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

Percentage Accurate: 69.9% → 77.0%

Time: 20.6s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.9% 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: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.74:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \cos y, \left(-\sin y\right) \cdot \sin \left(t \cdot \left(z \cdot -0.3333333333333333\right)\right)\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos y - t\_2\\ \end{array} \end{array} \]

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))) 0.74)
     (-
      (*
       t_1
       (fma
        (cos (* z (* t 0.3333333333333333)))
        (cos y)
        (* (- (sin y)) (sin (* t (* z -0.3333333333333333))))))
      t_2)
     (- (* t_1 (cos y)) 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))) <= 0.74) {
		tmp = (t_1 * fma(cos((z * (t * 0.3333333333333333))), cos(y), (-sin(y) * sin((t * (z * -0.3333333333333333)))))) - t_2;
	} else {
		tmp = (t_1 * cos(y)) - 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 (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 0.74)
		tmp = Float64(Float64(t_1 * fma(cos(Float64(z * Float64(t * 0.3333333333333333))), cos(y), Float64(Float64(-sin(y)) * sin(Float64(t * Float64(z * -0.3333333333333333)))))) - t_2);
	else
		tmp = Float64(Float64(t_1 * cos(y)) - t_2);
	end
	return 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[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.74], N[(N[(t$95$1 * N[(N[Cos[N[(z * N[(t * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[((-N[Sin[y], $MachinePrecision]) * N[Sin[N[(t * N[(z * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 0.73999999999999999

   1. Initial program 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. cos-sum N/A

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

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

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

      6. cos-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. div-inv N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-cos.f64 N/A

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

      18. lower-*.f64 N/A

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

   4. Applied egg-rr 72.0%

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

   if 0.73999999999999999 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 67.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-cos.f64 89.5

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

   5. Simplified 89.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.74:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \cos y, \left(-\sin y\right) \cdot \sin \left(t \cdot \left(z \cdot -0.3333333333333333\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := 2 \cdot \sqrt{x}\\ t_3 := z \cdot \left(t \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.74:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(\cos t\_3, \cos y, \sin y \cdot \sin t\_3\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \cos y - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b)))
        (t_2 (* 2.0 (sqrt x)))
        (t_3 (* z (* t 0.3333333333333333))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 0.74)
     (- (* t_2 (fma (cos t_3) (cos y) (* (sin y) (sin t_3)))) t_1)
     (- (* t_2 (cos y)) 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 t_3 = z * (t * 0.3333333333333333);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 0.74) {
		tmp = (t_2 * fma(cos(t_3), cos(y), (sin(y) * sin(t_3)))) - t_1;
	} else {
		tmp = (t_2 * cos(y)) - 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))
	t_3 = Float64(z * Float64(t * 0.3333333333333333))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 0.74)
		tmp = Float64(Float64(t_2 * fma(cos(t_3), cos(y), Float64(sin(y) * sin(t_3)))) - t_1);
	else
		tmp = Float64(Float64(t_2 * cos(y)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(t * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.74], N[(N[(t$95$2 * N[(N[Cos[t$95$3], $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 0.73999999999999999

   1. Initial program 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. lower-sin.f64 71.5

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. lower-*.f64 N/A

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

      22. div-inv N/A

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

      23. lower-*.f64 N/A

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

      24. metadata-eval 72.1

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

   4. Applied egg-rr 72.1%

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

   if 0.73999999999999999 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 67.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-cos.f64 89.5

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

   5. Simplified 89.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.74:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right), \cos y, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-156}:\\ \;\;\;\;t\_1 \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, \frac{-0.3333333333333333 \cdot a}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 2.0 (sqrt x))) (t_2 (/ a (* 3.0 b))))
   (if (<= t_2 -1e-33)
     (fma a (/ -0.3333333333333333 b) t_1)
     (if (<= t_2 5e-156)
       (* t_1 (cos (fma t (* z -0.3333333333333333) y)))
       (fma 2.0 (sqrt x) (/ (* -0.3333333333333333 a) b))))))

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 (t_2 <= -1e-33) {
		tmp = fma(a, (-0.3333333333333333 / b), t_1);
	} else if (t_2 <= 5e-156) {
		tmp = t_1 * cos(fma(t, (z * -0.3333333333333333), y));
	} else {
		tmp = fma(2.0, sqrt(x), ((-0.3333333333333333 * a) / b));
	}
	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 (t_2 <= -1e-33)
		tmp = fma(a, Float64(-0.3333333333333333 / b), t_1);
	elseif (t_2 <= 5e-156)
		tmp = Float64(t_1 * cos(fma(t, Float64(z * -0.3333333333333333), y)));
	else
		tmp = fma(2.0, sqrt(x), Float64(Float64(-0.3333333333333333 * a) / b));
	end
	return 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[t$95$2, -1e-33], N[(a * N[(-0.3333333333333333 / b), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 5e-156], N[(t$95$1 * N[Cos[N[(t * N[(z * -0.3333333333333333), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[x], $MachinePrecision] + N[(N[(-0.3333333333333333 * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.0000000000000001e-33

   1. Initial program 80.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-cos.f64 95.8

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

   5. Simplified 95.8%

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

   6. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-cos.f64 96.0

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

   8. Simplified 96.0%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 94.6

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

   11. Simplified 94.6%

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

   if -1.0000000000000001e-33 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 5.00000000000000007e-156

   1. Initial program 55.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 54.9%

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

   if 5.00000000000000007e-156 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 73.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-cos.f64 90.9

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

   5. Simplified 90.9%

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

   6. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 87.5

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

   8. Simplified 87.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, 2 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 5 \cdot 10^{-156}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, \frac{-0.3333333333333333 \cdot a}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-156}:\\ \;\;\;\;t\_1 \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, \frac{-0.3333333333333333 \cdot a}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 2.0 (sqrt x))) (t_2 (/ a (* 3.0 b))))
   (if (<= t_2 -1e-33)
     (fma a (/ -0.3333333333333333 b) t_1)
     (if (<= t_2 5e-156)
       (* t_1 (cos y))
       (fma 2.0 (sqrt x) (/ (* -0.3333333333333333 a) b))))))

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 (t_2 <= -1e-33) {
		tmp = fma(a, (-0.3333333333333333 / b), t_1);
	} else if (t_2 <= 5e-156) {
		tmp = t_1 * cos(y);
	} else {
		tmp = fma(2.0, sqrt(x), ((-0.3333333333333333 * a) / b));
	}
	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 (t_2 <= -1e-33)
		tmp = fma(a, Float64(-0.3333333333333333 / b), t_1);
	elseif (t_2 <= 5e-156)
		tmp = Float64(t_1 * cos(y));
	else
		tmp = fma(2.0, sqrt(x), Float64(Float64(-0.3333333333333333 * a) / b));
	end
	return 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[t$95$2, -1e-33], N[(a * N[(-0.3333333333333333 / b), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 5e-156], N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[x], $MachinePrecision] + N[(N[(-0.3333333333333333 * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.0000000000000001e-33

   1. Initial program 80.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-cos.f64 95.8

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

   5. Simplified 95.8%

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

   6. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-cos.f64 96.0

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

   8. Simplified 96.0%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 94.6

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

   11. Simplified 94.6%

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

   if -1.0000000000000001e-33 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 5.00000000000000007e-156

   1. Initial program 55.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-cos.f64 53.8

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

   5. Simplified 53.8%

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

   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

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-cos.f64 53.8

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

   8. Simplified 53.8%

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

   if 5.00000000000000007e-156 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 73.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-cos.f64 90.9

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

   5. Simplified 90.9%

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

   6. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 87.5

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

   8. Simplified 87.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, 2 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 5 \cdot 10^{-156}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, \frac{-0.3333333333333333 \cdot a}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 1.1× 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 68.7%

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

3. Taylor expanded in z around 0

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

   1. lower-cos.f64 79.3

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

5. Simplified 79.3%

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

6. Final simplification 79.3%

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

Alternative 6: 76.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in z around 0

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

   1. lower-cos.f64 79.3

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

5. Simplified 79.3%

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

6. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. lower-fma.f64 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. associate-*r* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. lower-sqrt.f64 N/A

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

   21. lower-cos.f64 79.3

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

8. Simplified 79.3%

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

Alternative 7: 76.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in z around 0

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

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

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 78.9

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

5. Simplified 78.9%

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

6. Final simplification 78.9%

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

Alternative 8: 59.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (<= t_1 -2e-13)
     (* a (/ -0.3333333333333333 b))
     (if (<= t_1 4e-72) (* 2.0 (sqrt x)) (/ (* -0.3333333333333333 a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if (t_1 <= -2e-13) {
		tmp = a * (-0.3333333333333333 / b);
	} else if (t_1 <= 4e-72) {
		tmp = 2.0 * sqrt(x);
	} else {
		tmp = (-0.3333333333333333 * a) / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (3.0d0 * b)
    if (t_1 <= (-2d-13)) then
        tmp = a * ((-0.3333333333333333d0) / b)
    else if (t_1 <= 4d-72) then
        tmp = 2.0d0 * sqrt(x)
    else
        tmp = ((-0.3333333333333333d0) * a) / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if (t_1 <= -2e-13) {
		tmp = a * (-0.3333333333333333 / b);
	} else if (t_1 <= 4e-72) {
		tmp = 2.0 * Math.sqrt(x);
	} else {
		tmp = (-0.3333333333333333 * a) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if t_1 <= -2e-13:
		tmp = a * (-0.3333333333333333 / b)
	elif t_1 <= 4e-72:
		tmp = 2.0 * math.sqrt(x)
	else:
		tmp = (-0.3333333333333333 * a) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (t_1 <= -2e-13)
		tmp = Float64(a * Float64(-0.3333333333333333 / b));
	elseif (t_1 <= 4e-72)
		tmp = Float64(2.0 * sqrt(x));
	else
		tmp = Float64(Float64(-0.3333333333333333 * a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	tmp = 0.0;
	if (t_1 <= -2e-13)
		tmp = a * (-0.3333333333333333 / b);
	elseif (t_1 <= 4e-72)
		tmp = 2.0 * sqrt(x);
	else
		tmp = (-0.3333333333333333 * a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-13], N[(a * N[(-0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-72], N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 * a), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2.0000000000000001e-13

   1. Initial program 80.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.5

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

   5. Simplified 93.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 93.7

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

   7. Applied egg-rr 93.7%

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

   if -2.0000000000000001e-13 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 3.9999999999999999e-72

   1. Initial program 58.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. sin-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 37.8

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

   5. Simplified 37.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right)\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right) + 2 \cdot \sqrt{x}\right)} - \frac{1}{3} \cdot \frac{a}{b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right) + \left(2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right), 2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}\right)} \]

   8. Simplified 34.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(0.6666666666666666, \sqrt{x} \cdot \left(y \cdot z\right), -0.1111111111111111 \cdot \left(\sqrt{x} \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)\right), \mathsf{fma}\left(2, \sqrt{x}, \frac{a \cdot -0.3333333333333333}{b}\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{x}} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{x}} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right) + 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, \frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right), 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

   11. Simplified 32.3%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.6666666666666666, \sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right), \left(-0.1111111111111111 \cdot \left(t \cdot \left(z \cdot z\right)\right)\right) \cdot \sqrt{\frac{1}{x}}\right), 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

   12. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 37.6

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

   14. Simplified 37.6%

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

   if 3.9999999999999999e-72 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 72.6%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 85.1

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

   5. Simplified 85.1%

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 86.1

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

   7. Applied egg-rr 86.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 4 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (<= t_1 -2e-13)
     (* a (/ -0.3333333333333333 b))
     (if (<= t_1 4e-72) (* 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 = a / (3.0 * b);
	double tmp;
	if (t_1 <= -2e-13) {
		tmp = a * (-0.3333333333333333 / b);
	} else if (t_1 <= 4e-72) {
		tmp = 2.0 * sqrt(x);
	} else {
		tmp = 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 = a / (3.0d0 * b)
    if (t_1 <= (-2d-13)) then
        tmp = a * ((-0.3333333333333333d0) / b)
    else if (t_1 <= 4d-72) then
        tmp = 2.0d0 * sqrt(x)
    else
        tmp = 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 = a / (3.0 * b);
	double tmp;
	if (t_1 <= -2e-13) {
		tmp = a * (-0.3333333333333333 / b);
	} else if (t_1 <= 4e-72) {
		tmp = 2.0 * Math.sqrt(x);
	} else {
		tmp = a / (b * -3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if t_1 <= -2e-13:
		tmp = a * (-0.3333333333333333 / b)
	elif t_1 <= 4e-72:
		tmp = 2.0 * math.sqrt(x)
	else:
		tmp = a / (b * -3.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (t_1 <= -2e-13)
		tmp = Float64(a * Float64(-0.3333333333333333 / b));
	elseif (t_1 <= 4e-72)
		tmp = Float64(2.0 * sqrt(x));
	else
		tmp = Float64(a / Float64(b * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	tmp = 0.0;
	if (t_1 <= -2e-13)
		tmp = a * (-0.3333333333333333 / b);
	elseif (t_1 <= 4e-72)
		tmp = 2.0 * sqrt(x);
	else
		tmp = a / (b * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-13], N[(a * N[(-0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-72], N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2.0000000000000001e-13

   1. Initial program 80.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.5

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

   5. Simplified 93.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 93.7

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

   7. Applied egg-rr 93.7%

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

   if -2.0000000000000001e-13 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 3.9999999999999999e-72

   1. Initial program 58.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. sin-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 37.8

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

   5. Simplified 37.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right)\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right) + 2 \cdot \sqrt{x}\right)} - \frac{1}{3} \cdot \frac{a}{b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right) + \left(2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right), 2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}\right)} \]

   8. Simplified 34.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(0.6666666666666666, \sqrt{x} \cdot \left(y \cdot z\right), -0.1111111111111111 \cdot \left(\sqrt{x} \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)\right), \mathsf{fma}\left(2, \sqrt{x}, \frac{a \cdot -0.3333333333333333}{b}\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{x}} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{x}} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right) + 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, \frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right), 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

   11. Simplified 32.3%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.6666666666666666, \sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right), \left(-0.1111111111111111 \cdot \left(t \cdot \left(z \cdot z\right)\right)\right) \cdot \sqrt{\frac{1}{x}}\right), 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

   12. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 37.6

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

   14. Simplified 37.6%

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

   if 3.9999999999999999e-72 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 72.6%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 85.1

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

   5. Simplified 85.1%

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. neg-mul-1 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 86.1

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

   7. Applied egg-rr 86.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 4 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (* a (/ -0.3333333333333333 b))))
   (if (<= t_1 -2e-13) t_2 (if (<= t_1 4e-72) (* 2.0 (sqrt x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = a * (-0.3333333333333333 / b);
	double tmp;
	if (t_1 <= -2e-13) {
		tmp = t_2;
	} else if (t_1 <= 4e-72) {
		tmp = 2.0 * sqrt(x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a / (3.0d0 * b)
    t_2 = a * ((-0.3333333333333333d0) / b)
    if (t_1 <= (-2d-13)) then
        tmp = t_2
    else if (t_1 <= 4d-72) then
        tmp = 2.0d0 * sqrt(x)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = a * (-0.3333333333333333 / b);
	double tmp;
	if (t_1 <= -2e-13) {
		tmp = t_2;
	} else if (t_1 <= 4e-72) {
		tmp = 2.0 * Math.sqrt(x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = a * (-0.3333333333333333 / b)
	tmp = 0
	if t_1 <= -2e-13:
		tmp = t_2
	elif t_1 <= 4e-72:
		tmp = 2.0 * math.sqrt(x)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(a * Float64(-0.3333333333333333 / b))
	tmp = 0.0
	if (t_1 <= -2e-13)
		tmp = t_2;
	elseif (t_1 <= 4e-72)
		tmp = Float64(2.0 * sqrt(x));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	t_2 = a * (-0.3333333333333333 / b);
	tmp = 0.0;
	if (t_1 <= -2e-13)
		tmp = t_2;
	elseif (t_1 <= 4e-72)
		tmp = 2.0 * sqrt(x);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(-0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-13], t$95$2, If[LessEqual[t$95$1, 4e-72], N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2.0000000000000001e-13 or 3.9999999999999999e-72 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 75.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 88.6

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

   5. Simplified 88.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 89.3

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

   7. Applied egg-rr 89.3%

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

   if -2.0000000000000001e-13 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 3.9999999999999999e-72

   1. Initial program 58.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. sin-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 37.8

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

   5. Simplified 37.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right)\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right) + 2 \cdot \sqrt{x}\right)} - \frac{1}{3} \cdot \frac{a}{b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right) + \left(2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right), 2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}\right)} \]

   8. Simplified 34.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(0.6666666666666666, \sqrt{x} \cdot \left(y \cdot z\right), -0.1111111111111111 \cdot \left(\sqrt{x} \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)\right), \mathsf{fma}\left(2, \sqrt{x}, \frac{a \cdot -0.3333333333333333}{b}\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{x}} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{x}} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right) + 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, \frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right), 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

   11. Simplified 32.3%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.6666666666666666, \sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right), \left(-0.1111111111111111 \cdot \left(t \cdot \left(z \cdot z\right)\right)\right) \cdot \sqrt{\frac{1}{x}}\right), 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

   12. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 37.6

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

   14. Simplified 37.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 4 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in z around 0

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

   1. lower-cos.f64 79.3

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

5. Simplified 79.3%

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

6. Taylor expanded in y around 0

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

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

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-*.f64 70.3

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

8. Simplified 70.3%

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

9. Final simplification 70.3%

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

Alternative 12: 17.5% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

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

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

   4. sin-neg N/A

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

   5. remove-double-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. cos-neg N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 53.4

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

5. Simplified 53.4%

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

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right)\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right) + 2 \cdot \sqrt{x}\right)} - \frac{1}{3} \cdot \frac{a}{b} \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right)\right) + \left(2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{x}\right) + \frac{2}{3} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right), 2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}\right)} \]

8. Simplified 43.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(0.6666666666666666, \sqrt{x} \cdot \left(y \cdot z\right), -0.1111111111111111 \cdot \left(\sqrt{x} \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)\right), \mathsf{fma}\left(2, \sqrt{x}, \frac{a \cdot -0.3333333333333333}{b}\right)\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{x}} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right)\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{x}} + t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right)\right)} \]

    2. +-commutative N/A

       \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(\frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right)\right) + 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

    3. lower-fma.f64 N/A

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, \frac{-1}{9} \cdot \left(\left(t \cdot {z}^{2}\right) \cdot \sqrt{\frac{1}{x}}\right) + \frac{2}{3} \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right)\right), 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

11. Simplified 20.2%

    \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(0.6666666666666666, \sqrt{\frac{1}{x}} \cdot \left(y \cdot z\right), \left(-0.1111111111111111 \cdot \left(t \cdot \left(z \cdot z\right)\right)\right) \cdot \sqrt{\frac{1}{x}}\right), 2 \cdot \sqrt{\frac{1}{x}}\right)} \]

12. Taylor expanded in t around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-sqrt.f64 18.8

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

14. Simplified 18.8%

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

15. Final simplification 18.8%

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

Developer Target 1: 74.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024207 
(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))))