Quadratic roots, wide range

Percentage Accurate: 18.3% → 97.6%

Time: 14.6s

Alternatives: 8

Speedup: 3.3×

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: 18.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \mathsf{fma}\left(b, b, 0\right)\\ \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(t\_0 \cdot t\_0\right)}, -0.25, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{b \cdot \left(b \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)\right)}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (fma b b 0.0))))
   (-
    (*
     (* a a)
     (fma
      (/ (* a (* (* c c) (* (* c c) 20.0))) (* b (* t_0 t_0)))
      -0.25
      (* (* c (* c c)) (/ -2.0 (* b (* b (fma b (fma b b 0.0) 0.0)))))))
    (/ (fma (* c c) (/ a (fma b b 0.0)) c) b))))

C

double code(double a, double b, double c) {
	double t_0 = b * fma(b, b, 0.0);
	return ((a * a) * fma(((a * ((c * c) * ((c * c) * 20.0))) / (b * (t_0 * t_0))), -0.25, ((c * (c * c)) * (-2.0 / (b * (b * fma(b, fma(b, b, 0.0), 0.0))))))) - (fma((c * c), (a / fma(b, b, 0.0)), c) / b);
}

Julia

function code(a, b, c)
	t_0 = Float64(b * fma(b, b, 0.0))
	return Float64(Float64(Float64(a * a) * fma(Float64(Float64(a * Float64(Float64(c * c) * Float64(Float64(c * c) * 20.0))) / Float64(b * Float64(t_0 * t_0))), -0.25, Float64(Float64(c * Float64(c * c)) * Float64(-2.0 / Float64(b * Float64(b * fma(b, fma(b, b, 0.0), 0.0))))))) - Float64(fma(Float64(c * c), Float64(a / fma(b, b, 0.0)), c) / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(a * a), $MachinePrecision] * N[(N[(N[(a * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25 + N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(-2.0 / N[(b * N[(b * N[(b * N[(b * b + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 17.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. 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. Simplified 98.1%

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

7. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right), 0\right)}, -0.25, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b}} \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \color{blue}{\left(\left(b \cdot b + 0\right) \cdot \left(\left(b \cdot b + 0\right) \cdot \left(b \cdot b + 0\right) + 0\right)\right)}}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   2. +-rgt-identity N/A

      \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot b + 0\right) \cdot \color{blue}{\left(\left(b \cdot b + 0\right) \cdot \left(b \cdot b + 0\right)\right)}\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   3. cube-unmult N/A

      \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \color{blue}{{\left(b \cdot b + 0\right)}^{3}}}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   4. +-rgt-identity N/A

      \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot {\color{blue}{\left(b \cdot b\right)}}^{3}}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   5. unpow-prod-down N/A

      \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)}}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \color{blue}{\left({b}^{3} \cdot {b}^{3}\right)}}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   7. cube-mult N/A

      \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{3}\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   8. +-rgt-identity N/A

      \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \color{blue}{\left(b \cdot b + 0\right)}\right) \cdot {b}^{3}\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\color{blue}{\left(b \cdot \left(b \cdot b + 0\right)\right)} \cdot {b}^{3}\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right) \cdot {b}^{3}\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   11. cube-mult N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   12. +-rgt-identity N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \color{blue}{\left(b \cdot b + 0\right)}\right)\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b + 0\right)\right)}\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

   14. accelerator-lowering-fma.f64 98.1

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)\right)}, -0.25, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

9. Applied egg-rr 98.1%

   \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \color{blue}{\left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}}, -0.25, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]
10. Step-by-step derivation

    1. +-rgt-identity N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\color{blue}{\left(\left(b \cdot b + 0\right) \cdot \left(b \cdot b + 0\right)\right)} \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

    2. +-rgt-identity N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\left(\left(b \cdot b + 0\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

    3. associate-*r* N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\color{blue}{\left(\left(\left(b \cdot b + 0\right) \cdot b\right) \cdot b\right)} \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

    4. *-commutative N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\left(\color{blue}{\left(b \cdot \left(b \cdot b + 0\right)\right)} \cdot b\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\color{blue}{\left(\left(b \cdot \left(b \cdot b + 0\right)\right) \cdot b\right)} \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

    6. +-rgt-identity N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\left(\left(b \cdot \color{blue}{\left(\left(b \cdot b + 0\right) + 0\right)}\right) \cdot b\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

    7. distribute-lft-in N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\left(\color{blue}{\left(b \cdot \left(b \cdot b + 0\right) + b \cdot 0\right)} \cdot b\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

    8. mul0-rgt N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\left(\left(b \cdot \left(b \cdot b + 0\right) + \color{blue}{0}\right) \cdot b\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

    9. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, \frac{-1}{4}, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\left(\color{blue}{\mathsf{fma}\left(b, b \cdot b + 0, 0\right)} \cdot b\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

    10. accelerator-lowering-fma.f64 98.1

        \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, -0.25, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\left(\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}, 0\right) \cdot b\right) \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

11. Applied egg-rr 98.1%

    \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, -0.25, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{\color{blue}{\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right) \cdot b\right)} \cdot b}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]

12. Final simplification 98.1%

    \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)}{b \cdot \left(\left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)}, -0.25, \left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-2}{b \cdot \left(b \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)\right)}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\mathsf{fma}\left(b, b, 0\right)}, c\right)}{b} \]
13. Add Preprocessing

Alternative 2: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(a \cdot \left(a \cdot -2\right)\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)} - \frac{c \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (-
    (*
     (* a (* a -2.0))
     (/ (* c (* c c)) (fma (fma b b 0.0) (fma b b 0.0) 0.0)))
    (/ (* c (* a c)) (fma b b 0.0)))
   c)
  b))

C

double code(double a, double b, double c) {
	return ((((a * (a * -2.0)) * ((c * (c * c)) / fma(fma(b, b, 0.0), fma(b, b, 0.0), 0.0))) - ((c * (a * c)) / fma(b, b, 0.0))) - c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(a * Float64(a * -2.0)) * Float64(Float64(c * Float64(c * c)) / fma(fma(b, b, 0.0), fma(b, b, 0.0), 0.0))) - Float64(Float64(c * Float64(a * c)) / fma(b, b, 0.0))) - c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a * N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + 0.0), $MachinePrecision] * N[(b * b + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 17.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. 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. /-lowering-/.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. Simplified 97.2%

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

   1. associate--r+ N/A

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

   2. --lowering--.f64 N/A

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

7. Applied egg-rr 97.2%

   \[\leadsto \frac{\color{blue}{\left(\left(a \cdot \left(a \cdot -2\right)\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)} - \frac{c \cdot \left(c \cdot a\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}}{b} \]

8. Final simplification 97.2%

   \[\leadsto \frac{\left(\left(a \cdot \left(a \cdot -2\right)\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)} - \frac{c \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]
9. Add Preprocessing

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot \left(a \cdot \left(a \cdot c\right)\right)\right)}{b \cdot b}, 0 - a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)} - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (/
   (fma -2.0 (/ (* c (* c (* a (* a c)))) (* b b)) (- 0.0 (* a (* c c))))
   (* b (* b b)))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (fma(-2.0, ((c * (c * (a * (a * c)))) / (b * b)), (0.0 - (a * (c * c)))) / (b * (b * b))) - (c / b);
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(-2.0, Float64(Float64(c * Float64(c * Float64(a * Float64(a * c)))) / Float64(b * b)), Float64(0.0 - Float64(a * Float64(c * c)))) / Float64(b * Float64(b * b))) - Float64(c / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-2.0 * N[(N[(c * N[(c * N[(a * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.0 - N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. 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. /-lowering-/.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. Simplified 97.2%

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

   1. associate--r+ N/A

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

   2. div-sub N/A

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

   3. --lowering--.f64 N/A

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

7. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)} - \frac{c \cdot \left(c \cdot a\right)}{\mathsf{fma}\left(b, b, 0\right)}}{b} - \frac{c}{b}} \]

8. Taylor expanded in b around inf

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

   1. /-lowering-/.f64 N/A

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

10. Simplified 97.2%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{\left(\left(\left(a \cdot c\right) \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}, 0 - a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}} - \frac{c}{b} \]

11. Final simplification 97.2%

    \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot \left(a \cdot \left(a \cdot c\right)\right)\right)}{b \cdot b}, 0 - a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)} - \frac{c}{b} \]
12. Add Preprocessing

Alternative 4: 96.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. sub-neg N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. *-lowering-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. *-lowering-*.f64 N/A

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

   13. metadata-eval 17.6

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

4. Applied egg-rr 17.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. +-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. +-rgt-identity N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, \color{blue}{b \cdot b + 0}\right)} - b}} \]

   12. accelerator-lowering-fma.f64 17.6

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}} \]

6. Applied egg-rr 17.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}} \]

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. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. --lowering--.f64 N/A

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

9. Simplified 97.0%

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

Alternative 5: 95.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(0.0 - N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. 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. neg-lowering-neg.f64 N/A

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

   5. /-lowering-/.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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.f64 95.5

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

5. Simplified 95.5%

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

6. Final simplification 95.5%

   \[\leadsto 0 - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]
7. Add Preprocessing

Alternative 6: 94.9% 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 17.6%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. sub-neg N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. *-lowering-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. *-lowering-*.f64 N/A

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

   13. metadata-eval 17.6

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

4. Applied egg-rr 17.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. +-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. +-rgt-identity N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, \color{blue}{b \cdot b + 0}\right)} - b}} \]

   12. accelerator-lowering-fma.f64 17.6

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, \color{blue}{\mathsf{fma}\left(b, b, 0\right)}\right)} - b}} \]

6. Applied egg-rr 17.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)} - b}}} \]

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. --lowering--.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. /-lowering-/.f64 95.3

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

9. Simplified 95.3%

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

Alternative 7: 90.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{c}{b} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 - (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 = 0.0d0 - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. 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. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   4. /-lowering-/.f64 90.8

      \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

5. Simplified 90.8%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. /-lowering-/.f64 90.8

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

7. Applied egg-rr 90.8%

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

8. Final simplification 90.8%

   \[\leadsto 0 - \frac{c}{b} \]
9. Add Preprocessing

Alternative 8: 1.7% accurate, 4.2× 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 / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. 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. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   4. /-lowering-/.f64 90.8

      \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

5. Simplified 90.8%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. /-lowering-/.f64 90.8

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

7. Applied egg-rr 90.8%

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

   1. div-inv N/A

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

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

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

   3. *-lowering-*.f64 N/A

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

   4. distribute-neg-frac N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 90.5

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

9. Applied egg-rr 90.5%

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

    1. clear-num N/A

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

    2. un-div-inv N/A

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

    3. clear-num N/A

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

    4. frac-2neg N/A

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

    5. metadata-eval N/A

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

    6. sub0-neg N/A

       \[\leadsto \frac{c}{\frac{1}{\frac{1}{\color{blue}{0 - b}}}} \]

    7. inv-pow N/A

       \[\leadsto \frac{c}{\frac{1}{\color{blue}{{\left(0 - b\right)}^{-1}}}} \]

    8. pow-flip N/A

       \[\leadsto \frac{c}{\color{blue}{{\left(0 - b\right)}^{\left(\mathsf{neg}\left(-1\right)\right)}}} \]

    9. metadata-eval N/A

       \[\leadsto \frac{c}{{\left(0 - b\right)}^{\color{blue}{1}}} \]

    10. metadata-eval N/A

        \[\leadsto \frac{c}{{\left(0 - b\right)}^{\color{blue}{\left(3 - 2\right)}}} \]

    11. pow-div N/A

        \[\leadsto \frac{c}{\color{blue}{\frac{{\left(0 - b\right)}^{3}}{{\left(0 - b\right)}^{2}}}} \]

    12. metadata-eval N/A

        \[\leadsto \frac{c}{\frac{{\left(0 - b\right)}^{\color{blue}{\left(2 \cdot \frac{3}{2}\right)}}}{{\left(0 - b\right)}^{2}}} \]

    13. metadata-eval N/A

        \[\leadsto \frac{c}{\frac{{\left(0 - b\right)}^{\left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot 3\right)}\right)}}{{\left(0 - b\right)}^{2}}} \]

    14. pow-pow N/A

        \[\leadsto \frac{c}{\frac{\color{blue}{{\left({\left(0 - b\right)}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}}{{\left(0 - b\right)}^{2}}} \]

    15. pow2 N/A

        \[\leadsto \frac{c}{\frac{{\color{blue}{\left(\left(0 - b\right) \cdot \left(0 - b\right)\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{{\left(0 - b\right)}^{2}}} \]

    16. sub0-neg N/A

        \[\leadsto \frac{c}{\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(0 - b\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{{\left(0 - b\right)}^{2}}} \]

    17. sub0-neg N/A

        \[\leadsto \frac{c}{\frac{{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{{\left(0 - b\right)}^{2}}} \]

    18. sqr-neg N/A

        \[\leadsto \frac{c}{\frac{{\color{blue}{\left(b \cdot b\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{{\left(0 - b\right)}^{2}}} \]

    19. pow2 N/A

        \[\leadsto \frac{c}{\frac{{\color{blue}{\left({b}^{2}\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{{\left(0 - b\right)}^{2}}} \]

    20. pow-pow N/A

        \[\leadsto \frac{c}{\frac{\color{blue}{{b}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}}{{\left(0 - b\right)}^{2}}} \]

    21. metadata-eval N/A

        \[\leadsto \frac{c}{\frac{{b}^{\left(2 \cdot \color{blue}{\frac{3}{2}}\right)}}{{\left(0 - b\right)}^{2}}} \]

    22. metadata-eval N/A

        \[\leadsto \frac{c}{\frac{{b}^{\color{blue}{3}}}{{\left(0 - b\right)}^{2}}} \]

    23. pow2 N/A

        \[\leadsto \frac{c}{\frac{{b}^{3}}{\color{blue}{\left(0 - b\right) \cdot \left(0 - b\right)}}} \]

    24. sub0-neg N/A

        \[\leadsto \frac{c}{\frac{{b}^{3}}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(0 - b\right)}} \]

    25. sub0-neg N/A

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

    26. sqr-neg N/A

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

    27. pow2 N/A

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

    28. pow-div N/A

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

11. Applied egg-rr 1.7%

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

Reproduce

herbie shell --seed 2024197 
(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)))