Cubic critical, medium range

Percentage Accurate: 31.3% → 99.6%

Time: 11.8s

Alternatives: 8

Speedup: 2.9×

• Could not converge on a ground truth

  1. a = 2.7437384088138098e-15
  2. b = 4455546869917.423
  3. c = 0.0002282595892415876

Specification

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

Math

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

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: 31.3% 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: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

4. Applied rewrites 31.9%

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

6. Applied rewrites 32.7%

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

8. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right) \cdot \frac{-3 \cdot a}{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. +-rgt-identity N/A

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

   8. clear-num N/A

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

   9. +-rgt-identity N/A

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

   10. lift-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

10. Applied rewrites 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot \frac{1}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ -1.0 (* (+ (sqrt (fma (* -3.0 c) a (* b b))) b) (/ 1.0 c))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

4. Applied rewrites 31.9%

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

6. Applied rewrites 32.7%

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in c around 0

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

    1. lower-/.f64 99.3

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

11. Applied rewrites 99.3%

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

12. Final simplification 99.3%

    \[\leadsto \frac{-1}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot \frac{1}{c}} \]
13. Add Preprocessing

Alternative 3: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.6666666666666666 \cdot b\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.3333333333333333
  (/ (fma (* (/ c b) a) 0.5 (* -0.6666666666666666 b)) c)))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / (fma(((c / b) * a), 0.5, (-0.6666666666666666 * b)) / c);
}

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / Float64(fma(Float64(Float64(c / b) * a), 0.5, Float64(-0.6666666666666666 * b)) / c))
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * 0.5 + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 33.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.5

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

4. Applied rewrites 33.5%

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

5. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 89.9

      \[\leadsto \frac{0.3333333333333333}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 0.5, \color{blue}{-0.6666666666666666 \cdot b}\right)}{c}} \]

7. Applied rewrites 89.9%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 0.5, -0.6666666666666666 \cdot b\right)}{c}}} \]

8. Final simplification 89.9%

   \[\leadsto \frac{0.3333333333333333}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.6666666666666666 \cdot b\right)}{c}} \]
9. Add Preprocessing

Alternative 4: 90.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (fma (/ a b) 1.5 (* -2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma((a / b), 1.5, (-2.0 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(Float64(a / b), 1.5, Float64(-2.0 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.5%

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

4. Applied rewrites 31.9%

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

6. Applied rewrites 32.7%

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. lower-/.f64 89.8

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \color{blue}{\frac{b}{c}} \cdot -2\right)} \]

11. Applied rewrites 89.8%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}} \]

12. Final simplification 89.8%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)} \]
13. Add Preprocessing

Alternative 5: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.3333333333333333 (fma (/ a b) 0.5 (* (/ b c) -0.6666666666666666))))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / fma((a / b), 0.5, ((b / c) * -0.6666666666666666));
}

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.6666666666666666)))
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 33.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.5

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

4. Applied rewrites 33.5%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 89.8

      \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \color{blue}{\frac{b}{c}} \cdot -0.6666666666666666\right)} \]

7. Applied rewrites 89.8%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}} \]
8. Add Preprocessing

Alternative 6: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5))))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / fma(-0.6666666666666666, (b / c), ((a / b) * 0.5));
}

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)))
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 33.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.5

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

4. Applied rewrites 33.5%

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

5. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. distribute-neg-frac N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 89.8

       \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, \frac{0.5}{b}, \color{blue}{\frac{-0.6666666666666666}{c}}\right) \cdot b} \]

7. Applied rewrites 89.8%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, \frac{0.5}{b}, \frac{-0.6666666666666666}{c}\right) \cdot b}} \]
8. Step-by-step derivation

9. Applied rewrites 89.7%

   \[\leadsto \frac{0.3333333333333333}{\frac{\mathsf{fma}\left(-0.6666666666666666, b \cdot b, c \cdot \left(0.5 \cdot a\right)\right)}{c \cdot \left(b \cdot b\right)} \cdot b} \]

10. Taylor expanded in a around 0

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

    1. lower-fma.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower-/.f64 89.8

       \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \color{blue}{\frac{a}{b}} \cdot 0.5\right)} \]

12. Applied rewrites 89.8%

    \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}} \]
13. Add Preprocessing

Alternative 7: 81.2% 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 33.5%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
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 79.8

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

5. Applied rewrites 79.8%

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

6. Final simplification 79.8%

   \[\leadsto -0.5 \cdot \frac{c}{b} \]
7. Add Preprocessing

Alternative 8: 81.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
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 79.8

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

5. Applied rewrites 79.8%

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

7. Applied rewrites 79.6%

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

8. Final simplification 79.6%

   \[\leadsto \frac{-0.5}{b} \cdot c \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))