Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, K

Percentage Accurate: 70.7% → 78.2%

Time: 19.5s

Alternatives: 12

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 78.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t\_2 \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - t\_1 \leq 10^{+201}:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(\cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \cos y, \left(-\sin y\right) \cdot \sin \left(t \cdot \left(z \cdot -0.3333333333333333\right)\right)\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2 \cdot \left(b \cdot \cos y\right), \sqrt{x}, a \cdot -0.3333333333333333\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (* 2.0 (sqrt x))))
   (if (<= (- (* t_2 (cos (- y (/ (* z t) 3.0)))) t_1) 1e+201)
     (-
      (*
       t_2
       (fma
        (cos (* z (* t 0.3333333333333333)))
        (cos y)
        (* (- (sin y)) (sin (* t (* z -0.3333333333333333))))))
      t_1)
     (/ (fma (* 2.0 (* b (cos y))) (sqrt x) (* a -0.3333333333333333)) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = 2.0 * sqrt(x);
	double tmp;
	if (((t_2 * cos((y - ((z * t) / 3.0)))) - t_1) <= 1e+201) {
		tmp = (t_2 * fma(cos((z * (t * 0.3333333333333333))), cos(y), (-sin(y) * sin((t * (z * -0.3333333333333333)))))) - t_1;
	} else {
		tmp = fma((2.0 * (b * cos(y))), sqrt(x), (a * -0.3333333333333333)) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (Float64(Float64(t_2 * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - t_1) <= 1e+201)
		tmp = Float64(Float64(t_2 * fma(cos(Float64(z * Float64(t * 0.3333333333333333))), cos(y), Float64(Float64(-sin(y)) * sin(Float64(t * Float64(z * -0.3333333333333333)))))) - t_1);
	else
		tmp = Float64(fma(Float64(2.0 * Float64(b * cos(y))), sqrt(x), Float64(a * -0.3333333333333333)) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], 1e+201], N[(N[(t$95$2 * N[(N[Cos[N[(z * N[(t * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[((-N[Sin[y], $MachinePrecision]) * N[Sin[N[(t * N[(z * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(2.0 * N[(b * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision] + N[(a * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64)))) < 1.00000000000000004e201

   1. Initial program 81.9%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{z \cdot t}{3}\right)} - \frac{a}{b \cdot 3} \]

      2. lift--.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(y - \frac{z \cdot t}{3}\right)} - \frac{a}{b \cdot 3} \]

      3. sub-neg N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(y + \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right)} - \frac{a}{b \cdot 3} \]

      4. cos-sum N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) - \sin y \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right)} - \frac{a}{b \cdot 3} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right)} - \frac{a}{b \cdot 3} \]

      6. cos-neg N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(\frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      7. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\cos \left(\frac{z \cdot t}{3}\right) \cdot \cos y} + \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\frac{z \cdot t}{3}\right), \cos y, \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right)} - \frac{a}{b \cdot 3} \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \left(\frac{z \cdot t}{3}\right)}, \cos y, \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      10. lift-/.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\frac{z \cdot t}{3}\right)}, \cos y, \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \left(\frac{\color{blue}{z \cdot t}}{3}\right), \cos y, \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      12. associate-/l* N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \color{blue}{\left(z \cdot \frac{t}{3}\right)}, \cos y, \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \color{blue}{\left(z \cdot \frac{t}{3}\right)}, \cos y, \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      14. div-inv N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \left(z \cdot \color{blue}{\left(t \cdot \frac{1}{3}\right)}\right), \cos y, \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \left(z \cdot \color{blue}{\left(t \cdot \frac{1}{3}\right)}\right), \cos y, \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      16. metadata-eval N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \left(z \cdot \left(t \cdot \color{blue}{\frac{1}{3}}\right)\right), \cos y, \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      17. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \left(z \cdot \left(t \cdot \frac{1}{3}\right)\right), \color{blue}{\cos y}, \left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3} \]

      18. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \left(z \cdot \left(t \cdot \frac{1}{3}\right)\right), \cos y, \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right) \cdot \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)}\right) - \frac{a}{b \cdot 3} \]

   4. Applied rewrites 83.5%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \cos y, \left(-\sin y\right) \cdot \sin \left(t \cdot \left(z \cdot -0.3333333333333333\right)\right)\right)} - \frac{a}{b \cdot 3} \]

   if 1.00000000000000004e201 < (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64))))

   1. Initial program 45.4%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 75.1

         \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333\right) \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b} \cdot -0.3333333333333333\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{-1}{3} \cdot a + 2 \cdot \left(\left(b \cdot \cos y\right) \cdot \sqrt{x}\right)}{\color{blue}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.2%

      \[\leadsto \frac{\mathsf{fma}\left(2 \cdot \left(b \cdot \cos y\right), \sqrt{x}, a \cdot -0.3333333333333333\right)}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{3 \cdot b} \leq 10^{+201}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \cos y, \left(-\sin y\right) \cdot \sin \left(t \cdot \left(z \cdot -0.3333333333333333\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2 \cdot \left(b \cdot \cos y\right), \sqrt{x}, a \cdot -0.3333333333333333\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(t \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9666:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sqrt{x}, \mathsf{fma}\left(\cos t\_1, \cos y, \sin y \cdot \sin t\_1\right), \frac{a \cdot -0.3333333333333333}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b \cdot -3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (* t 0.3333333333333333))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 0.9666)
     (fma
      (* 2.0 (sqrt x))
      (fma (cos t_1) (cos y) (* (sin y) (sin t_1)))
      (/ (* a -0.3333333333333333) b))
     (fma 2.0 (* (sqrt x) (cos y)) (/ a (* b -3.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (t * 0.3333333333333333);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 0.9666) {
		tmp = fma((2.0 * sqrt(x)), fma(cos(t_1), cos(y), (sin(y) * sin(t_1))), ((a * -0.3333333333333333) / b));
	} else {
		tmp = fma(2.0, (sqrt(x) * cos(y)), (a / (b * -3.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(t * 0.3333333333333333))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 0.9666)
		tmp = fma(Float64(2.0 * sqrt(x)), fma(cos(t_1), cos(y), Float64(sin(y) * sin(t_1))), Float64(Float64(a * -0.3333333333333333) / b));
	else
		tmp = fma(2.0, Float64(sqrt(x) * cos(y)), Float64(a / Float64(b * -3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(t * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.9666], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.3333333333333333), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 0.966600000000000015

   1. Initial program 77.1%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      6. cos-diff N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) \cdot \cos \left(\frac{z \cdot t}{3}\right)} + \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)\right) \]

   4. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y, \cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \mathsf{fma}\left(2 \cdot \sqrt{x}, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \frac{a}{b \cdot -3}\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b} + \left(2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) + \frac{-1}{3} \cdot \frac{a}{b}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} + 2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) + \frac{-1}{3} \cdot \frac{a}{b} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}\right) + \frac{-1}{3} \cdot \frac{a}{b} \]

      4. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} + \frac{-1}{3} \cdot \frac{a}{b} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right), \frac{-1}{3} \cdot \frac{a}{b}\right)} \]

   7. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \mathsf{fma}\left(\cos \left(\left(0.3333333333333333 \cdot t\right) \cdot z\right), \cos y, \sin y \cdot \sin \left(\left(0.3333333333333333 \cdot t\right) \cdot z\right)\right), \frac{a \cdot -0.3333333333333333}{b}\right)} \]

   if 0.966600000000000015 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 68.9%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 85.1

         \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333\right) \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b} \cdot -0.3333333333333333\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.2%

      \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b \cdot -3}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9666:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sqrt{x}, \mathsf{fma}\left(\cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \cos y, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right), \frac{a \cdot -0.3333333333333333}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b \cdot -3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, 1, \frac{a}{b \cdot -3}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-61}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{b}, a, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (* 2.0 (sqrt x))))
   (if (<= t_1 -1e-17)
     (fma t_2 1.0 (/ a (* b -3.0)))
     (if (<= t_1 1e-61)
       (* 2.0 (* (sqrt x) (cos y)))
       (fma (/ -0.3333333333333333 b) a t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = 2.0 * sqrt(x);
	double tmp;
	if (t_1 <= -1e-17) {
		tmp = fma(t_2, 1.0, (a / (b * -3.0)));
	} else if (t_1 <= 1e-61) {
		tmp = 2.0 * (sqrt(x) * cos(y));
	} else {
		tmp = fma((-0.3333333333333333 / b), a, t_2);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (t_1 <= -1e-17)
		tmp = fma(t_2, 1.0, Float64(a / Float64(b * -3.0)));
	elseif (t_1 <= 1e-61)
		tmp = Float64(2.0 * Float64(sqrt(x) * cos(y)));
	else
		tmp = fma(Float64(-0.3333333333333333 / b), a, t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-17], N[(t$95$2 * 1.0 + N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-61], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / b), $MachinePrecision] * a + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.00000000000000007e-17

   1. Initial program 85.9%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.4

         \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333\right) \]

   5. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b} \cdot -0.3333333333333333\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.5%

      \[\leadsto \mathsf{fma}\left(\frac{-0.3333333333333333}{b}, \color{blue}{a}, \sqrt{x} \cdot \left(2 \cdot \cos y\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 96.5%

      \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \color{blue}{\cos y}, \frac{a}{b \cdot -3}\right) \]

   10. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, 1, \frac{a}{b \cdot -3}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 96.5%

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

   if -1.00000000000000007e-17 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 1e-61

   1. Initial program 60.2%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 62.4

         \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b} \cdot -0.3333333333333333\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \cos y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \cos y\right)} \]

   if 1e-61 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 82.6%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 90.3

         \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333\right) \]

   5. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b} \cdot -0.3333333333333333\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(\frac{-0.3333333333333333}{b}, \color{blue}{a}, \sqrt{x} \cdot \left(2 \cdot \cos y\right)\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{3}}{b}, a, 2 \cdot \sqrt{x}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 87.9%

       \[\leadsto \mathsf{fma}\left(\frac{-0.3333333333333333}{b}, a, 2 \cdot \sqrt{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -1 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sqrt{x}, 1, \frac{a}{b \cdot -3}\right)\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 10^{-61}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{b}, a, 2 \cdot \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-0.3333333333333333}{b}, a, \sqrt{x} \cdot \left(2 \cdot \cos y\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (/ -0.3333333333333333 b) a (* (sqrt x) (* 2.0 (cos y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((-0.3333333333333333 / b), a, (sqrt(x) * (2.0 * cos(y))));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(-0.3333333333333333 / b), a, Float64(sqrt(x) * Float64(2.0 * cos(y))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(-0.3333333333333333 / b), $MachinePrecision] * a + N[(N[Sqrt[x], $MachinePrecision] * N[(2.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 79.8

      \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333\right) \]

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b} \cdot -0.3333333333333333\right)} \]
6. Step-by-step derivation

7. Applied rewrites 79.8%

   \[\leadsto \mathsf{fma}\left(\frac{-0.3333333333333333}{b}, \color{blue}{a}, \sqrt{x} \cdot \left(2 \cdot \cos y\right)\right) \]
8. Add Preprocessing

Alternative 5: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a}{b \cdot -3}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (* 2.0 (sqrt x)) (cos y) (/ a (* b -3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((2.0 * sqrt(x)), cos(y), (a / (b * -3.0)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(2.0 * sqrt(x)), cos(y), Float64(a / Float64(b * -3.0)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 79.8

      \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333\right) \]

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b} \cdot -0.3333333333333333\right)} \]
6. Step-by-step derivation

7. Applied rewrites 79.8%

   \[\leadsto \mathsf{fma}\left(\frac{-0.3333333333333333}{b}, \color{blue}{a}, \sqrt{x} \cdot \left(2 \cdot \cos y\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 79.8%

   \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \color{blue}{\cos y}, \frac{a}{b \cdot -3}\right) \]
10. Add Preprocessing

Alternative 6: 77.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, -0.3333333333333333 \cdot \frac{a}{b}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma 2.0 (* (sqrt x) (cos y)) (* -0.3333333333333333 (/ a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(2.0, (sqrt(x) * cos(y)), (-0.3333333333333333 * (a / b)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(2.0, Float64(sqrt(x) * cos(y)), Float64(-0.3333333333333333 * Float64(a / b)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 79.8

      \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333\right) \]

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b} \cdot -0.3333333333333333\right)} \]

6. Final simplification 79.8%

   \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, -0.3333333333333333 \cdot \frac{a}{b}\right) \]
7. Add Preprocessing

Alternative 7: 60.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := \frac{\frac{a}{-3}}{b}\\ \mathbf{if}\;t\_1 \leq -3 \cdot 10^{-124}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-62}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (/ (/ a -3.0) b)))
   (if (<= t_1 -3e-124) t_2 (if (<= t_1 2e-62) (* 2.0 (sqrt x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = (a / -3.0) / b;
	double tmp;
	if (t_1 <= -3e-124) {
		tmp = t_2;
	} else if (t_1 <= 2e-62) {
		tmp = 2.0 * sqrt(x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a / (3.0d0 * b)
    t_2 = (a / (-3.0d0)) / b
    if (t_1 <= (-3d-124)) then
        tmp = t_2
    else if (t_1 <= 2d-62) then
        tmp = 2.0d0 * sqrt(x)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = (a / -3.0) / b;
	double tmp;
	if (t_1 <= -3e-124) {
		tmp = t_2;
	} else if (t_1 <= 2e-62) {
		tmp = 2.0 * Math.sqrt(x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = (a / -3.0) / b
	tmp = 0
	if t_1 <= -3e-124:
		tmp = t_2
	elif t_1 <= 2e-62:
		tmp = 2.0 * math.sqrt(x)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(Float64(a / -3.0) / b)
	tmp = 0.0
	if (t_1 <= -3e-124)
		tmp = t_2;
	elseif (t_1 <= 2e-62)
		tmp = Float64(2.0 * sqrt(x));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	t_2 = (a / -3.0) / b;
	tmp = 0.0;
	if (t_1 <= -3e-124)
		tmp = t_2;
	elseif (t_1 <= 2e-62)
		tmp = 2.0 * sqrt(x);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / -3.0), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$1, -3e-124], t$95$2, If[LessEqual[t$95$1, 2e-62], N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -3e-124 or 2.0000000000000001e-62 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 80.9%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 79.7

         \[\leadsto \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333 \]

   5. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\frac{a}{b} \cdot -0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.8%

      \[\leadsto \frac{\frac{a}{-3}}{\color{blue}{b}} \]

   if -3e-124 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.0000000000000001e-62

   1. Initial program 60.0%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      6. cos-diff N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) \cdot \cos \left(\frac{z \cdot t}{3}\right)} + \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)\right) \]

   4. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y, \cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \mathsf{fma}\left(2 \cdot \sqrt{x}, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \frac{a}{b \cdot -3}\right)\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

      4. distribute-lft-out N/A

         \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto x \cdot \left(2 \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto x \cdot \left(2 \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) \]

   7. Applied rewrites 60.3%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \mathsf{fma}\left(\cos \left(\left(0.3333333333333333 \cdot t\right) \cdot z\right), \cos y, \sin y \cdot \sin \left(\left(0.3333333333333333 \cdot t\right) \cdot z\right)\right)\right)\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{2}{3} \cdot \left(\left(t \cdot \left(z \cdot \sin y\right)\right) \cdot \sqrt{x}\right) + \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.9%

       \[\leadsto \mathsf{fma}\left(0.6666666666666666, \color{blue}{\sqrt{x} \cdot \left(\left(t \cdot z\right) \cdot \sin y\right)}, \left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) \]

   11. Taylor expanded in y around 0

       \[\leadsto 2 \cdot \sqrt{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 30.7%

       \[\leadsto 2 \cdot \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -3 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{a}{-3}}{b}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-62}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{-3}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_1 \leq -3 \cdot 10^{-124}:\\ \;\;\;\;\frac{a}{b \cdot -3}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-62}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (<= t_1 -3e-124)
     (/ a (* b -3.0))
     (if (<= t_1 2e-62) (* 2.0 (sqrt x)) (* a (/ -0.3333333333333333 b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if (t_1 <= -3e-124) {
		tmp = a / (b * -3.0);
	} else if (t_1 <= 2e-62) {
		tmp = 2.0 * sqrt(x);
	} else {
		tmp = a * (-0.3333333333333333 / b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (3.0d0 * b)
    if (t_1 <= (-3d-124)) then
        tmp = a / (b * (-3.0d0))
    else if (t_1 <= 2d-62) then
        tmp = 2.0d0 * sqrt(x)
    else
        tmp = a * ((-0.3333333333333333d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if (t_1 <= -3e-124) {
		tmp = a / (b * -3.0);
	} else if (t_1 <= 2e-62) {
		tmp = 2.0 * Math.sqrt(x);
	} else {
		tmp = a * (-0.3333333333333333 / b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if t_1 <= -3e-124:
		tmp = a / (b * -3.0)
	elif t_1 <= 2e-62:
		tmp = 2.0 * math.sqrt(x)
	else:
		tmp = a * (-0.3333333333333333 / b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (t_1 <= -3e-124)
		tmp = Float64(a / Float64(b * -3.0));
	elseif (t_1 <= 2e-62)
		tmp = Float64(2.0 * sqrt(x));
	else
		tmp = Float64(a * Float64(-0.3333333333333333 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	tmp = 0.0;
	if (t_1 <= -3e-124)
		tmp = a / (b * -3.0);
	elseif (t_1 <= 2e-62)
		tmp = 2.0 * sqrt(x);
	else
		tmp = a * (-0.3333333333333333 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -3e-124], N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-62], N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(a * N[(-0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -3e-124

   1. Initial program 78.8%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 75.1

         \[\leadsto \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333 \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{a}{b} \cdot -0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.2%

      \[\leadsto \frac{a}{\color{blue}{b \cdot -3}} \]

   if -3e-124 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.0000000000000001e-62

   1. Initial program 60.0%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      6. cos-diff N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) \cdot \cos \left(\frac{z \cdot t}{3}\right)} + \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)\right) \]

   4. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y, \cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \mathsf{fma}\left(2 \cdot \sqrt{x}, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \frac{a}{b \cdot -3}\right)\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

      4. distribute-lft-out N/A

         \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto x \cdot \left(2 \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto x \cdot \left(2 \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) \]

   7. Applied rewrites 60.3%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \mathsf{fma}\left(\cos \left(\left(0.3333333333333333 \cdot t\right) \cdot z\right), \cos y, \sin y \cdot \sin \left(\left(0.3333333333333333 \cdot t\right) \cdot z\right)\right)\right)\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{2}{3} \cdot \left(\left(t \cdot \left(z \cdot \sin y\right)\right) \cdot \sqrt{x}\right) + \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.9%

       \[\leadsto \mathsf{fma}\left(0.6666666666666666, \color{blue}{\sqrt{x} \cdot \left(\left(t \cdot z\right) \cdot \sin y\right)}, \left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) \]

   11. Taylor expanded in y around 0

       \[\leadsto 2 \cdot \sqrt{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 30.7%

       \[\leadsto 2 \cdot \sqrt{x} \]

   if 2.0000000000000001e-62 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 82.8%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.7

         \[\leadsto \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333 \]

   5. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\frac{a}{b} \cdot -0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.7%

      \[\leadsto \frac{-0.3333333333333333}{b} \cdot \color{blue}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -3 \cdot 10^{-124}:\\ \;\;\;\;\frac{a}{b \cdot -3}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-62}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{if}\;t\_1 \leq -3 \cdot 10^{-124}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-62}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (* a (/ -0.3333333333333333 b))))
   (if (<= t_1 -3e-124) t_2 (if (<= t_1 2e-62) (* 2.0 (sqrt x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = a * (-0.3333333333333333 / b);
	double tmp;
	if (t_1 <= -3e-124) {
		tmp = t_2;
	} else if (t_1 <= 2e-62) {
		tmp = 2.0 * sqrt(x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a / (3.0d0 * b)
    t_2 = a * ((-0.3333333333333333d0) / b)
    if (t_1 <= (-3d-124)) then
        tmp = t_2
    else if (t_1 <= 2d-62) then
        tmp = 2.0d0 * sqrt(x)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = a * (-0.3333333333333333 / b);
	double tmp;
	if (t_1 <= -3e-124) {
		tmp = t_2;
	} else if (t_1 <= 2e-62) {
		tmp = 2.0 * Math.sqrt(x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = a * (-0.3333333333333333 / b)
	tmp = 0
	if t_1 <= -3e-124:
		tmp = t_2
	elif t_1 <= 2e-62:
		tmp = 2.0 * math.sqrt(x)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(a * Float64(-0.3333333333333333 / b))
	tmp = 0.0
	if (t_1 <= -3e-124)
		tmp = t_2;
	elseif (t_1 <= 2e-62)
		tmp = Float64(2.0 * sqrt(x));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	t_2 = a * (-0.3333333333333333 / b);
	tmp = 0.0;
	if (t_1 <= -3e-124)
		tmp = t_2;
	elseif (t_1 <= 2e-62)
		tmp = 2.0 * sqrt(x);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(-0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -3e-124], t$95$2, If[LessEqual[t$95$1, 2e-62], N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -3e-124 or 2.0000000000000001e-62 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 80.9%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 79.7

         \[\leadsto \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333 \]

   5. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\frac{a}{b} \cdot -0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.7%

      \[\leadsto \frac{-0.3333333333333333}{b} \cdot \color{blue}{a} \]

   if -3e-124 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.0000000000000001e-62

   1. Initial program 60.0%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      6. cos-diff N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) \cdot \cos \left(\frac{z \cdot t}{3}\right)} + \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)\right) \]

   4. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y, \cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \mathsf{fma}\left(2 \cdot \sqrt{x}, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \frac{a}{b \cdot -3}\right)\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

      4. distribute-lft-out N/A

         \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto x \cdot \left(2 \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto x \cdot \left(2 \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) \]

   7. Applied rewrites 60.3%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \mathsf{fma}\left(\cos \left(\left(0.3333333333333333 \cdot t\right) \cdot z\right), \cos y, \sin y \cdot \sin \left(\left(0.3333333333333333 \cdot t\right) \cdot z\right)\right)\right)\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{2}{3} \cdot \left(\left(t \cdot \left(z \cdot \sin y\right)\right) \cdot \sqrt{x}\right) + \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.9%

       \[\leadsto \mathsf{fma}\left(0.6666666666666666, \color{blue}{\sqrt{x} \cdot \left(\left(t \cdot z\right) \cdot \sin y\right)}, \left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) \]

   11. Taylor expanded in y around 0

       \[\leadsto 2 \cdot \sqrt{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 30.7%

       \[\leadsto 2 \cdot \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -3 \cdot 10^{-124}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-62}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-0.3333333333333333}{b}, a, 2 \cdot \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (/ -0.3333333333333333 b) a (* 2.0 (sqrt x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((-0.3333333333333333 / b), a, (2.0 * sqrt(x)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(-0.3333333333333333 / b), a, Float64(2.0 * sqrt(x)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(-0.3333333333333333 / b), $MachinePrecision] * a + N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 79.8

      \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333\right) \]

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b} \cdot -0.3333333333333333\right)} \]
6. Step-by-step derivation

7. Applied rewrites 79.8%

   \[\leadsto \mathsf{fma}\left(\frac{-0.3333333333333333}{b}, \color{blue}{a}, \sqrt{x} \cdot \left(2 \cdot \cos y\right)\right) \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{3}}{b}, a, 2 \cdot \sqrt{x}\right) \]
9. Step-by-step derivation

10. Applied rewrites 66.6%

    \[\leadsto \mathsf{fma}\left(\frac{-0.3333333333333333}{b}, a, 2 \cdot \sqrt{x}\right) \]
11. Add Preprocessing

Alternative 11: 66.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, \sqrt{x}, \frac{a \cdot -0.3333333333333333}{b}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma 2.0 (sqrt x) (/ (* a -0.3333333333333333) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(2.0, sqrt(x), ((a * -0.3333333333333333) / b));
}

Julia

function code(x, y, z, t, a, b)
	return fma(2.0, sqrt(x), Float64(Float64(a * -0.3333333333333333) / b))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(2.0 * N[Sqrt[x], $MachinePrecision] + N[(N[(a * -0.3333333333333333), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 79.8

      \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \color{blue}{\frac{a}{b}} \cdot -0.3333333333333333\right) \]

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{a}{b} \cdot -0.3333333333333333\right)} \]

6. Taylor expanded in y around 0

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

8. Applied rewrites 66.6%

   \[\leadsto \mathsf{fma}\left(2, \color{blue}{\sqrt{x}}, \frac{a \cdot -0.3333333333333333}{b}\right) \]
9. Add Preprocessing

Alternative 12: 17.9% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* 2.0 (sqrt x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 2.0 * sqrt(x);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 2.0d0 * sqrt(x)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return 2.0 * Math.sqrt(x);
}

Python

def code(x, y, z, t, a, b):
	return 2.0 * math.sqrt(x)

Julia

function code(x, y, z, t, a, b)
	return Float64(2.0 * sqrt(x))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = 2.0 * sqrt(x);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   4. lift-cos.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   5. lift--.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(y - \frac{z \cdot t}{3}\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   6. cos-diff N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   7. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)\right)} \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) \cdot \cos \left(\frac{z \cdot t}{3}\right)} + \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)\right) \]

4. Applied rewrites 75.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y, \cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \mathsf{fma}\left(2 \cdot \sqrt{x}, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \frac{a}{b \cdot -3}\right)\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

   2. distribute-lft-out N/A

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sqrt{\frac{1}{x}} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)} \]

   4. distribute-lft-out N/A

      \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto x \cdot \left(2 \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) \]

   7. lower-/.f64 N/A

      \[\leadsto x \cdot \left(2 \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) \]

7. Applied rewrites 28.3%

   \[\leadsto \color{blue}{x \cdot \left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \mathsf{fma}\left(\cos \left(\left(0.3333333333333333 \cdot t\right) \cdot z\right), \cos y, \sin y \cdot \sin \left(\left(0.3333333333333333 \cdot t\right) \cdot z\right)\right)\right)\right)} \]

8. Taylor expanded in t around 0

   \[\leadsto \frac{2}{3} \cdot \left(\left(t \cdot \left(z \cdot \sin y\right)\right) \cdot \sqrt{x}\right) + \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 28.4%

    \[\leadsto \mathsf{fma}\left(0.6666666666666666, \color{blue}{\sqrt{x} \cdot \left(\left(t \cdot z\right) \cdot \sin y\right)}, \left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) \]

11. Taylor expanded in y around 0

    \[\leadsto 2 \cdot \sqrt{x} \]
12. Step-by-step derivation

13. Applied rewrites 15.0%

    \[\leadsto 2 \cdot \sqrt{x} \]
14. Add Preprocessing

Developer Target 1: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{0.3333333333333333}{z}}{t}\\ t_2 := \frac{\frac{a}{3}}{b}\\ t_3 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;t\_3 \cdot \cos \left(\frac{1}{y} - t\_1\right) - t\_2\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - t\_1\right) \cdot t\_3 - \frac{\frac{a}{b}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ 0.3333333333333333 z) t))
        (t_2 (/ (/ a 3.0) b))
        (t_3 (* 2.0 (sqrt x))))
   (if (< z -1.3793337487235141e+129)
     (- (* t_3 (cos (- (/ 1.0 y) t_1))) t_2)
     (if (< z 3.516290613555987e+106)
       (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) t_2)
       (- (* (cos (- y t_1)) t_3) (/ (/ a b) 3.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (0.3333333333333333d0 / z) / t
    t_2 = (a / 3.0d0) / b
    t_3 = 2.0d0 * sqrt(x)
    if (z < (-1.3793337487235141d+129)) then
        tmp = (t_3 * cos(((1.0d0 / y) - t_1))) - t_2
    else if (z < 3.516290613555987d+106) then
        tmp = ((sqrt(x) * 2.0d0) * cos((y - ((t / 3.0d0) * z)))) - t_2
    else
        tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * Math.sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * Math.cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((Math.sqrt(x) * 2.0) * Math.cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (Math.cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (0.3333333333333333 / z) / t
	t_2 = (a / 3.0) / b
	t_3 = 2.0 * math.sqrt(x)
	tmp = 0
	if z < -1.3793337487235141e+129:
		tmp = (t_3 * math.cos(((1.0 / y) - t_1))) - t_2
	elif z < 3.516290613555987e+106:
		tmp = ((math.sqrt(x) * 2.0) * math.cos((y - ((t / 3.0) * z)))) - t_2
	else:
		tmp = (math.cos((y - t_1)) * t_3) - ((a / b) / 3.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(0.3333333333333333 / z) / t)
	t_2 = Float64(Float64(a / 3.0) / b)
	t_3 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (z < -1.3793337487235141e+129)
		tmp = Float64(Float64(t_3 * cos(Float64(Float64(1.0 / y) - t_1))) - t_2);
	elseif (z < 3.516290613555987e+106)
		tmp = Float64(Float64(Float64(sqrt(x) * 2.0) * cos(Float64(y - Float64(Float64(t / 3.0) * z)))) - t_2);
	else
		tmp = Float64(Float64(cos(Float64(y - t_1)) * t_3) - Float64(Float64(a / b) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (0.3333333333333333 / z) / t;
	t_2 = (a / 3.0) / b;
	t_3 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (z < -1.3793337487235141e+129)
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	elseif (z < 3.516290613555987e+106)
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	else
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.3793337487235141e+129], N[(N[(t$95$3 * N[Cos[N[(N[(1.0 / y), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[z, 3.516290613555987e+106], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(y - N[(N[(t / 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(N[Cos[N[(y - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2024233 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -1379333748723514100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 3333333333333333/10000000000000000 z) t)))) (/ (/ a 3) b)) (if (< z 35162906135559870000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 3333333333333333/10000000000000000 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3)))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))