Cubic critical, wide range

Percentage Accurate: 17.8% → 99.7%

Time: 14.9s

Alternatives: 8

Speedup: 2.9×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{a \cdot 3}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (fma c (* a -3.0) 0.0) (* a 3.0))
  (+ b (sqrt (fma c (* a -3.0) (* b b))))))

C

double code(double a, double b, double c) {
	return (fma(c, (a * -3.0), 0.0) / (a * 3.0)) / (b + sqrt(fma(c, (a * -3.0), (b * b))));
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(c, Float64(a * -3.0), 0.0) / Float64(a * 3.0)) / Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(a * -3.0), $MachinePrecision] + 0.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.0%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. div-sub N/A

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

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

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

4. Applied egg-rr 17.7%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

   3. sub-div N/A

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

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

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

6. Applied egg-rr 17.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} \cdot 0.3333333333333333 - b \cdot 0.3333333333333333}{a}} \]

8. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{a \cdot 3}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}} \]
9. Add Preprocessing

Alternative 2: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma c (* a -3.0) 0.0)
  (* (* a 3.0) (+ b (sqrt (fma c (* a -3.0) (* b b)))))))

C

double code(double a, double b, double c) {
	return fma(c, (a * -3.0), 0.0) / ((a * 3.0) * (b + sqrt(fma(c, (a * -3.0), (b * b)))));
}

Julia

function code(a, b, c)
	return Float64(fma(c, Float64(a * -3.0), 0.0) / Float64(Float64(a * 3.0) * Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(a * -3.0), $MachinePrecision] + 0.0), $MachinePrecision] / N[(N[(a * 3.0), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.0%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. div-sub N/A

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

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

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

4. Applied egg-rr 17.7%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

   3. sub-div N/A

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

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

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

6. Applied egg-rr 17.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} \cdot 0.3333333333333333 - b \cdot 0.3333333333333333}{a}} \]

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \]
10. Add Preprocessing

Alternative 3: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, a, \frac{c}{b \cdot -2}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma (/ (* (* c c) -0.375) (* b (* b b))) a (/ c (* b -2.0))))

C

double code(double a, double b, double c) {
	return fma((((c * c) * -0.375) / (b * (b * b))), a, (c / (b * -2.0)));
}

Julia

function code(a, b, c)
	return fma(Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b))), a, Float64(c / Float64(b * -2.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(c / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. distribute-lft-in N/A

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

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

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

   9. associate-*r/ N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r* N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. associate-*l* N/A

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

5. Simplified 94.8%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{-0.5}{b}\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

   7. associate-*l* N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

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

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

   11. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

   16. *-commutative N/A

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

   17. associate-*l/ N/A

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

   18. associate-*r/ N/A

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

   19. clear-num N/A

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

   20. un-div-inv N/A

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

7. Applied egg-rr 95.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, a, \frac{c}{b \cdot -2}\right)} \]
8. Add Preprocessing

Alternative 4: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b))

C

double code(double a, double b, double c) {
	return fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in b around inf

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

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

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

5. Simplified 95.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
6. Add Preprocessing

Alternative 5: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (fma a (* -0.375 (/ c (* b (* b b)))) (/ -0.5 b))))

C

double code(double a, double b, double c) {
	return c * fma(a, (-0.375 * (c / (b * (b * b)))), (-0.5 / b));
}

Julia

function code(a, b, c)
	return Float64(c * fma(a, Float64(-0.375 * Float64(c / Float64(b * Float64(b * b)))), Float64(-0.5 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(a * N[(-0.375 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. distribute-lft-in N/A

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

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

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

   9. associate-*r/ N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r* N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. associate-*l* N/A

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

5. Simplified 94.8%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{-0.5}{b}\right)} \]

6. Final simplification 94.8%

   \[\leadsto c \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right) \]
7. Add Preprocessing

Alternative 6: 95.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{\mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ (fma a (* -0.375 (/ c (* b b))) -0.5) b)))

C

double code(double a, double b, double c) {
	return c * (fma(a, (-0.375 * (c / (b * b))), -0.5) / b);
}

Julia

function code(a, b, c)
	return Float64(c * Float64(fma(a, Float64(-0.375 * Float64(c / Float64(b * b))), -0.5) / b))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(a * N[(-0.375 * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. distribute-lft-in N/A

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

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

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

   9. associate-*r/ N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r* N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. associate-*l* N/A

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

5. Simplified 94.8%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{-0.5}{b}\right)} \]

6. Taylor expanded in b around inf

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

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

      \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b}} \]

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

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

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 94.8

       \[\leadsto c \cdot \frac{\mathsf{fma}\left(a, \frac{c}{\color{blue}{b \cdot b}} \cdot -0.375, -0.5\right)}{b} \]

8. Simplified 94.8%

   \[\leadsto c \cdot \color{blue}{\frac{\mathsf{fma}\left(a, \frac{c}{b \cdot b} \cdot -0.375, -0.5\right)}{b}} \]

9. Final simplification 94.8%

   \[\leadsto c \cdot \frac{\mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot b}, -0.5\right)}{b} \]
10. Add Preprocessing

Alternative 7: 90.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return -0.5 * (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 = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

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

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in b around inf

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

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

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

   2. /-lowering-/.f64 90.2

      \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

5. Simplified 90.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Alternative 8: 3.3% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 18.0%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. div-sub N/A

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

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

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

4. Applied egg-rr 17.7%

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

   1. sub-neg N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

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

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

6. Applied egg-rr 19.3%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.3

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

9. Simplified 3.3%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))