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

Percentage Accurate: 70.5% → 78.0%

Time: 19.7s

Alternatives: 12

Speedup: 1.0×

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: 70.5% 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: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9996:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \cos y \cdot \cos \left(t \cdot \frac{z}{3}\right), 2 \cdot \left(\left(\sqrt{x} \cdot \sin y\right) \cdot \sin \left(\frac{t}{\frac{3}{z}}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \frac{\frac{a}{b}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 2.0 (sqrt x))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 0.9996)
     (-
      (fma
       t_1
       (* (cos y) (cos (* t (/ z 3.0))))
       (* 2.0 (* (* (sqrt x) (sin y)) (sin (/ t (/ 3.0 z))))))
      (/ a (* 3.0 b)))
     (- t_1 (/ (/ a b) 3.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * sqrt(x);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 0.9996) {
		tmp = fma(t_1, (cos(y) * cos((t * (z / 3.0)))), (2.0 * ((sqrt(x) * sin(y)) * sin((t / (3.0 / z)))))) - (a / (3.0 * b));
	} else {
		tmp = t_1 - ((a / b) / 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 0.9996)
		tmp = Float64(fma(t_1, Float64(cos(y) * cos(Float64(t * Float64(z / 3.0)))), Float64(2.0 * Float64(Float64(sqrt(x) * sin(y)) * sin(Float64(t / Float64(3.0 / z)))))) - Float64(a / Float64(3.0 * b)));
	else
		tmp = Float64(t_1 - Float64(Float64(a / b) / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.9996], N[(N[(t$95$1 * N[(N[Cos[y], $MachinePrecision] * N[Cos[N[(t * N[(z / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[Sqrt[x], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(t / N[(3.0 / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 0.99960000000000004

   1. Initial program 70.5%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right), \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\left(2 \cdot \sqrt{x}\right), \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(\frac{z \cdot t}{3}\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(\frac{t \cdot z}{3}\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(t \cdot \frac{z}{3}\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{z}{3}\right)\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \sqrt{x}\right), \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   4. Applied egg-rr 72.5%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \left(2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(t \cdot \frac{z}{3}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \left(\left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(t \cdot \frac{z}{3}\right)\right)\right) \cdot 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(t \cdot \frac{z}{3}\right)\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\sqrt{x} \cdot \sin y\right) \cdot \sin \left(t \cdot \frac{z}{3}\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x} \cdot \sin y\right), \sin \left(t \cdot \frac{z}{3}\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \sin y\right), \sin \left(t \cdot \frac{z}{3}\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \sin y\right), \sin \left(t \cdot \frac{z}{3}\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sin.f64}\left(y\right)\right), \sin \left(t \cdot \frac{z}{3}\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sin.f64}\left(y\right)\right), \sin \left(\frac{t \cdot z}{3}\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sin.f64}\left(y\right)\right), \sin \left(\frac{z \cdot t}{3}\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sin.f64}\left(y\right)\right), \mathsf{sin.f64}\left(\left(\frac{z \cdot t}{3}\right)\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sin.f64}\left(y\right)\right), \mathsf{sin.f64}\left(\left(\frac{t \cdot z}{3}\right)\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sin.f64}\left(y\right)\right), \mathsf{sin.f64}\left(\left(t \cdot \frac{z}{3}\right)\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      14. clear-num N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sin.f64}\left(y\right)\right), \mathsf{sin.f64}\left(\left(t \cdot \frac{1}{\frac{3}{z}}\right)\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      15. un-div-inv N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sin.f64}\left(y\right)\right), \mathsf{sin.f64}\left(\left(\frac{t}{\frac{3}{z}}\right)\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sin.f64}\left(y\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(t, \left(\frac{3}{z}\right)\right)\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      17. /-lowering-/.f64 72.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sin.f64}\left(y\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(3, z\right)\right)\right)\right), 2\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   6. Applied egg-rr 72.8%

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

   if 0.99960000000000004 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 62.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\cos y}\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 80.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   5. Simplified 80.8%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \left(\frac{\frac{a}{b}}{\color{blue}{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(\left(\frac{a}{b}\right), \color{blue}{3}\right)\right) \]

      3. /-lowering-/.f64 80.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, b\right), 3\right)\right) \]

   7. Applied egg-rr 80.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(2 \cdot \sqrt{x}\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, b\right), 3\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(a, b\right)}, 3\right)\right) \]

      2. sqrt-lowering-sqrt.f64 81.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, \color{blue}{b}\right), 3\right)\right) \]

   10. Simplified 81.6%

       \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{\frac{a}{b}}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9996:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y \cdot \cos \left(t \cdot \frac{z}{3}\right), 2 \cdot \left(\left(\sqrt{x} \cdot \sin y\right) \cdot \sin \left(\frac{t}{\frac{3}{z}}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \left(z \cdot t\right) \cdot 0.3333333333333333\\ t_3 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot t\_1 - t\_3 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\frac{a}{b} \cdot -0.3333333333333333 + t\_1 \cdot \left(\cos y \cdot \cos t\_2 + \sin y \cdot \sin t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 2.0 (sqrt x)))
        (t_2 (* (* z t) 0.3333333333333333))
        (t_3 (/ a (* 3.0 b))))
   (if (<= (- (* (cos (- y (/ (* z t) 3.0))) t_1) t_3) 2e+149)
     (+
      (* (/ a b) -0.3333333333333333)
      (* t_1 (+ (* (cos y) (cos t_2)) (* (sin y) (sin t_2)))))
     (- t_1 t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * sqrt(x);
	double t_2 = (z * t) * 0.3333333333333333;
	double t_3 = a / (3.0 * b);
	double tmp;
	if (((cos((y - ((z * t) / 3.0))) * t_1) - t_3) <= 2e+149) {
		tmp = ((a / b) * -0.3333333333333333) + (t_1 * ((cos(y) * cos(t_2)) + (sin(y) * sin(t_2))));
	} else {
		tmp = t_1 - t_3;
	}
	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 = 2.0d0 * sqrt(x)
    t_2 = (z * t) * 0.3333333333333333d0
    t_3 = a / (3.0d0 * b)
    if (((cos((y - ((z * t) / 3.0d0))) * t_1) - t_3) <= 2d+149) then
        tmp = ((a / b) * (-0.3333333333333333d0)) + (t_1 * ((cos(y) * cos(t_2)) + (sin(y) * sin(t_2))))
    else
        tmp = t_1 - t_3
    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 = 2.0 * Math.sqrt(x);
	double t_2 = (z * t) * 0.3333333333333333;
	double t_3 = a / (3.0 * b);
	double tmp;
	if (((Math.cos((y - ((z * t) / 3.0))) * t_1) - t_3) <= 2e+149) {
		tmp = ((a / b) * -0.3333333333333333) + (t_1 * ((Math.cos(y) * Math.cos(t_2)) + (Math.sin(y) * Math.sin(t_2))));
	} else {
		tmp = t_1 - t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 2.0 * math.sqrt(x)
	t_2 = (z * t) * 0.3333333333333333
	t_3 = a / (3.0 * b)
	tmp = 0
	if ((math.cos((y - ((z * t) / 3.0))) * t_1) - t_3) <= 2e+149:
		tmp = ((a / b) * -0.3333333333333333) + (t_1 * ((math.cos(y) * math.cos(t_2)) + (math.sin(y) * math.sin(t_2))))
	else:
		tmp = t_1 - t_3
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(2.0 * sqrt(x))
	t_2 = Float64(Float64(z * t) * 0.3333333333333333)
	t_3 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (Float64(Float64(cos(Float64(y - Float64(Float64(z * t) / 3.0))) * t_1) - t_3) <= 2e+149)
		tmp = Float64(Float64(Float64(a / b) * -0.3333333333333333) + Float64(t_1 * Float64(Float64(cos(y) * cos(t_2)) + Float64(sin(y) * sin(t_2)))));
	else
		tmp = Float64(t_1 - t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 2.0 * sqrt(x);
	t_2 = (z * t) * 0.3333333333333333;
	t_3 = a / (3.0 * b);
	tmp = 0.0;
	if (((cos((y - ((z * t) / 3.0))) * t_1) - t_3) <= 2e+149)
		tmp = ((a / b) * -0.3333333333333333) + (t_1 * ((cos(y) * cos(t_2)) + (sin(y) * sin(t_2))));
	else
		tmp = t_1 - t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$3 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$3), $MachinePrecision], 2e+149], N[(N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(t$95$1 * N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t$95$3), $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)))) < 2.0000000000000001e149

   1. Initial program 74.1%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right), \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\left(2 \cdot \sqrt{x}\right), \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(\frac{z \cdot t}{3}\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(\frac{t \cdot z}{3}\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(t \cdot \frac{z}{3}\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{z}{3}\right)\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \sqrt{x}\right), \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{3} \cdot \frac{a}{b}\right), \color{blue}{\left(2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{a}{b} \cdot \frac{-1}{3}\right), \left(\color{blue}{2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)} + 2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{a}{b}\right), \frac{-1}{3}\right), \left(\color{blue}{2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)} + 2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(a, b\right), \frac{-1}{3}\right), \left(\color{blue}{2} \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) + 2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(a, b\right), \frac{-1}{3}\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \color{blue}{2} \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(a, b\right), \frac{-1}{3}\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}\right)\right) \]

   7. Simplified 75.8%

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

   if 2.0000000000000001e149 < (-.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 51.5%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\cos y}\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 74.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   5. Simplified 74.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(2 \cdot \sqrt{x}\right)}, \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 75.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   8. Simplified 75.5%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{3 \cdot b} \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\frac{a}{b} \cdot -0.3333333333333333 + \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333\right) + \sin y \cdot \sin \left(\left(z \cdot t\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot t\_1 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;t\_1 \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{3}\right) - \sin y \cdot \sin \left(\frac{z \cdot t}{-3}\right)\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 2.0 (sqrt x))) (t_2 (/ a (* 3.0 b))))
   (if (<= (* (cos (- y (/ (* z t) 3.0))) t_1) 2e+149)
     (-
      (*
       t_1
       (-
        (* (cos y) (cos (* t (/ z 3.0))))
        (* (sin y) (sin (/ (* z t) -3.0)))))
      t_2)
     (- t_1 t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * sqrt(x);
	double t_2 = a / (3.0 * b);
	double tmp;
	if ((cos((y - ((z * t) / 3.0))) * t_1) <= 2e+149) {
		tmp = (t_1 * ((cos(y) * cos((t * (z / 3.0)))) - (sin(y) * sin(((z * t) / -3.0))))) - t_2;
	} else {
		tmp = t_1 - 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 = 2.0d0 * sqrt(x)
    t_2 = a / (3.0d0 * b)
    if ((cos((y - ((z * t) / 3.0d0))) * t_1) <= 2d+149) then
        tmp = (t_1 * ((cos(y) * cos((t * (z / 3.0d0)))) - (sin(y) * sin(((z * t) / (-3.0d0)))))) - t_2
    else
        tmp = t_1 - 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 = 2.0 * Math.sqrt(x);
	double t_2 = a / (3.0 * b);
	double tmp;
	if ((Math.cos((y - ((z * t) / 3.0))) * t_1) <= 2e+149) {
		tmp = (t_1 * ((Math.cos(y) * Math.cos((t * (z / 3.0)))) - (Math.sin(y) * Math.sin(((z * t) / -3.0))))) - t_2;
	} else {
		tmp = t_1 - t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 2.0 * math.sqrt(x)
	t_2 = a / (3.0 * b)
	tmp = 0
	if (math.cos((y - ((z * t) / 3.0))) * t_1) <= 2e+149:
		tmp = (t_1 * ((math.cos(y) * math.cos((t * (z / 3.0)))) - (math.sin(y) * math.sin(((z * t) / -3.0))))) - t_2
	else:
		tmp = t_1 - t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(2.0 * sqrt(x))
	t_2 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (Float64(cos(Float64(y - Float64(Float64(z * t) / 3.0))) * t_1) <= 2e+149)
		tmp = Float64(Float64(t_1 * Float64(Float64(cos(y) * cos(Float64(t * Float64(z / 3.0)))) - Float64(sin(y) * sin(Float64(Float64(z * t) / -3.0))))) - t_2);
	else
		tmp = Float64(t_1 - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 2.0 * sqrt(x);
	t_2 = a / (3.0 * b);
	tmp = 0.0;
	if ((cos((y - ((z * t) / 3.0))) * t_1) <= 2e+149)
		tmp = (t_1 * ((cos(y) * cos((t * (z / 3.0)))) - (sin(y) * sin(((z * t) / -3.0))))) - t_2;
	else
		tmp = t_1 - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], 2e+149], N[(N[(t$95$1 * N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[N[(t * N[(z / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * N[Sin[N[(N[(z * t), $MachinePrecision] / -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(t$95$1 - t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) < 2.0000000000000001e149

   1. Initial program 78.3%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \cos \left(y + \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \cos \left(\left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) + y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      3. cos-sum N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \left(\cos \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \cos y - \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      4. cos-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \left(\cos \left(\frac{z \cdot t}{3}\right) \cdot \cos y - \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) - \sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(\frac{z \cdot t}{3}\right)\right)\right), \left(\sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(\frac{t \cdot z}{3}\right)\right)\right), \left(\sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(t \cdot \frac{z}{3}\right)\right)\right), \left(\sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{z}{3}\right)\right)\right)\right), \left(\sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \left(\sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right) \cdot \sin y\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\sin \left(\mathsf{neg}\left(\frac{z \cdot t}{3}\right)\right), \sin y\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   4. Applied egg-rr 79.9%

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

   if 2.0000000000000001e149 < (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))))

   1. Initial program 8.1%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\cos y}\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 51.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   5. Simplified 51.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(2 \cdot \sqrt{x}\right)}, \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 53.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   8. Simplified 53.7%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot \left(2 \cdot \sqrt{x}\right) \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{3}\right) - \sin y \cdot \sin \left(\frac{z \cdot t}{-3}\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\frac{3}{z}}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9996:\\ \;\;\;\;2 \cdot \left(\left(\sqrt{x} \cdot \sin y\right) \cdot \sin t\_1 + \left(\sqrt{x} \cdot \cos y\right) \cdot \cos t\_1\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (/ 3.0 z))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 0.9996)
     (-
      (*
       2.0
       (+
        (* (* (sqrt x) (sin y)) (sin t_1))
        (* (* (sqrt x) (cos y)) (cos t_1))))
      (/ a (* 3.0 b)))
     (- (* 2.0 (sqrt x)) (/ (/ a b) 3.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (3.0 / z);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 0.9996) {
		tmp = (2.0 * (((sqrt(x) * sin(y)) * sin(t_1)) + ((sqrt(x) * cos(y)) * cos(t_1)))) - (a / (3.0 * b));
	} else {
		tmp = (2.0 * sqrt(x)) - ((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) :: tmp
    t_1 = t / (3.0d0 / z)
    if (cos((y - ((z * t) / 3.0d0))) <= 0.9996d0) then
        tmp = (2.0d0 * (((sqrt(x) * sin(y)) * sin(t_1)) + ((sqrt(x) * cos(y)) * cos(t_1)))) - (a / (3.0d0 * b))
    else
        tmp = (2.0d0 * sqrt(x)) - ((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 = t / (3.0 / z);
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 0.9996) {
		tmp = (2.0 * (((Math.sqrt(x) * Math.sin(y)) * Math.sin(t_1)) + ((Math.sqrt(x) * Math.cos(y)) * Math.cos(t_1)))) - (a / (3.0 * b));
	} else {
		tmp = (2.0 * Math.sqrt(x)) - ((a / b) / 3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (3.0 / z)
	tmp = 0
	if math.cos((y - ((z * t) / 3.0))) <= 0.9996:
		tmp = (2.0 * (((math.sqrt(x) * math.sin(y)) * math.sin(t_1)) + ((math.sqrt(x) * math.cos(y)) * math.cos(t_1)))) - (a / (3.0 * b))
	else:
		tmp = (2.0 * math.sqrt(x)) - ((a / b) / 3.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(3.0 / z))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 0.9996)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(sqrt(x) * sin(y)) * sin(t_1)) + Float64(Float64(sqrt(x) * cos(y)) * cos(t_1)))) - Float64(a / Float64(3.0 * b)));
	else
		tmp = Float64(Float64(2.0 * sqrt(x)) - Float64(Float64(a / b) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (3.0 / z);
	tmp = 0.0;
	if (cos((y - ((z * t) / 3.0))) <= 0.9996)
		tmp = (2.0 * (((sqrt(x) * sin(y)) * sin(t_1)) + ((sqrt(x) * cos(y)) * cos(t_1)))) - (a / (3.0 * b));
	else
		tmp = (2.0 * sqrt(x)) - ((a / b) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(3.0 / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.9996], N[(N[(2.0 * N[(N[(N[(N[Sqrt[x], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 0.99960000000000004

   1. Initial program 70.5%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y \cdot \cos \left(\frac{z \cdot t}{3}\right), \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\left(2 \cdot \sqrt{x}\right), \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \cos \left(\frac{z \cdot t}{3}\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(\frac{z \cdot t}{3}\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(\frac{t \cdot z}{3}\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\left(t \cdot \frac{z}{3}\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{z}{3}\right)\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \sqrt{x}\right), \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(z, 3\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   4. Applied egg-rr 72.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(t \cdot \frac{z}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{3}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(t \cdot \frac{z}{3}\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{3}\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(t \cdot \frac{z}{3}\right)\right)\right) + 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(2 \cdot \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(t \cdot \frac{z}{3}\right)\right) + \sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(t \cdot \frac{z}{3}\right)\right) + \sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{3}\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\sqrt{x} \cdot \left(\sin y \cdot \sin \left(t \cdot \frac{z}{3}\right)\right)\right), \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{3}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   6. Applied egg-rr 72.4%

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

   if 0.99960000000000004 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 62.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\cos y}\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 80.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   5. Simplified 80.8%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \left(\frac{\frac{a}{b}}{\color{blue}{3}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(\left(\frac{a}{b}\right), \color{blue}{3}\right)\right) \]

      3. /-lowering-/.f64 80.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, b\right), 3\right)\right) \]

   7. Applied egg-rr 80.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(2 \cdot \sqrt{x}\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, b\right), 3\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(a, b\right)}, 3\right)\right) \]

      2. sqrt-lowering-sqrt.f64 81.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, \color{blue}{b}\right), 3\right)\right) \]

   10. Simplified 81.6%

       \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{\frac{a}{b}}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9996:\\ \;\;\;\;2 \cdot \left(\left(\sqrt{x} \cdot \sin y\right) \cdot \sin \left(\frac{t}{\frac{3}{z}}\right) + \left(\sqrt{x} \cdot \cos y\right) \cdot \cos \left(\frac{t}{\frac{3}{z}}\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-206}:\\ \;\;\;\;t\_2 + \frac{a \cdot -0.3333333333333333}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-41}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(t \cdot \left(z \cdot 0.3333333333333333\right) - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 - t\_1\\ \end{array} \end{array} \]

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 -2e-206)
     (+ t_2 (/ (* a -0.3333333333333333) b))
     (if (<= t_1 2e-41)
       (* 2.0 (* (sqrt x) (cos (- (* t (* z 0.3333333333333333)) y))))
       (- t_2 t_1)))))

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 <= -2e-206) {
		tmp = t_2 + ((a * -0.3333333333333333) / b);
	} else if (t_1 <= 2e-41) {
		tmp = 2.0 * (sqrt(x) * cos(((t * (z * 0.3333333333333333)) - y)));
	} else {
		tmp = t_2 - t_1;
	}
	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 = 2.0d0 * sqrt(x)
    if (t_1 <= (-2d-206)) then
        tmp = t_2 + ((a * (-0.3333333333333333d0)) / b)
    else if (t_1 <= 2d-41) then
        tmp = 2.0d0 * (sqrt(x) * cos(((t * (z * 0.3333333333333333d0)) - y)))
    else
        tmp = t_2 - t_1
    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 = 2.0 * Math.sqrt(x);
	double tmp;
	if (t_1 <= -2e-206) {
		tmp = t_2 + ((a * -0.3333333333333333) / b);
	} else if (t_1 <= 2e-41) {
		tmp = 2.0 * (Math.sqrt(x) * Math.cos(((t * (z * 0.3333333333333333)) - y)));
	} else {
		tmp = t_2 - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = 2.0 * math.sqrt(x)
	tmp = 0
	if t_1 <= -2e-206:
		tmp = t_2 + ((a * -0.3333333333333333) / b)
	elif t_1 <= 2e-41:
		tmp = 2.0 * (math.sqrt(x) * math.cos(((t * (z * 0.3333333333333333)) - y)))
	else:
		tmp = t_2 - t_1
	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 <= -2e-206)
		tmp = Float64(t_2 + Float64(Float64(a * -0.3333333333333333) / b));
	elseif (t_1 <= 2e-41)
		tmp = Float64(2.0 * Float64(sqrt(x) * cos(Float64(Float64(t * Float64(z * 0.3333333333333333)) - y))));
	else
		tmp = Float64(t_2 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	t_2 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (t_1 <= -2e-206)
		tmp = t_2 + ((a * -0.3333333333333333) / b);
	elseif (t_1 <= 2e-41)
		tmp = 2.0 * (sqrt(x) * cos(((t * (z * 0.3333333333333333)) - y)));
	else
		tmp = t_2 - t_1;
	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[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-206], N[(t$95$2 + N[(N[(a * -0.3333333333333333), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-41], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[N[(N[(t * N[(z * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2.00000000000000006e-206

   1. Initial program 67.2%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) + \frac{a}{b \cdot 3}}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right) \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3} \cdot \frac{a}{b \cdot 3}}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right) \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3} \cdot \frac{a}{b \cdot 3}}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) + \frac{a}{b \cdot 3}}}}\right)\right) \]

   4. Applied egg-rr 67.1%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(\sqrt{x} \cdot \cos y\right)\right)}, \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \sqrt{x}\right), \cos y\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \cos y\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \cos y\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 77.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]

   7. Simplified 77.2%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{a}{b}\right)\right)}\right) \]

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

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

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

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

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

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

      11. *-lowering-*.f64 70.4%

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

   10. Simplified 70.4%

       \[\leadsto \color{blue}{2 \cdot \sqrt{x} + \frac{-0.3333333333333333 \cdot a}{b}} \]

   if -2.00000000000000006e-206 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.00000000000000001e-41

   1. Initial program 61.8%

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

   3. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x} \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \cos \color{blue}{\left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)}\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \cos \left(y + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)\right) \]

      5. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot y\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right)\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \cos \left(-1 \cdot y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\left(-1 \cdot y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\left(\frac{1}{3} \cdot \left(t \cdot z\right) + -1 \cdot y\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\left(\frac{1}{3} \cdot \left(t \cdot z\right) + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\left(\frac{1}{3} \cdot \left(t \cdot z\right) - y\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{3} \cdot \left(t \cdot z\right)\right), y\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\left(\left(t \cdot z\right) \cdot \frac{1}{3}\right), y\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\left(t \cdot \left(z \cdot \frac{1}{3}\right)\right), y\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\left(t \cdot \left(z \cdot \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), y\right)\right)\right)\right) \]

      17. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\left(t \cdot \left(\mathsf{neg}\left(z \cdot \frac{-1}{3}\right)\right)\right), y\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\left(t \cdot \left(\mathsf{neg}\left(\frac{-1}{3} \cdot z\right)\right)\right), y\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\left(t \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot z\right)\right), y\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\left(t \cdot \left(\frac{1}{3} \cdot z\right)\right), y\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{1}{3} \cdot z\right)\right), y\right)\right)\right)\right) \]

      22. *-lowering-*.f64 60.7%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\frac{1}{3}, z\right)\right), y\right)\right)\right)\right) \]

   5. Simplified 60.7%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(t \cdot \left(0.3333333333333333 \cdot z\right) - y\right)\right)} \]

   if 2.00000000000000001e-41 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 78.3%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\cos y}\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 88.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   5. Simplified 88.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(2 \cdot \sqrt{x}\right)}, \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 84.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   8. Simplified 84.8%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-206}:\\ \;\;\;\;2 \cdot \sqrt{x} + \frac{a \cdot -0.3333333333333333}{b}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(t \cdot \left(z \cdot 0.3333333333333333\right) - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := 2 \cdot \sqrt{x}\\ t_3 := t\_2 - t\_1\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-20}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-41}:\\ \;\;\;\;t\_2 \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (* 2.0 (sqrt x))) (t_3 (- t_2 t_1)))
   (if (<= t_1 -4e-20) t_3 (if (<= t_1 2e-41) (* t_2 (cos y)) t_3))))

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 t_3 = t_2 - t_1;
	double tmp;
	if (t_1 <= -4e-20) {
		tmp = t_3;
	} else if (t_1 <= 2e-41) {
		tmp = t_2 * cos(y);
	} else {
		tmp = t_3;
	}
	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 = a / (3.0d0 * b)
    t_2 = 2.0d0 * sqrt(x)
    t_3 = t_2 - t_1
    if (t_1 <= (-4d-20)) then
        tmp = t_3
    else if (t_1 <= 2d-41) then
        tmp = t_2 * cos(y)
    else
        tmp = t_3
    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 = 2.0 * Math.sqrt(x);
	double t_3 = t_2 - t_1;
	double tmp;
	if (t_1 <= -4e-20) {
		tmp = t_3;
	} else if (t_1 <= 2e-41) {
		tmp = t_2 * Math.cos(y);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = 2.0 * math.sqrt(x)
	t_3 = t_2 - t_1
	tmp = 0
	if t_1 <= -4e-20:
		tmp = t_3
	elif t_1 <= 2e-41:
		tmp = t_2 * math.cos(y)
	else:
		tmp = t_3
	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))
	t_3 = Float64(t_2 - t_1)
	tmp = 0.0
	if (t_1 <= -4e-20)
		tmp = t_3;
	elseif (t_1 <= 2e-41)
		tmp = Float64(t_2 * cos(y));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	t_2 = 2.0 * sqrt(x);
	t_3 = t_2 - t_1;
	tmp = 0.0;
	if (t_1 <= -4e-20)
		tmp = t_3;
	elseif (t_1 <= 2e-41)
		tmp = t_2 * cos(y);
	else
		tmp = t_3;
	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[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-20], t$95$3, If[LessEqual[t$95$1, 2e-41], N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -3.99999999999999978e-20 or 2.00000000000000001e-41 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 77.3%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\cos y}\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 89.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   5. Simplified 89.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(2 \cdot \sqrt{x}\right)}, \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 84.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   8. Simplified 84.9%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]

   if -3.99999999999999978e-20 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.00000000000000001e-41

   1. Initial program 57.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\cos y}\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 58.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   5. Simplified 58.8%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \sqrt{x}\right), \color{blue}{\cos y}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \cos \color{blue}{y}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \cos y\right) \]

      5. cos-lowering-cos.f64 54.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right) \]

   8. Simplified 54.7%

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -4 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) - (a / (3.0d0 * b))
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)) - (a / (3.0 * b));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

3. Taylor expanded in z around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\cos y}\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 73.9%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

5. Simplified 73.9%

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

6. Final simplification 73.9%

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

Alternative 8: 66.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} - \frac{a}{3 \cdot b} \end{array} \]

FPCore

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

C

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

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)) - (a / (3.0d0 * b))
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)) - (a / (3.0 * b));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

3. Taylor expanded in z around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\cos y}\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 73.9%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

5. Simplified 73.9%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(2 \cdot \sqrt{x}\right)}, \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

   2. sqrt-lowering-sqrt.f64 60.3%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, 3\right)\right)\right) \]

8. Simplified 60.3%

   \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]

9. Final simplification 60.3%

   \[\leadsto 2 \cdot \sqrt{x} - \frac{a}{3 \cdot b} \]
10. Add Preprocessing

Alternative 9: 66.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} + \frac{a \cdot -0.3333333333333333}{b} \end{array} \]

FPCore

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

C

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

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)) + ((a * (-0.3333333333333333d0)) / b)
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)) + ((a * -0.3333333333333333) / b);
}

Python

def code(x, y, z, t, a, b):
	return (2.0 * math.sqrt(x)) + ((a * -0.3333333333333333) / b)

Julia

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (2.0 * sqrt(x)) + ((a * -0.3333333333333333) / b);
end

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) + \frac{a}{b \cdot 3}}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right) \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3} \cdot \frac{a}{b \cdot 3}}\right)}\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right) \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3} \cdot \frac{a}{b \cdot 3}}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) + \frac{a}{b \cdot 3}}}}\right)\right) \]

4. Applied egg-rr 67.5%

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

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(\sqrt{x} \cdot \cos y\right)\right)}, \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \sqrt{x}\right), \cos y\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{x}\right)\right), \cos y\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \cos y\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]

   5. cos-lowering-cos.f64 73.9%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, -3\right)\right)\right)\right)\right) \]

7. Simplified 73.9%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

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

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{a}{b}\right)\right)}\right) \]

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

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

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

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

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

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

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

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

   11. *-lowering-*.f64 60.3%

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

10. Simplified 60.3%

    \[\leadsto \color{blue}{2 \cdot \sqrt{x} + \frac{-0.3333333333333333 \cdot a}{b}} \]

11. Final simplification 60.3%

    \[\leadsto 2 \cdot \sqrt{x} + \frac{a \cdot -0.3333333333333333}{b} \]
12. Add Preprocessing

Alternative 10: 51.5% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{a}{-3}}{b} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ (/ a -3.0) b))

C

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

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 = (a / (-3.0d0)) / b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (a / -3.0) / b;
}

Python

def code(x, y, z, t, a, b):
	return (a / -3.0) / b

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a / -3.0), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 67.6%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

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

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

   3. /-lowering-/.f64 44.6%

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

5. Simplified 44.6%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

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

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

   6. /-lowering-/.f64 45.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, -3\right), b\right) \]

7. Applied egg-rr 45.0%

   \[\leadsto \color{blue}{\frac{\frac{a}{-3}}{b}} \]
8. Add Preprocessing

Alternative 11: 51.5% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \frac{a}{b \cdot -3} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return 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 = a / (b * (-3.0d0))
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return a / (b * -3.0)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.6%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

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

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

   3. /-lowering-/.f64 44.6%

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

5. Simplified 44.6%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

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

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

   5. *-lowering-*.f64 44.9%

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

7. Applied egg-rr 44.9%

   \[\leadsto \color{blue}{\frac{a}{b \cdot -3}} \]
8. Add Preprocessing

Alternative 12: 51.4% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ a \cdot \frac{-0.3333333333333333}{b} \end{array} \]

FPCore

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

C

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

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 = a * ((-0.3333333333333333d0) / b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return a * (-0.3333333333333333 / b);
}

Python

def code(x, y, z, t, a, b):
	return a * (-0.3333333333333333 / b)

Julia

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = a * (-0.3333333333333333 / b);
end

Wolfram

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

Derivation

1. Initial program 67.6%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

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

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

   3. /-lowering-/.f64 44.6%

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

5. Simplified 44.6%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

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

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. /-lowering-/.f64 44.9%

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

7. Applied egg-rr 44.9%

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

8. Final simplification 44.9%

   \[\leadsto a \cdot \frac{-0.3333333333333333}{b} \]
9. Add Preprocessing

Developer Target 1: 74.6% accurate, 1.0× 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 2024158 
(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))))