Quadratic roots, wide range

Percentage Accurate: 18.0% → 97.7%

Time: 14.3s

Alternatives: 10

Speedup: 3.6×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{{c}^{4} \cdot 20}{{b}^{6}} \cdot \frac{a}{b}, -0.25, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -2}{{b}^{5}}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   (* a a)
   (fma
    (* (/ (* (pow c 4.0) 20.0) (pow b 6.0)) (/ a b))
    -0.25
    (/ (* (* c (* c c)) -2.0) (pow b 5.0))))
  (/ (fma (* c c) (/ a (* b b)) c) b)))

C

double code(double a, double b, double c) {
	return ((a * a) * fma((((pow(c, 4.0) * 20.0) / pow(b, 6.0)) * (a / b)), -0.25, (((c * (c * c)) * -2.0) / pow(b, 5.0)))) - (fma((c * c), (a / (b * b)), c) / b);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * a) * fma(Float64(Float64(Float64((c ^ 4.0) * 20.0) / (b ^ 6.0)) * Float64(a / b)), -0.25, Float64(Float64(Float64(c * Float64(c * c)) * -2.0) / (b ^ 5.0)))) - Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * a), $MachinePrecision] * N[(N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision] * -0.25 + N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.4%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 97.7%

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

6. Final simplification 97.7%

   \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{{c}^{4} \cdot 20}{{b}^{6}} \cdot \frac{a}{b}, -0.25, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -2}{{b}^{5}}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]
7. Add Preprocessing

Alternative 2: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-2}{\left(b \cdot b\right) \cdot t\_0}, \frac{-0.25 \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 20\right)\right)}{t\_0 \cdot \left(b \cdot t\_0\right)}\right), a \cdot a, \frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (fma
    (fma
     c
     (* (* c c) (/ -2.0 (* (* b b) t_0)))
     (/ (* -0.25 (* (* c (* c (* c c))) (* a 20.0))) (* t_0 (* b t_0))))
    (* a a)
    (/ (fma c (* c (/ a (* b b))) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma(fma(c, ((c * c) * (-2.0 / ((b * b) * t_0))), ((-0.25 * ((c * (c * (c * c))) * (a * 20.0))) / (t_0 * (b * t_0)))), (a * a), (fma(c, (c * (a / (b * b))), c) / -b));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(fma(c, Float64(Float64(c * c) * Float64(-2.0 / Float64(Float64(b * b) * t_0))), Float64(Float64(-0.25 * Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * 20.0))) / Float64(t_0 * Float64(b * t_0)))), Float64(a * a), Float64(fma(c, Float64(c * Float64(a / Float64(b * b))), c) / Float64(-b)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * N[(N[(c * c), $MachinePrecision] * N[(-2.0 / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(c * N[(c * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.4%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 97.7%

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

8. Final simplification 97.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-2}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-0.25 \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 20\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), a \cdot a, \frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\right) \]
9. Add Preprocessing

Alternative 3: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -2 \cdot \left(-\frac{\mathsf{fma}\left(a, \frac{2 \cdot \left(c \cdot c\right) - c \cdot c}{b \cdot b}, c \cdot 0.5\right)}{b \cdot \left(b \cdot b\right)}\right), \frac{1}{b}\right), \frac{-b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma
   a
   (fma
    a
    (*
     -2.0
     (-
      (/
       (fma a (/ (- (* 2.0 (* c c)) (* c c)) (* b b)) (* c 0.5))
       (* b (* b b)))))
    (/ 1.0 b))
   (/ (- b) c))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(a, fma(a, (-2.0 * -(fma(a, (((2.0 * (c * c)) - (c * c)) / (b * b)), (c * 0.5)) / (b * (b * b)))), (1.0 / b)), (-b / c));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(a, fma(a, Float64(-2.0 * Float64(-Float64(fma(a, Float64(Float64(Float64(2.0 * Float64(c * c)) - Float64(c * c)) / Float64(b * b)), Float64(c * 0.5)) / Float64(b * Float64(b * b))))), Float64(1.0 / b)), Float64(Float64(-b) / c)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(a * N[(a * N[(-2.0 * (-N[(N[(a * N[(N[(N[(2.0 * N[(c * c), $MachinePrecision]), $MachinePrecision] - N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * 0.5), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[((-b) / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 19.4

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 19.4

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

4. Applied rewrites 19.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 19.4

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 19.4

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 19.4

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

6. Applied rewrites 19.4%

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

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-2 \cdot \left(a \cdot \left(-1 \cdot \frac{c \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-1}{8} \cdot \frac{b \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) + -2 \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}} \]

9. Applied rewrites 97.5%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -2 \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(c, -\frac{\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5}{b \cdot b}, \mathsf{fma}\left(-0.125, b \cdot \frac{\frac{{c}^{4} \cdot 20}{{b}^{6}}}{c \cdot c}, \frac{c \cdot c}{{b}^{5}}\right)\right), \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right), \frac{1}{b}\right), \frac{b}{-c}\right)}} \]

10. Taylor expanded in b around -inf

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

12. Applied rewrites 97.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -2 \cdot \left(-\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot 2 - c \cdot c}{b \cdot b}, 0.5 \cdot c\right)}{b \cdot \left(b \cdot b\right)}\right), \frac{1}{b}\right), \frac{b}{-c}\right)} \]

13. Final simplification 97.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -2 \cdot \left(-\frac{\mathsf{fma}\left(a, \frac{2 \cdot \left(c \cdot c\right) - c \cdot c}{b \cdot b}, c \cdot 0.5\right)}{b \cdot \left(b \cdot b\right)}\right), \frac{1}{b}\right), \frac{-b}{c}\right)} \]
14. Add Preprocessing

Alternative 4: 96.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 97.0%

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

6. Final simplification 97.0%

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

Alternative 5: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2, a \cdot \left(-0.5 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right), \frac{1}{b}\right), \frac{-b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma a (fma -2.0 (* a (* -0.5 (/ c (* b (* b b))))) (/ 1.0 b)) (/ (- b) c))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(a, fma(-2.0, (a * (-0.5 * (c / (b * (b * b))))), (1.0 / b)), (-b / c));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(a, fma(-2.0, Float64(a * Float64(-0.5 * Float64(c / Float64(b * Float64(b * b))))), Float64(1.0 / b)), Float64(Float64(-b) / c)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(a * N[(-2.0 * N[(a * N[(-0.5 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[((-b) / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 19.4

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 19.4

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

4. Applied rewrites 19.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 19.4

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 19.4

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 19.4

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

6. Applied rewrites 19.4%

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

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. distribute-rgt-out N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. cube-mult N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

9. Applied rewrites 96.8%

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

10. Final simplification 96.8%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2, a \cdot \left(-0.5 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right), \frac{1}{b}\right), \frac{-b}{c}\right)} \]
11. Add Preprocessing

Alternative 6: 96.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 19.4

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 19.4

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

4. Applied rewrites 19.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 19.4

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 19.4

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 19.4

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

6. Applied rewrites 19.4%

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

7. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

9. Applied rewrites 96.7%

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

10. Taylor expanded in b around inf

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

12. Applied rewrites 96.7%

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

Alternative 7: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 19.4

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 19.4

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

4. Applied rewrites 19.4%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-out N/A

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

   5. lower-neg.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 95.3

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

7. Applied rewrites 95.3%

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

Alternative 8: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

3. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.3

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

5. Applied rewrites 95.3%

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

6. Final simplification 95.3%

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

Alternative 9: 95.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Python

def code(a, b, c):
	return 1.0 / ((a / b) - (b / c))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 19.4

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 19.4

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

4. Applied rewrites 19.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 19.4

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 19.4

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 19.4

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

6. Applied rewrites 19.4%

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

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 95.1

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

9. Applied rewrites 95.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
10. Add Preprocessing

Alternative 10: 90.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   4. lower-neg.f64 89.4

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

5. Applied rewrites 89.4%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))