Quadratic roots, medium range

Percentage Accurate: 31.4% → 95.7%

Time: 17.2s

Alternatives: 8

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.4% 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: 95.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * (((c ** 4.0d0) * (((-5.0d0) * ((a ** 2.0d0) / (b ** 7.0d0))) + ((-2.0d0) * (a / (c * (b ** 5.0d0)))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

   1. *-commutative 31.6%

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

3. Simplified 31.6%

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

5. Taylor expanded in a around 0 95.0%

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

   1. +-commutative 95.0%

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

   2. mul-1-neg 95.0%

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

   3. unsub-neg 95.0%

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

7. Simplified 95.0%

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

8. Taylor expanded in c around inf 95.0%

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

9. Final simplification 95.0%

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

Alternative 2: 95.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((c * ((c * (((-5.0d0) * ((c * (a ** 3.0d0)) / (b ** 7.0d0))) + ((-2.0d0) * ((a ** 2.0d0) / (b ** 5.0d0))))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((c * ((-5.0 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 7.0))) + (-2.0 * (Math.pow(a, 2.0) / Math.pow(b, 5.0))))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * ((c * ((-5.0 * ((c * math.pow(a, 3.0)) / math.pow(b, 7.0))) + (-2.0 * (math.pow(a, 2.0) / math.pow(b, 5.0))))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((c * ((-5.0 * ((c * (a ^ 3.0)) / (b ^ 7.0))) + (-2.0 * ((a ^ 2.0) / (b ^ 5.0))))) - (a / (b ^ 3.0)))) + (-1.0 / b));
end

Wolfram

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

Derivation

1. Initial program 31.6%

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

   1. *-commutative 31.6%

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

3. Simplified 31.6%

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

   1. frac-2neg 31.6%

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

   2. div-inv 31.6%

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

   3. sub-neg 31.6%

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

   4. distribute-neg-in 31.6%

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

   5. pow2 31.6%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 1.6%

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

   8. sqr-neg 1.6%

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

   9. sqrt-prod 1.6%

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

   10. add-sqr-sqrt 1.6%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 31.6%

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

   13. sqr-neg 31.6%

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

   14. sqrt-prod 31.5%

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

   15. add-sqr-sqrt 31.6%

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

   16. distribute-rgt-neg-in 31.6%

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

   17. metadata-eval 31.6%

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

6. Applied egg-rr 31.6%

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

   1. flip-+ 31.4%

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

   2. pow2 31.4%

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

   3. unpow2 31.4%

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

8. Applied egg-rr 31.4%

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

   1. unpow2 31.4%

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

   2. sqr-neg 31.4%

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

   3. rem-square-sqrt 32.3%

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

   4. fma-define 32.3%

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

   5. associate-*r* 32.3%

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

   6. fma-define 32.3%

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

   7. fma-define 32.3%

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

   8. associate-*r* 32.3%

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

   9. fma-define 32.3%

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

10. Simplified 32.3%

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

11. Taylor expanded in c around 0 94.8%

    \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + 2 \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{3}}{{b}^{7}} + -0.5 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right)\right)\right) - \frac{1}{b}\right)} \]
12. Step-by-step derivation

    1. fmm-def 94.8%

       \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + 2 \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{3}}{{b}^{7}} + -0.5 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right)\right), -\frac{1}{b}\right)} \]

13. Simplified 94.8%

    \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, \left(2 \cdot c\right) \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot -2.5\right)\right) - \frac{a}{{b}^{3}}, -\frac{1}{b}\right)} \]

14. Taylor expanded in c around 0 94.8%

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

15. Final simplification 94.8%

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

Alternative 3: 94.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * (((-2.0d0) * (a * ((c ** 3.0d0) / (b ** 5.0d0)))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

   1. *-commutative 31.6%

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

3. Simplified 31.6%

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

5. Taylor expanded in a around 0 93.5%

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

   1. +-commutative 93.5%

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

   2. mul-1-neg 93.5%

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

   3. unsub-neg 93.5%

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

   4. mul-1-neg 93.5%

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

   5. unsub-neg 93.5%

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

   6. associate-/l* 93.5%

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

7. Simplified 93.5%

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

8. Final simplification 93.5%

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

Alternative 4: 93.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((c * (((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

   1. *-commutative 31.6%

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

3. Simplified 31.6%

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

5. Taylor expanded in c around 0 93.2%

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

6. Final simplification 93.2%

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

Alternative 5: 90.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

   1. *-commutative 31.6%

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

3. Simplified 31.6%

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

5. Taylor expanded in a around 0 90.9%

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

   1. mul-1-neg 90.9%

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

   2. unsub-neg 90.9%

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

   3. mul-1-neg 90.9%

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

   4. distribute-neg-frac2 90.9%

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

   5. associate-/l* 90.9%

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

7. Simplified 90.9%

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

8. Taylor expanded in b around inf 90.9%

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

   1. distribute-lft-out 90.9%

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

   2. associate-*r/ 90.9%

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

   3. mul-1-neg 90.9%

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

   4. distribute-neg-frac2 90.9%

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

   5. +-commutative 90.9%

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

   6. associate-/l* 90.9%

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

   7. fma-define 90.9%

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

   8. unpow2 90.9%

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

   9. unpow2 90.9%

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

   10. times-frac 90.9%

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

   11. sqr-neg 90.9%

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

   12. distribute-frac-neg2 90.9%

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

   13. distribute-frac-neg2 90.9%

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

   14. unpow2 90.9%

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

   15. distribute-frac-neg2 90.9%

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

   16. distribute-frac-neg 90.9%

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

10. Simplified 90.9%

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

11. Final simplification 90.9%

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

Alternative 6: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-1.0d0) / b) - ((a * c) / (b ** 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

   1. *-commutative 31.6%

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

3. Simplified 31.6%

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

5. Taylor expanded in c around 0 90.6%

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

   1. associate-*r/ 90.6%

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

   2. neg-mul-1 90.6%

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

   3. distribute-rgt-neg-in 90.6%

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

7. Simplified 90.6%

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

8. Final simplification 90.6%

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

Alternative 7: 81.3% 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 31.6%

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

   1. *-commutative 31.6%

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

3. Simplified 31.6%

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

5. Taylor expanded in b around inf 81.4%

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

   1. associate-*r/ 81.4%

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

   2. mul-1-neg 81.4%

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

7. Simplified 81.4%

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

8. Final simplification 81.4%

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

Alternative 8: 3.2% 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 31.6%

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

   1. *-commutative 31.6%

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

3. Simplified 31.6%

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

   1. frac-2neg 31.6%

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

   2. div-inv 31.6%

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

   3. sub-neg 31.6%

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

   4. distribute-neg-in 31.6%

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

   5. pow2 31.6%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 1.6%

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

   8. sqr-neg 1.6%

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

   9. sqrt-prod 1.6%

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

   10. add-sqr-sqrt 1.6%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 31.6%

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

   13. sqr-neg 31.6%

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

   14. sqrt-prod 31.5%

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

   15. add-sqr-sqrt 31.6%

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

   16. distribute-rgt-neg-in 31.6%

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

   17. metadata-eval 31.6%

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

6. Applied egg-rr 31.6%

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

   1. add-log-exp 23.0%

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

   2. un-div-inv 23.0%

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

   3. neg-mul-1 23.0%

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

   4. fma-define 23.0%

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

8. Applied egg-rr 23.0%

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

9. Taylor expanded in a around 0 3.2%

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

    1. associate-*r/ 3.2%

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

    2. distribute-rgt1-in 3.2%

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

    3. metadata-eval 3.2%

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

    4. mul0-lft 3.2%

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

    5. metadata-eval 3.2%

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

11. Simplified 3.2%

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

12. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2024076 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))