Cubic critical, narrow range

Percentage Accurate: 56.4% → 99.3%

Time: 17.9s

Alternatives: 8

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 56.4% 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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c * a) / (a * ((0.0d0 - b) - sqrt(((b * b) + (c * (a * (-3.0d0)))))))
end function

Java

public static double code(double a, double b, double c) {
	return (c * a) / (a * ((0.0 - b) - Math.sqrt(((b * b) + (c * (a * -3.0))))));
}

Python

def code(a, b, c):
	return (c * a) / (a * ((0.0 - b) - math.sqrt(((b * b) + (c * (a * -3.0))))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (c * a) / (a * ((0.0 - b) - sqrt(((b * b) + (c * (a * -3.0))))));
end

Wolfram

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

Derivation

1. Initial program 56.4%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{3 \cdot a}\right), \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{3}}{a}\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{3}\right), a\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, a\right), \left(\left(\mathsf{neg}\left(\color{blue}{b}\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, a\right), \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right) \]

   8. unsub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, a\right), \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - \color{blue}{b}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, a\right), \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{b}\right)\right) \]

4. Applied egg-rr 56.4%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b\right)} \]
5. Step-by-step derivation

   1. *-commutative N/A

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

   2. flip-- N/A

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

   3. frac-times N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b \cdot b\right) \cdot \frac{1}{3}\right), \color{blue}{\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} + b\right) \cdot a\right)}\right) \]

6. Applied egg-rr 58.0%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right), a\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(a \cdot c\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)}, a\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(c \cdot a\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)}\right), a\right)\right) \]

   3. *-lowering-*.f64 99.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)}\right), a\right)\right) \]

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 2: 90.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \frac{c}{t\_0}\\ t_2 := c \cdot t\_1\\ a \cdot \left(\left(c \cdot -0.375\right) \cdot t\_1 + a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot t\_0} + \frac{\left(a \cdot -0.16666666666666666\right) \cdot \left(t\_2 \cdot t\_2\right)}{\frac{b}{6.328125}}\right)\right) - \frac{c \cdot 0.5}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (/ c t_0)) (t_2 (* c t_1)))
   (-
    (*
     a
     (+
      (* (* c -0.375) t_1)
      (*
       a
       (+
        (/ (* c (* (* c c) -0.5625)) (* (* b b) t_0))
        (/ (* (* a -0.16666666666666666) (* t_2 t_2)) (/ b 6.328125))))))
    (/ (* c 0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = c / t_0;
	double t_2 = c * t_1;
	return (a * (((c * -0.375) * t_1) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + (((a * -0.16666666666666666) * (t_2 * t_2)) / (b / 6.328125)))))) - ((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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = b * (b * b)
    t_1 = c / t_0
    t_2 = c * t_1
    code = (a * (((c * (-0.375d0)) * t_1) + (a * (((c * ((c * c) * (-0.5625d0))) / ((b * b) * t_0)) + (((a * (-0.16666666666666666d0)) * (t_2 * t_2)) / (b / 6.328125d0)))))) - ((c * 0.5d0) / b)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = c / t_0;
	double t_2 = c * t_1;
	return (a * (((c * -0.375) * t_1) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + (((a * -0.16666666666666666) * (t_2 * t_2)) / (b / 6.328125)))))) - ((c * 0.5) / b);
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = c / t_0
	t_2 = c * t_1
	return (a * (((c * -0.375) * t_1) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + (((a * -0.16666666666666666) * (t_2 * t_2)) / (b / 6.328125)))))) - ((c * 0.5) / b)

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(c / t_0)
	t_2 = Float64(c * t_1)
	return Float64(Float64(a * Float64(Float64(Float64(c * -0.375) * t_1) + Float64(a * Float64(Float64(Float64(c * Float64(Float64(c * c) * -0.5625)) / Float64(Float64(b * b) * t_0)) + Float64(Float64(Float64(a * -0.16666666666666666) * Float64(t_2 * t_2)) / Float64(b / 6.328125)))))) - Float64(Float64(c * 0.5) / b))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = c / t_0;
	t_2 = c * t_1;
	tmp = (a * (((c * -0.375) * t_1) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + (((a * -0.16666666666666666) * (t_2 * t_2)) / (b / 6.328125)))))) - ((c * 0.5) / b);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(c * t$95$1), $MachinePrecision]}, N[(N[(a * N[(N[(N[(c * -0.375), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(a * N[(N[(N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * -0.16666666666666666), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(b / 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in a around 0

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

5. Simplified 91.3%

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

7. Applied egg-rr 91.3%

   \[\leadsto \color{blue}{a \cdot \left(\frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot -0.16666666666666666\right) \cdot \left(\left(c \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right) \cdot \left(c \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right)\right)}{\frac{b}{6.328125}}\right)\right) - \frac{c \cdot 0.5}{b}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{\left(c \cdot \frac{-3}{8}\right) \cdot c}{b \cdot \left(b \cdot b\right)}\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(b, \frac{405}{64}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(c \cdot \frac{-3}{8}\right) \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(b, \frac{405}{64}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(c \cdot \frac{-3}{8}\right), \left(\frac{c}{b \cdot \left(b \cdot b\right)}\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(b, \frac{405}{64}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \frac{-3}{8}\right), \left(\frac{c}{b \cdot \left(b \cdot b\right)}\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(b, \frac{405}{64}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \frac{-3}{8}\right), \mathsf{/.f64}\left(c, \left(b \cdot \left(b \cdot b\right)\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(b, \frac{405}{64}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \frac{-3}{8}\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \left(b \cdot b\right)\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(b, \frac{405}{64}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\right)\right) \]

   7. *-lowering-*.f64 91.3%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \frac{-3}{8}\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{6}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(b, \frac{405}{64}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\right)\right) \]

9. Applied egg-rr 91.3%

   \[\leadsto a \cdot \left(\color{blue}{\left(c \cdot -0.375\right) \cdot \frac{c}{b \cdot \left(b \cdot b\right)}} + a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot -0.16666666666666666\right) \cdot \left(\left(c \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right) \cdot \left(c \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right)\right)}{\frac{b}{6.328125}}\right)\right) - \frac{c \cdot 0.5}{b} \]
10. Add Preprocessing

Alternative 3: 87.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} + \frac{\left(c \cdot c\right) \cdot \left(\frac{-0.5625 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{b \cdot b} + a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (/ (* c -0.5) b)
  (/
   (* (* c c) (+ (/ (* -0.5625 (* c (* a a))) (* b b)) (* a -0.375)))
   (* b (* b b)))))

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (((c * c) * (((-0.5625 * (c * (a * a))) / (b * b)) + (a * -0.375))) / (b * (b * 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) + (((c * c) * ((((-0.5625d0) * (c * (a * a))) / (b * b)) + (a * (-0.375d0)))) / (b * (b * b)))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (((c * c) * (((-0.5625 * (c * (a * a))) / (b * b)) + (a * -0.375))) / (b * (b * b)));
}

Python

def code(a, b, c):
	return ((c * -0.5) / b) + (((c * c) * (((-0.5625 * (c * (a * a))) / (b * b)) + (a * -0.375))) / (b * (b * b)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * -0.5) / b) + Float64(Float64(Float64(c * c) * Float64(Float64(Float64(-0.5625 * Float64(c * Float64(a * a))) / Float64(b * b)) + Float64(a * -0.375))) / Float64(b * Float64(b * b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) / b) + (((c * c) * (((-0.5625 * (c * (a * a))) / (b * b)) + (a * -0.375))) / (b * (b * b)));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(N[(N[(-0.5625 * N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in a around 0

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

5. Simplified 91.3%

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

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{3}}\right)}\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)\right), \color{blue}{\left({b}^{3}\right)}\right)\right) \]

8. Simplified 88.0%

   \[\leadsto \frac{c \cdot -0.5}{b} + \color{blue}{\frac{\left(-0.375 \cdot a\right) \cdot \left(c \cdot c\right) + \frac{-0.5625 \cdot \left(a \cdot a\right)}{b} \cdot \frac{c \cdot \left(c \cdot c\right)}{b}}{b \cdot \left(b \cdot b\right)}} \]

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\color{blue}{\left({c}^{2} \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a\right)\right)}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({c}^{2}\right), \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a\right)\right), \mathsf{*.f64}\left(\color{blue}{b}, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c \cdot c\right), \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

    5. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right)}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \left({a}^{2} \cdot c\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \left(c \cdot {a}^{2}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(c, \left({a}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(c, \left(a \cdot a\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, a\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, a\right)\right)\right), \left(b \cdot b\right)\right), \left(\frac{-3}{8} \cdot a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-3}{8} \cdot a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

    14. *-lowering-*.f64 88.0%

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, a\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

11. Simplified 88.0%

    \[\leadsto \frac{c \cdot -0.5}{b} + \frac{\color{blue}{\left(c \cdot c\right) \cdot \left(\frac{-0.5625 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{b \cdot b} + -0.375 \cdot a\right)}}{b \cdot \left(b \cdot b\right)} \]

12. Final simplification 88.0%

    \[\leadsto \frac{c \cdot -0.5}{b} + \frac{\left(c \cdot c\right) \cdot \left(\frac{-0.5625 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{b \cdot b} + a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)} \]
13. Add Preprocessing

Alternative 4: 80.7% accurate, 6.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + ((-0.375 * (c * (c * a))) / (b * (b * 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) + (((-0.375d0) * (c * (c * a))) / (b * (b * b)))
end function

Java

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

Python

def code(a, b, c):
	return ((c * -0.5) / b) + ((-0.375 * (c * (c * a))) / (b * (b * b)))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) / b) + ((-0.375 * (c * (c * a))) / (b * (b * b)));
end

Wolfram

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

Derivation

1. Initial program 56.4%

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

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{c}{b}\right), \color{blue}{\left(a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot c}{b}\right), \left(\color{blue}{a} \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), b\right), \left(\color{blue}{a} \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right), \left(a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \left(a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot \color{blue}{\frac{-3}{8}}\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \left(\left(a \cdot \frac{{c}^{2}}{{b}^{3}}\right) \cdot \color{blue}{\frac{-3}{8}}\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \left(\frac{a \cdot {c}^{2}}{{b}^{3}} \cdot \frac{-3}{8}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \left(\frac{-3}{8} \cdot \color{blue}{\frac{a \cdot {c}^{2}}{{b}^{3}}}\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \left(\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{\color{blue}{{b}^{3}}}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{/.f64}\left(\left(\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)\right), \color{blue}{\left({b}^{3}\right)}\right)\right) \]

5. Simplified 81.6%

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

Alternative 5: 80.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5 + \frac{a \cdot -0.375}{b} \cdot \frac{c \cdot c}{b}}{b} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * -0.5) + (((a * -0.375) / b) * ((c * c) / b))) / 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)) + (((a * (-0.375d0)) / b) * ((c * c) / b))) / b
end function

Java

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

Python

def code(a, b, c):
	return ((c * -0.5) + (((a * -0.375) / b) * ((c * c) / b))) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) + (((a * -0.375) / b) * ((c * c) / b))) / b;
end

Wolfram

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

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in a around 0

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

5. Simplified 91.3%

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

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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right), \color{blue}{b}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot c\right), \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \left(\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right)\right), b\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \left(\frac{\left(\frac{-3}{8} \cdot a\right) \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \left(\frac{\left(\frac{-3}{8} \cdot a\right) \cdot {c}^{2}}{b \cdot b}\right)\right), b\right) \]

   7. times-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \left(\frac{\frac{-3}{8} \cdot a}{b} \cdot \frac{{c}^{2}}{b}\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{*.f64}\left(\left(\frac{\frac{-3}{8} \cdot a}{b}\right), \left(\frac{{c}^{2}}{b}\right)\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-3}{8} \cdot a\right), b\right), \left(\frac{{c}^{2}}{b}\right)\right)\right), b\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, a\right), b\right), \left(\frac{{c}^{2}}{b}\right)\right)\right), b\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, a\right), b\right), \mathsf{/.f64}\left(\left({c}^{2}\right), b\right)\right)\right), b\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, a\right), b\right), \mathsf{/.f64}\left(\left(c \cdot c\right), b\right)\right)\right), b\right) \]

   13. *-lowering-*.f64 81.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, a\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, c\right), b\right)\right)\right), b\right) \]

8. Simplified 81.6%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + \frac{-0.375 \cdot a}{b} \cdot \frac{c \cdot c}{b}}{b}} \]

9. Final simplification 81.6%

   \[\leadsto \frac{c \cdot -0.5 + \frac{a \cdot -0.375}{b} \cdot \frac{c \cdot c}{b}}{b} \]
10. Add Preprocessing

Alternative 6: 80.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((-0.5 / b) + (-0.375 * (a * (((c / b) / b) / 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) + ((-0.375d0) * (a * (((c / b) / b) / b))))
end function

Java

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

Python

def code(a, b, c):
	return c * ((-0.5 / b) + (-0.375 * (a * (((c / b) / b) / b))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.5 / b) + (-0.375 * (a * (((c / b) / b) / b))));
end

Wolfram

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

Derivation

1. Initial program 56.4%

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

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)}\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) + \color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right), \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)}\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{b}\right)\right), \left(\left(\color{blue}{\frac{-3}{8}} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b}\right)\right), \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

   11. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b}\right), \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \left(\left(\color{blue}{\frac{-3}{8}} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right)\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c\right)\right)\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{\color{blue}{{b}^{3}}}\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{\color{blue}{b}}^{3}}\right)\right)\right) \]

   17. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{-3}{8} \cdot \color{blue}{\frac{a \cdot c}{{b}^{3}}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{*.f64}\left(\frac{-3}{8}, \color{blue}{\left(\frac{a \cdot c}{{b}^{3}}\right)}\right)\right)\right) \]

   19. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left(a \cdot \color{blue}{\frac{c}{{b}^{3}}}\right)\right)\right)\right) \]

5. Simplified 81.5%

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

Alternative 7: 63.5% accurate, 23.2× speedup

Math

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

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(Float64(c * -0.5) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. associate-*r/ N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

   4. *-lowering-*.f64 63.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

5. Simplified 63.4%

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

Alternative 8: 63.4% accurate, 23.2× 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 56.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. associate-*r/ N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

   4. *-lowering-*.f64 63.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

5. Simplified 63.4%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \color{blue}{c}\right) \]

   4. /-lowering-/.f64 63.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), c\right) \]

7. Applied egg-rr 63.3%

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

8. Final simplification 63.3%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))