Cubic critical, wide range

Percentage Accurate: 17.9% → 99.4%

Time: 8.0s

Alternatives: 7

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: 17.9% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 20.3

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

4. Applied rewrites 20.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. lift-*.f64 N/A

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

   11. fp-cancel-sub-sign-inv N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*l* N/A

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

6. Applied rewrites 21.2%

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

   1. lift-fma.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. flip-+ N/A

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

   6. lower-/.f64 N/A

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

8. Applied rewrites 20.8%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

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

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. +-inverses 99.4

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 20.3

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

4. Applied rewrites 20.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. lift-*.f64 N/A

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

   11. fp-cancel-sub-sign-inv N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*l* N/A

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

6. Applied rewrites 21.2%

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

   1. lift-fma.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. flip-+ N/A

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

   6. lower-/.f64 N/A

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

8. Applied rewrites 20.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-inverses N/A

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

   11. lower-*.f64 99.1

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

10. Applied rewrites 99.1%

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

11. Final simplification 99.1%

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

Alternative 3: 95.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

5. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 94.8

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

8. Applied rewrites 94.8%

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

Alternative 4: 95.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 94.8

       \[\leadsto \mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{{b}^{3}}, \color{blue}{\frac{c}{b}} \cdot -0.5\right) \]

5. Applied rewrites 94.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{{b}^{3}}, \frac{c}{b} \cdot -0.5\right)} \]
6. Step-by-step derivation

7. Applied rewrites 94.8%

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

Alternative 5: 95.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 94.8

       \[\leadsto \mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{{b}^{3}}, \color{blue}{\frac{c}{b}} \cdot -0.5\right) \]

5. Applied rewrites 94.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot -0.375, c \cdot \frac{c}{{b}^{3}}, \frac{c}{b} \cdot -0.5\right)} \]
6. Step-by-step derivation

7. Applied rewrites 94.8%

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

9. Applied rewrites 94.4%

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

Alternative 6: 90.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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) * (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 89.1

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

5. Applied rewrites 89.1%

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

Alternative 7: 90.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 89.1

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

5. Applied rewrites 89.1%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 88.7%

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

Reproduce

herbie shell --seed 2024339 
(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)))