap-sqr 2.9%

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

   19. sqrt-unprod 2.9%

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

   20. add-sqr-sqrt 2.9%

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

   21. *-commutative 2.9%

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

   22. add-sqr-sqrt 2.9%

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

   23. sqrt-unprod 2.9%

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

   24. swap-sqr 2.9%

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

6. Applied egg-rr 20.4%

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

   1. +-commutative 20.4%

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

   2. fma-undefine 20.4%

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

   3. associate-+l+ 20.4%

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

   4. *-commutative 20.4%

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

   5. associate-*l* 20.4%

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

   6. +-commutative 20.4%

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

   7. fma-undefine 20.3%

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

   8. unpow2 20.3%

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

   9. associate-+l+ 20.4%

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

   10. distribute-rgt-out 20.4%

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

   11. distribute-lft-out 20.4%

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

   12. metadata-eval 20.4%

       \[\leadsto \frac{\left(-b\right) + \sqrt{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot \color{blue}{-8}\right)\right)}}{a \cdot 2} \]

8. Simplified 20.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot -8\right)\right)}}}{a \cdot 2} \]

9. Taylor expanded in b around inf 96.8%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{0.0625 \cdot {\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{4} + {\left(-0.125 \cdot {\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2}}{{b}^{7}} + \left(-0.125 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{b}\right)\right)}}{a \cdot 2} \]

10. Final simplification 96.8%

    \[\leadsto \frac{-0.5 \cdot \frac{0.0625 \cdot {\left(\left(a \cdot c\right) \cdot -8 + 4 \cdot \left(a \cdot c\right)\right)}^{4} + {\left(-0.125 \cdot {\left(\left(a \cdot c\right) \cdot -8 + 4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2}}{{b}^{7}} + \left(-0.125 \cdot \frac{{\left(\left(a \cdot c\right) \cdot -8 + 4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(\left(a \cdot c\right) \cdot -8 + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{\left(a \cdot c\right) \cdot -8 + 4 \cdot \left(a \cdot c\right)}{b}\right)\right)}{a \cdot 2} \]
11. Add Preprocessing

Alternative 4: 96.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (- (/ (* -2.0 (pow a 2.0)) (/ (pow b 5.0) (pow c 3.0))) (/ 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(b, 5.0) / pow(c, 3.0))) - (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)) / ((b ** 5.0d0) / (c ** 3.0d0))) - (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(b, 5.0) / Math.pow(c, 3.0))) - (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(b, 5.0) / math.pow(c, 3.0))) - (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 * (a ^ 2.0)) / Float64((b ^ 5.0) / (c ^ 3.0))) - 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)) / ((b ^ 5.0) / (c ^ 3.0))) - (c / b)) - (a / ((b ^ 3.0) / (c ^ 2.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $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 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 96.1%

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

   1. associate-+r+ 96.1%

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

   2. mul-1-neg 96.1%

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

   3. unsub-neg 96.1%

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

   4. mul-1-neg 96.1%

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

   5. unsub-neg 96.1%

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

   6. associate-/l* 96.1%

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

   7. associate-*r/ 96.1%

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

   8. associate-/l* 96.1%

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

7. Simplified 96.1%

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

8. Final simplification 96.1%

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

Alternative 5: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot -8 + 4 \cdot \left(a \cdot c\right)\\ t_1 := a \cdot \left(c \cdot -4\right)\\ \frac{\left(0.0625 \cdot \frac{{t\_0}^{3}}{{b}^{5}} + 0.5 \cdot \frac{t\_0}{b}\right) + -0.125 \cdot \frac{t\_1 \cdot t\_1}{{b}^{3}}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* (* a c) -8.0) (* 4.0 (* a c)))) (t_1 (* a (* c -4.0))))
   (/
    (+
     (+ (* 0.0625 (/ (pow t_0 3.0) (pow b 5.0))) (* 0.5 (/ t_0 b)))
     (* -0.125 (/ (* t_1 t_1) (pow b 3.0))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = ((a * c) * -8.0) + (4.0 * (a * c));
	double t_1 = a * (c * -4.0);
	return (((0.0625 * (pow(t_0, 3.0) / pow(b, 5.0))) + (0.5 * (t_0 / b))) + (-0.125 * ((t_1 * t_1) / pow(b, 3.0)))) / (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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = ((a * c) * (-8.0d0)) + (4.0d0 * (a * c))
    t_1 = a * (c * (-4.0d0))
    code = (((0.0625d0 * ((t_0 ** 3.0d0) / (b ** 5.0d0))) + (0.5d0 * (t_0 / b))) + ((-0.125d0) * ((t_1 * t_1) / (b ** 3.0d0)))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = ((a * c) * -8.0) + (4.0 * (a * c));
	double t_1 = a * (c * -4.0);
	return (((0.0625 * (Math.pow(t_0, 3.0) / Math.pow(b, 5.0))) + (0.5 * (t_0 / b))) + (-0.125 * ((t_1 * t_1) / Math.pow(b, 3.0)))) / (a * 2.0);
}

Python

def code(a, b, c):
	t_0 = ((a * c) * -8.0) + (4.0 * (a * c))
	t_1 = a * (c * -4.0)
	return (((0.0625 * (math.pow(t_0, 3.0) / math.pow(b, 5.0))) + (0.5 * (t_0 / b))) + (-0.125 * ((t_1 * t_1) / math.pow(b, 3.0)))) / (a * 2.0)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(a * c) * -8.0) + Float64(4.0 * Float64(a * c)))
	t_1 = Float64(a * Float64(c * -4.0))
	return Float64(Float64(Float64(Float64(0.0625 * Float64((t_0 ^ 3.0) / (b ^ 5.0))) + Float64(0.5 * Float64(t_0 / b))) + Float64(-0.125 * Float64(Float64(t_1 * t_1) / (b ^ 3.0)))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = ((a * c) * -8.0) + (4.0 * (a * c));
	t_1 = a * (c * -4.0);
	tmp = (((0.0625 * ((t_0 ^ 3.0) / (b ^ 5.0))) + (0.5 * (t_0 / b))) + (-0.125 * ((t_1 * t_1) / (b ^ 3.0)))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] * -8.0), $MachinePrecision] + N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(0.0625 * N[(N[Power[t$95$0, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

   1. *-commutative 20.3%

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

   2. prod-diff 20.4%

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

   3. *-commutative 20.4%

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

   4. distribute-rgt-neg-in 20.4%

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

   5. distribute-lft-neg-in 20.4%

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

   6. metadata-eval 20.4%

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

   7. associate-*r* 20.4%

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

   8. distribute-lft-neg-in 20.4%

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

   9. metadata-eval 20.4%

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

   10. *-commutative 20.4%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 2.9%

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

   13. *-commutative 2.9%

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

   14. *-commutative 2.9%

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

   15. swap-sqr 2.9%

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

   16. metadata-eval 2.9%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right) + \mathsf{fma}\left(\sqrt{\color{blue}{16} \cdot \left(a \cdot a\right)}, c, \left(4 \cdot a\right) \cdot c\right)}}{a \cdot 2} \]

   17. metadata-eval 2.9%

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

   18. swap-sqr 2.9%

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

   19. sqrt-unprod 2.9%

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

   20. add-sqr-sqrt 2.9%

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

   21. *-commutative 2.9%

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

   22. add-sqr-sqrt 2.9%

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

   23. sqrt-unprod 2.9%

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

   24. swap-sqr 2.9%

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

6. Applied egg-rr 20.4%

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

   1. +-commutative 20.4%

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

   2. fma-undefine 20.4%

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

   3. associate-+l+ 20.4%

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

   4. *-commutative 20.4%

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

   5. associate-*l* 20.4%

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

   6. +-commutative 20.4%

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

   7. fma-undefine 20.3%

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

   8. unpow2 20.3%

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

   9. associate-+l+ 20.4%

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

   10. distribute-rgt-out 20.4%

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

   11. distribute-lft-out 20.4%

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

   12. metadata-eval 20.4%

       \[\leadsto \frac{\left(-b\right) + \sqrt{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot \color{blue}{-8}\right)\right)}}{a \cdot 2} \]

8. Simplified 20.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot -8\right)\right)}}}{a \cdot 2} \]

9. Taylor expanded in b around inf 95.8%

   \[\leadsto \frac{\color{blue}{-0.125 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{b}\right)}}{a \cdot 2} \]
10. Step-by-step derivation

    1. unpow2 95.8%

       \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right) \cdot \left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

    2. distribute-rgt-out 95.8%

       \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(-8 + 4\right)\right)} \cdot \left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

    3. metadata-eval 95.8%

       \[\leadsto \frac{-0.125 \cdot \frac{\left(\left(a \cdot c\right) \cdot \color{blue}{-4}\right) \cdot \left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

    4. associate-*r* 95.8%

       \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{\left(a \cdot \left(c \cdot -4\right)\right)} \cdot \left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

    5. distribute-rgt-out 95.8%

       \[\leadsto \frac{-0.125 \cdot \frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(-8 + 4\right)\right)}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

    6. metadata-eval 95.8%

       \[\leadsto \frac{-0.125 \cdot \frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(\left(a \cdot c\right) \cdot \color{blue}{-4}\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

    7. associate-*r* 95.8%

       \[\leadsto \frac{-0.125 \cdot \frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot \color{blue}{\left(a \cdot \left(c \cdot -4\right)\right)}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

11. Applied egg-rr 95.8%

    \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

12. Final simplification 95.8%

    \[\leadsto \frac{\left(0.0625 \cdot \frac{{\left(\left(a \cdot c\right) \cdot -8 + 4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{\left(a \cdot c\right) \cdot -8 + 4 \cdot \left(a \cdot c\right)}{b}\right) + -0.125 \cdot \frac{\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)}{{b}^{3}}}{a \cdot 2} \]
13. Add Preprocessing

Alternative 6: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -5e-20) t_0 (/ (- c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -5e-20) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-5d-20)) then
        tmp = t_0
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -5e-20) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -5e-20:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-20)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -5e-20)
		tmp = t_0;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-20], t$95$0, N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -4.9999999999999999e-20

   1. Initial program 70.2%

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

   if -4.9999999999999999e-20 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 6.1%

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

      1. *-commutative 6.1%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in b around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. distribute-neg-frac 98.3%

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

   7. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.9% 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 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 93.7%

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

   1. mul-1-neg 93.7%

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

   2. unsub-neg 93.7%

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

   3. mul-1-neg 93.7%

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

   4. distribute-neg-frac 93.7%

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

   5. associate-/l* 93.7%

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

7. Simplified 93.7%

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

8. Final simplification 93.7%

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

Alternative 8: 94.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 93.3%

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

   1. distribute-lft-out 93.3%

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

   2. associate-/l* 93.3%

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

   3. associate-/l* 93.3%

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

7. Simplified 93.3%

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

8. Taylor expanded in a around 0 93.7%

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

   1. mul-1-neg 93.7%

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

   2. unsub-neg 93.7%

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

   3. associate-*r/ 93.7%

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

   4. mul-1-neg 93.7%

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

10. Simplified 93.7%

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

11. Final simplification 93.7%

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

Alternative 9: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 93.3%

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

   1. distribute-lft-out 93.3%

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

   2. associate-/l* 93.3%

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

   3. associate-/l* 93.3%

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

7. Simplified 93.3%

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

8. Taylor expanded in a around 0 93.3%

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

   1. unpow2 93.3%

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

   2. unpow2 93.3%

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

   3. swap-sqr 93.3%

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

   4. unpow2 93.3%

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

10. Simplified 93.3%

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

    1. associate-/r/ 93.3%

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

12. Applied egg-rr 93.3%

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

    1. unpow2 93.3%

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

14. Applied egg-rr 93.3%

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

15. Final simplification 93.3%

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

Alternative 10: 89.7% 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 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

5. Taylor expanded in b around inf 88.1%

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

   1. mul-1-neg 88.1%

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

   2. distribute-neg-frac 88.1%

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

7. Simplified 88.1%

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

8. Final simplification 88.1%

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

Alternative 11: 3.3% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

   1. *-commutative 20.3%

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

3. Simplified 20.3%

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

   1. *-commutative 20.3%

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

   2. prod-diff 20.4%

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

   3. *-commutative 20.4%

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

   4. distribute-rgt-neg-in 20.4%

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

   5. distribute-lft-neg-in 20.4%

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

   6. metadata-eval 20.4%

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

   7. associate-*r* 20.4%

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

   8. distribute-lft-neg-in 20.4%

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

   9. metadata-eval 20.4%

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

   10. *-commutative 20.4%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 2.9%

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

   13. *-commutative 2.9%

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

   14. *-commutative 2.9%

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

   15. swap-sqr 2.9%

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

   16. metadata-eval 2.9%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right) + \mathsf{fma}\left(\sqrt{\color{blue}{16} \cdot \left(a \cdot a\right)}, c, \left(4 \cdot a\right) \cdot c\right)}}{a \cdot 2} \]

   17. metadata-eval 2.9%

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

   18. swap-sqr 2.9%

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

   19. sqrt-unprod 2.9%

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

   20. add-sqr-sqrt 2.9%

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

   21. *-commutative 2.9%

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

   22. add-sqr-sqrt 2.9%

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

   23. sqrt-unprod 2.9%

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

   24. swap-sqr 2.9%

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

6. Applied egg-rr 20.4%

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

   1. +-commutative 20.4%

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

   2. fma-undefine 20.4%

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

   3. associate-+l+ 20.4%

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

   4. *-commutative 20.4%

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

   5. associate-*l* 20.4%

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

   6. +-commutative 20.4%

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

   7. fma-undefine 20.3%

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

   8. unpow2 20.3%

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

   9. associate-+l+ 20.4%

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

   10. distribute-rgt-out 20.4%

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

   11. distribute-lft-out 20.4%

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

   12. metadata-eval 20.4%

       \[\leadsto \frac{\left(-b\right) + \sqrt{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot \color{blue}{-8}\right)\right)}}{a \cdot 2} \]

8. Simplified 20.4%

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

   1. add-cube-cbrt 20.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot -8\right)\right)}}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot -8\right)\right)}}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot -8\right)\right)}}{a \cdot 2}}} \]

   2. pow3 20.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{a \cdot \left(4 \cdot c\right) + \left({b}^{2} + a \cdot \left(c \cdot -8\right)\right)}}{a \cdot 2}}\right)}^{3}} \]

10. Applied egg-rr 20.4%

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot 4, \mathsf{fma}\left(a, c \cdot -8, {b}^{2}\right)\right)}\right)}{a \cdot 2}}\right)}^{3}} \]

11. Taylor expanded in c around 0 3.3%

    \[\leadsto \color{blue}{0.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{b + -1 \cdot b}{a}\right)} \]
12. Step-by-step derivation

    1. pow-base-1 3.3%

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

    2. associate-*r* 3.3%

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

    3. metadata-eval 3.3%

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

    4. associate-*r/ 3.3%

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

    5. distribute-rgt1-in 3.3%

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

    6. metadata-eval 3.3%

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

    7. mul0-lft 3.3%

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

    8. metadata-eval 3.3%

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

13. Simplified 3.3%

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

14. Final simplification 3.3%

    \[\leadsto \frac{0}{a} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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)))