Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1

Percentage Accurate: 27.2% → 29.9%

Time: 13.7s

Alternatives: 7

Speedup: 44.8×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
  (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

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 = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

Python

def code(x, y, z, t, a, b):
	return (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 27.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
  (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

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 = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

Python

def code(x, y, z, t, a, b):
	return (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 29.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{+51}:\\ \;\;\;\;\cos \left(\left(-0.0625 \cdot t\_m\right) \cdot z\right) \cdot \left(\cos \left(0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\_m\right) \cdot b\right)\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t\_m}, \frac{a}{b}, \frac{-32}{b \cdot t\_m}\right), a, \frac{16}{b \cdot t\_m}\right)}\right) \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<= t_m 2.9e+51)
   (*
    (cos (* (* -0.0625 t_m) z))
    (* (cos (* 0.0625 (* (* (fma a 2.0 1.0) t_m) b))) x))
   (*
    (cos
     (/
      1.0
      (fma
       (fma (/ 64.0 t_m) (/ a b) (/ -32.0 (* b t_m)))
       a
       (/ 16.0 (* b t_m)))))
    (* 1.0 x))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (t_m <= 2.9e+51) {
		tmp = cos(((-0.0625 * t_m) * z)) * (cos((0.0625 * ((fma(a, 2.0, 1.0) * t_m) * b))) * x);
	} else {
		tmp = cos((1.0 / fma(fma((64.0 / t_m), (a / b), (-32.0 / (b * t_m))), a, (16.0 / (b * t_m))))) * (1.0 * x);
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (t_m <= 2.9e+51)
		tmp = Float64(cos(Float64(Float64(-0.0625 * t_m) * z)) * Float64(cos(Float64(0.0625 * Float64(Float64(fma(a, 2.0, 1.0) * t_m) * b))) * x));
	else
		tmp = Float64(cos(Float64(1.0 / fma(fma(Float64(64.0 / t_m), Float64(a / b), Float64(-32.0 / Float64(b * t_m))), a, Float64(16.0 / Float64(b * t_m))))) * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[t$95$m, 2.9e+51], N[(N[Cos[N[(N[(-0.0625 * t$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(0.0625 * N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(1.0 / N[(N[(N[(64.0 / t$95$m), $MachinePrecision] * N[(a / b), $MachinePrecision] + N[(-32.0 / N[(b * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(16.0 / N[(b * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.8999999999999998e51

   1. Initial program 32.4%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot \frac{1}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. lower-*.f64 34.3

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Applied rewrites 34.3%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(z \cdot t\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\left(2 \cdot a + 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\cos \left(\left(\left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right)} \cdot \left(t \cdot z\right)\right) \]

   8. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x\right) \cdot \cos \left(\left(-0.0625 \cdot t\right) \cdot z\right)} \]

   if 2.8999999999999998e51 < t

   1. Initial program 7.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 12.4%

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. clear-num N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{16}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)} \cdot t}}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\left(\color{blue}{\left(a \cdot 2 + 1\right)} \cdot b\right) \cdot t}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot b\right) \cdot t}}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\left(\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot b\right) \cdot t}}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\color{blue}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right)} \cdot t}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\color{blue}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right)} \cdot t}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\color{blue}{\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left({\left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)}^{-1}\right)} \]

      11. exp-to-pow N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(e^{\log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right) \cdot -1}\right)} \]

      12. lift-log.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(e^{\color{blue}{\log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)} \cdot -1}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(e^{\color{blue}{-1 \cdot \log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)}}\right) \]

      14. exp-prod N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left({\left(e^{-1}\right)}^{\log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)}\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left({\left(e^{-1}\right)}^{\log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)}\right)} \]

      16. lower-exp.f64 3.2

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left({\color{blue}{\left(e^{-1}\right)}}^{\log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)}\right) \]

   7. Applied rewrites 3.2%

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left({\left(e^{-1}\right)}^{\log \left(\frac{16}{\left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right) \cdot t}\right)}\right)} \]

   9. Applied rewrites 2.9%

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\frac{1}{{\left(e^{-1}\right)}^{\log \left(\left(\left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot t\right) \cdot 0.0625\right)}}\right)} \]

   10. Taylor expanded in a around 0

       \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\color{blue}{a \cdot \left(64 \cdot \frac{a}{b \cdot t} - 32 \cdot \frac{1}{b \cdot t}\right) + 16 \cdot \frac{1}{b \cdot t}}}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\color{blue}{\left(64 \cdot \frac{a}{b \cdot t} - 32 \cdot \frac{1}{b \cdot t}\right) \cdot a} + 16 \cdot \frac{1}{b \cdot t}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\color{blue}{\mathsf{fma}\left(64 \cdot \frac{a}{b \cdot t} - 32 \cdot \frac{1}{b \cdot t}, a, 16 \cdot \frac{1}{b \cdot t}\right)}}\right) \]

       3. sub-neg N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\color{blue}{64 \cdot \frac{a}{b \cdot t} + \left(\mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right)}, a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       4. associate-*r/ N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\color{blue}{\frac{64 \cdot a}{b \cdot t}} + \left(\mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\frac{64 \cdot a}{\color{blue}{t \cdot b}} + \left(\mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       6. times-frac N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\color{blue}{\frac{64}{t} \cdot \frac{a}{b}} + \left(\mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right)}, a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{64}{t}}, \frac{a}{b}, \mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       9. lower-/.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \color{blue}{\frac{a}{b}}, \mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       10. associate-*r/ N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \mathsf{neg}\left(\color{blue}{\frac{32 \cdot 1}{b \cdot t}}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \mathsf{neg}\left(\frac{\color{blue}{32}}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       12. distribute-neg-frac N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \color{blue}{\frac{\mathsf{neg}\left(32\right)}{b \cdot t}}\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{\color{blue}{-32}}{b \cdot t}\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       14. lower-/.f64 N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \color{blue}{\frac{-32}{b \cdot t}}\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       15. *-commutative N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{\color{blue}{t \cdot b}}\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{\color{blue}{t \cdot b}}\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       17. associate-*r/ N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{t \cdot b}\right), a, \color{blue}{\frac{16 \cdot 1}{b \cdot t}}\right)}\right) \]

       18. metadata-eval N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{t \cdot b}\right), a, \frac{\color{blue}{16}}{b \cdot t}\right)}\right) \]

       19. lower-/.f64 N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{t \cdot b}\right), a, \color{blue}{\frac{16}{b \cdot t}}\right)}\right) \]

   12. Applied rewrites 13.9%

       \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{t \cdot b}\right), a, \frac{16}{t \cdot b}\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+51}:\\ \;\;\;\;\cos \left(\left(-0.0625 \cdot t\right) \cdot z\right) \cdot \left(\cos \left(0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{b \cdot t}\right), a, \frac{16}{b \cdot t}\right)}\right) \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \cdot \left(\cos \left(\frac{\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\_m}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\cos \left(\frac{t\_m}{\frac{16}{\mathsf{fma}\left(2, a, 1\right) \cdot b}}\right) \cdot \left(1 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0))
       (* (cos (/ (* (* (+ (* 2.0 y) 1.0) z) t_m) 16.0)) x))
      5e+305)
   (* (cos (/ t_m (/ 16.0 (* (fma 2.0 a 1.0) b)))) (* 1.0 x))
   (* 1.0 x)))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if ((cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0)) * (cos((((((2.0 * y) + 1.0) * z) * t_m) / 16.0)) * x)) <= 5e+305) {
		tmp = cos((t_m / (16.0 / (fma(2.0, a, 1.0) * b)))) * (1.0 * x);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0)) * Float64(cos(Float64(Float64(Float64(Float64(Float64(2.0 * y) + 1.0) * z) * t_m) / 16.0)) * x)) <= 5e+305)
		tmp = Float64(cos(Float64(t_m / Float64(16.0 / Float64(fma(2.0, a, 1.0) * b)))) * Float64(1.0 * x));
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(N[(N[(N[(N[(2.0 * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], 5e+305], N[(N[Cos[N[(t$95$m / N[(16.0 / N[(N[(2.0 * a + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 * x), $MachinePrecision]), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5.00000000000000009e305

   1. Initial program 45.6%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 44.2%

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}{16}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\color{blue}{t \cdot \left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{16}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t \cdot \color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)}}{16}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t \cdot \left(\color{blue}{\left(a \cdot 2 + 1\right)} \cdot b\right)}{16}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t \cdot \left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot b\right)}{16}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t \cdot \left(\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot b\right)}{16}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right)}}{16}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right)}}{16}\right) \]

      10. associate-/l* N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(t \cdot \frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{16}\right)} \]

      11. clear-num N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(t \cdot \color{blue}{\frac{1}{\frac{16}{b \cdot \mathsf{fma}\left(a, 2, 1\right)}}}\right) \]

      12. div-inv N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\frac{t}{\frac{16}{b \cdot \mathsf{fma}\left(a, 2, 1\right)}}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\frac{t}{\frac{16}{b \cdot \mathsf{fma}\left(a, 2, 1\right)}}\right)} \]

      14. lower-/.f64 44.4

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t}{\color{blue}{\frac{16}{b \cdot \mathsf{fma}\left(a, 2, 1\right)}}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t}{\frac{16}{\color{blue}{b \cdot \mathsf{fma}\left(a, 2, 1\right)}}}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t}{\frac{16}{\color{blue}{\mathsf{fma}\left(a, 2, 1\right) \cdot b}}}\right) \]

      17. lower-*.f64 44.4

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t}{\frac{16}{\color{blue}{\mathsf{fma}\left(a, 2, 1\right) \cdot b}}}\right) \]

      18. lift-fma.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t}{\frac{16}{\color{blue}{\left(a \cdot 2 + 1\right)} \cdot b}}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t}{\frac{16}{\left(\color{blue}{2 \cdot a} + 1\right) \cdot b}}\right) \]

      20. lower-fma.f64 44.4

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{t}{\frac{16}{\color{blue}{\mathsf{fma}\left(2, a, 1\right)} \cdot b}}\right) \]

   7. Applied rewrites 44.4%

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\frac{t}{\frac{16}{\mathsf{fma}\left(2, a, 1\right) \cdot b}}\right)} \]

   if 5.00000000000000009e305 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot \frac{1}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. lower-*.f64 4.5

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Applied rewrites 4.5%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(z \cdot t\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\left(2 \cdot a + 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\cos \left(\left(\left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right)} \cdot \left(t \cdot z\right)\right) \]

   8. Applied rewrites 6.1%

      \[\leadsto \color{blue}{\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x\right) \cdot \cos \left(\left(-0.0625 \cdot t\right) \cdot z\right)} \]

   9. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\cos \left(\frac{-1}{16} \cdot \left(t \cdot z\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 8.8%

       \[\leadsto \cos \left(\left(z \cdot t\right) \cdot -0.0625\right) \cdot \color{blue}{x} \]

   12. Taylor expanded in t around 0

       \[\leadsto 1 \cdot x \]
   13. Step-by-step derivation

   14. Applied rewrites 9.8%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\cos \left(\frac{t}{\frac{16}{\mathsf{fma}\left(2, a, 1\right) \cdot b}}\right) \cdot \left(1 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \cdot \left(\cos \left(\frac{\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t\_m}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right) \cdot t\_m\right) \cdot -0.0625\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<=
      (*
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0))
       (* (cos (/ (* (* (+ (* 2.0 y) 1.0) z) t_m) 16.0)) x))
      5e+305)
   (* (* (cos (* (* (* (fma 2.0 a 1.0) b) t_m) -0.0625)) x) 1.0)
   (* 1.0 x)))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if ((cos((((((a * 2.0) + 1.0) * b) * t_m) / 16.0)) * (cos((((((2.0 * y) + 1.0) * z) * t_m) / 16.0)) * x)) <= 5e+305) {
		tmp = (cos((((fma(2.0, a, 1.0) * b) * t_m) * -0.0625)) * x) * 1.0;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t_m) / 16.0)) * Float64(cos(Float64(Float64(Float64(Float64(Float64(2.0 * y) + 1.0) * z) * t_m) / 16.0)) * x)) <= 5e+305)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(fma(2.0, a, 1.0) * b) * t_m) * -0.0625)) * x) * 1.0);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[N[(N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(N[(N[(N[(N[(2.0 * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], 5e+305], N[(N[(N[Cos[N[(N[(N[(N[(2.0 * a + 1.0), $MachinePrecision] * b), $MachinePrecision] * t$95$m), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5.00000000000000009e305

   1. Initial program 45.6%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 44.2%

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \cdot \left(x \cdot 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \cdot \color{blue}{\left(x \cdot 1\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \cdot x\right) \cdot 1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \cdot x\right) \cdot 1} \]

   7. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\left(\cos \left(\left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right) \cdot t\right) \cdot -0.0625\right) \cdot x\right) \cdot 1} \]

   if 5.00000000000000009e305 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot \frac{1}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. lower-*.f64 4.5

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Applied rewrites 4.5%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(z \cdot t\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\left(2 \cdot a + 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\cos \left(\left(\left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right)} \cdot \left(t \cdot z\right)\right) \]

   8. Applied rewrites 6.1%

      \[\leadsto \color{blue}{\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x\right) \cdot \cos \left(\left(-0.0625 \cdot t\right) \cdot z\right)} \]

   9. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\cos \left(\frac{-1}{16} \cdot \left(t \cdot z\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 8.8%

       \[\leadsto \cos \left(\left(z \cdot t\right) \cdot -0.0625\right) \cdot \color{blue}{x} \]

   12. Taylor expanded in t around 0

       \[\leadsto 1 \cdot x \]
   13. Step-by-step derivation

   14. Applied rewrites 9.8%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{\left(\left(2 \cdot y + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right) \cdot t\right) \cdot -0.0625\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 29.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 9.8 \cdot 10^{+27}:\\ \;\;\;\;\left(\cos \left(0.125 \cdot \left(\left(b \cdot t\_m\right) \cdot a\right)\right) \cdot x\right) \cdot \cos \left(\left(-0.0625 \cdot t\_m\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t\_m}, \frac{a}{b}, \frac{-32}{b \cdot t\_m}\right), a, \frac{16}{b \cdot t\_m}\right)}\right) \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<= t_m 9.8e+27)
   (* (* (cos (* 0.125 (* (* b t_m) a))) x) (cos (* (* -0.0625 t_m) z)))
   (*
    (cos
     (/
      1.0
      (fma
       (fma (/ 64.0 t_m) (/ a b) (/ -32.0 (* b t_m)))
       a
       (/ 16.0 (* b t_m)))))
    (* 1.0 x))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (t_m <= 9.8e+27) {
		tmp = (cos((0.125 * ((b * t_m) * a))) * x) * cos(((-0.0625 * t_m) * z));
	} else {
		tmp = cos((1.0 / fma(fma((64.0 / t_m), (a / b), (-32.0 / (b * t_m))), a, (16.0 / (b * t_m))))) * (1.0 * x);
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (t_m <= 9.8e+27)
		tmp = Float64(Float64(cos(Float64(0.125 * Float64(Float64(b * t_m) * a))) * x) * cos(Float64(Float64(-0.0625 * t_m) * z)));
	else
		tmp = Float64(cos(Float64(1.0 / fma(fma(Float64(64.0 / t_m), Float64(a / b), Float64(-32.0 / Float64(b * t_m))), a, Float64(16.0 / Float64(b * t_m))))) * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[t$95$m, 9.8e+27], N[(N[(N[Cos[N[(0.125 * N[(N[(b * t$95$m), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Cos[N[(N[(-0.0625 * t$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(1.0 / N[(N[(N[(64.0 / t$95$m), $MachinePrecision] * N[(a / b), $MachinePrecision] + N[(-32.0 / N[(b * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(16.0 / N[(b * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.8000000000000003e27

   1. Initial program 32.3%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot \frac{1}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. lower-*.f64 34.3

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Applied rewrites 34.3%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(z \cdot t\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\left(2 \cdot a + 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\cos \left(\left(\left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right)} \cdot \left(t \cdot z\right)\right) \]

   8. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x\right) \cdot \cos \left(\left(-0.0625 \cdot t\right) \cdot z\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \left(\cos \left(\frac{1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right)\right) \cdot x\right) \cdot \cos \left(\left(\color{blue}{\frac{-1}{16}} \cdot t\right) \cdot z\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 33.8%

       \[\leadsto \left(\cos \left(\left(\left(t \cdot b\right) \cdot a\right) \cdot 0.125\right) \cdot x\right) \cdot \cos \left(\left(\color{blue}{-0.0625} \cdot t\right) \cdot z\right) \]

   if 9.8000000000000003e27 < t

   1. Initial program 10.1%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 14.5%

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)} \]

      2. clear-num N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{16}{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\color{blue}{\left(\left(a \cdot 2 + 1\right) \cdot b\right)} \cdot t}}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\left(\color{blue}{\left(a \cdot 2 + 1\right)} \cdot b\right) \cdot t}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot b\right) \cdot t}}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\left(\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot b\right) \cdot t}}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\color{blue}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right)} \cdot t}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\frac{16}{\color{blue}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right)} \cdot t}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\color{blue}{\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}}}\right) \]

      10. inv-pow N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left({\left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)}^{-1}\right)} \]

      11. exp-to-pow N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(e^{\log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right) \cdot -1}\right)} \]

      12. lift-log.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(e^{\color{blue}{\log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)} \cdot -1}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(e^{\color{blue}{-1 \cdot \log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)}}\right) \]

      14. exp-prod N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left({\left(e^{-1}\right)}^{\log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)}\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left({\left(e^{-1}\right)}^{\log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)}\right)} \]

      16. lower-exp.f64 3.7

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left({\color{blue}{\left(e^{-1}\right)}}^{\log \left(\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}\right)}\right) \]

   7. Applied rewrites 3.7%

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left({\left(e^{-1}\right)}^{\log \left(\frac{16}{\left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right) \cdot t}\right)}\right)} \]

   9. Applied rewrites 3.4%

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\frac{1}{{\left(e^{-1}\right)}^{\log \left(\left(\left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot t\right) \cdot 0.0625\right)}}\right)} \]

   10. Taylor expanded in a around 0

       \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\color{blue}{a \cdot \left(64 \cdot \frac{a}{b \cdot t} - 32 \cdot \frac{1}{b \cdot t}\right) + 16 \cdot \frac{1}{b \cdot t}}}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\color{blue}{\left(64 \cdot \frac{a}{b \cdot t} - 32 \cdot \frac{1}{b \cdot t}\right) \cdot a} + 16 \cdot \frac{1}{b \cdot t}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\color{blue}{\mathsf{fma}\left(64 \cdot \frac{a}{b \cdot t} - 32 \cdot \frac{1}{b \cdot t}, a, 16 \cdot \frac{1}{b \cdot t}\right)}}\right) \]

       3. sub-neg N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\color{blue}{64 \cdot \frac{a}{b \cdot t} + \left(\mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right)}, a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       4. associate-*r/ N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\color{blue}{\frac{64 \cdot a}{b \cdot t}} + \left(\mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\frac{64 \cdot a}{\color{blue}{t \cdot b}} + \left(\mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       6. times-frac N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\color{blue}{\frac{64}{t} \cdot \frac{a}{b}} + \left(\mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right)}, a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{64}{t}}, \frac{a}{b}, \mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       9. lower-/.f64 N/A

          \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \color{blue}{\frac{a}{b}}, \mathsf{neg}\left(32 \cdot \frac{1}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       10. associate-*r/ N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \mathsf{neg}\left(\color{blue}{\frac{32 \cdot 1}{b \cdot t}}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \mathsf{neg}\left(\frac{\color{blue}{32}}{b \cdot t}\right)\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       12. distribute-neg-frac N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \color{blue}{\frac{\mathsf{neg}\left(32\right)}{b \cdot t}}\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{\color{blue}{-32}}{b \cdot t}\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       14. lower-/.f64 N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \color{blue}{\frac{-32}{b \cdot t}}\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       15. *-commutative N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{\color{blue}{t \cdot b}}\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{\color{blue}{t \cdot b}}\right), a, 16 \cdot \frac{1}{b \cdot t}\right)}\right) \]

       17. associate-*r/ N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{t \cdot b}\right), a, \color{blue}{\frac{16 \cdot 1}{b \cdot t}}\right)}\right) \]

       18. metadata-eval N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{t \cdot b}\right), a, \frac{\color{blue}{16}}{b \cdot t}\right)}\right) \]

       19. lower-/.f64 N/A

           \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{t \cdot b}\right), a, \color{blue}{\frac{16}{b \cdot t}}\right)}\right) \]

   12. Applied rewrites 16.0%

       \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{t \cdot b}\right), a, \frac{16}{t \cdot b}\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.8 \cdot 10^{+27}:\\ \;\;\;\;\left(\cos \left(0.125 \cdot \left(\left(b \cdot t\right) \cdot a\right)\right) \cdot x\right) \cdot \cos \left(\left(-0.0625 \cdot t\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{t}, \frac{a}{b}, \frac{-32}{b \cdot t}\right), a, \frac{16}{b \cdot t}\right)}\right) \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 28.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \cos \left(0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\_m\right) \cdot b\right)\right) \cdot x \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (* (cos (* 0.0625 (* (* (fma a 2.0 1.0) t_m) b))) x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return cos((0.0625 * ((fma(a, 2.0, 1.0) * t_m) * b))) * x;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return Float64(cos(Float64(0.0625 * Float64(Float64(fma(a, 2.0, 1.0) * t_m) * b))) * x)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := N[(N[Cos[N[(0.0625 * N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.1%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   3. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot \frac{1}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   4. lower-*.f64 28.8

      \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

5. Applied rewrites 28.8%

   \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(z \cdot t\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x} \]

   3. lower-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x \]

   4. *-commutative N/A

      \[\leadsto \cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x \]

   6. *-commutative N/A

      \[\leadsto \cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x \]

   7. lower-*.f64 N/A

      \[\leadsto \cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   9. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   10. +-commutative N/A

       \[\leadsto \cos \left(\left(\left(\color{blue}{\left(2 \cdot a + 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   11. *-commutative N/A

       \[\leadsto \cos \left(\left(\left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   12. lower-fma.f64 28.5

       \[\leadsto \cos \left(\left(\left(\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x \]

8. Applied rewrites 28.5%

   \[\leadsto \color{blue}{\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x} \]

9. Final simplification 28.5%

   \[\leadsto \cos \left(0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x \]
10. Add Preprocessing

Alternative 6: 29.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \cos \left(\left(z \cdot t\_m\right) \cdot -0.0625\right) \cdot x \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b) :precision binary64 (* (cos (* (* z t_m) -0.0625)) x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return cos(((z * t_m) * -0.0625)) * x;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = cos(((z * t_m) * (-0.0625d0))) * x
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m, double a, double b) {
	return Math.cos(((z * t_m) * -0.0625)) * x;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	return math.cos(((z * t_m) * -0.0625)) * x

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return Float64(cos(Float64(Float64(z * t_m) * -0.0625)) * x)
end

MATLAB

t_m = abs(t);
function tmp = code(x, y, z, t_m, a, b)
	tmp = cos(((z * t_m) * -0.0625)) * x;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := N[(N[Cos[N[(N[(z * t$95$m), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.1%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   3. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot \frac{1}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   4. lower-*.f64 28.8

      \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

5. Applied rewrites 28.8%

   \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(z \cdot t\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   5. lower-cos.f64 N/A

      \[\leadsto \left(\color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\left(2 \cdot a + 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \left(\cos \left(\left(\left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right)} \cdot \left(t \cdot z\right)\right) \]

8. Applied rewrites 29.1%

   \[\leadsto \color{blue}{\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x\right) \cdot \cos \left(\left(-0.0625 \cdot t\right) \cdot z\right)} \]

9. Taylor expanded in b around 0

   \[\leadsto x \cdot \color{blue}{\cos \left(\frac{-1}{16} \cdot \left(t \cdot z\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 28.5%

    \[\leadsto \cos \left(\left(z \cdot t\right) \cdot -0.0625\right) \cdot \color{blue}{x} \]
12. Add Preprocessing

Alternative 7: 30.4% accurate, 44.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ 1 \cdot x \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b) :precision binary64 (* 1.0 x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return 1.0 * x;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 * x
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m, double a, double b) {
	return 1.0 * x;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	return 1.0 * x

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return Float64(1.0 * x)
end

MATLAB

t_m = abs(t);
function tmp = code(x, y, z, t_m, a, b)
	tmp = 1.0 * x;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 27.1%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{16}\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   3. *-commutative N/A

      \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot \frac{1}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   4. lower-*.f64 28.8

      \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

5. Applied rewrites 28.8%

   \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(z \cdot t\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x\right)} \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   5. lower-cos.f64 N/A

      \[\leadsto \left(\color{blue}{\cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\cos \color{blue}{\left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right)} \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\color{blue}{\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right)} \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\cos \left(\left(\color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot t\right)} \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\left(2 \cdot a + 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \left(\cos \left(\left(\left(\left(\color{blue}{a \cdot 2} + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \left(\cos \left(\left(\left(\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\frac{1}{16} \cdot \left(t \cdot z\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right)} \cdot \left(t \cdot z\right)\right) \]

8. Applied rewrites 29.1%

   \[\leadsto \color{blue}{\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x\right) \cdot \cos \left(\left(-0.0625 \cdot t\right) \cdot z\right)} \]

9. Taylor expanded in b around 0

   \[\leadsto x \cdot \color{blue}{\cos \left(\frac{-1}{16} \cdot \left(t \cdot z\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 28.5%

    \[\leadsto \cos \left(\left(z \cdot t\right) \cdot -0.0625\right) \cdot \color{blue}{x} \]

12. Taylor expanded in t around 0

    \[\leadsto 1 \cdot x \]
13. Step-by-step derivation

14. Applied rewrites 27.6%

    \[\leadsto 1 \cdot x \]
15. Add Preprocessing

Developer Target 1: 30.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + pow((a * 2.0), 2.0)))));
}

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 = x * cos(((b / 16.0d0) * (t / ((1.0d0 - (a * 2.0d0)) + ((a * 2.0d0) ** 2.0d0)))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + Math.pow((a * 2.0), 2.0)))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + math.pow((a * 2.0), 2.0)))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * cos(Float64(Float64(b / 16.0) * Float64(t / Float64(Float64(1.0 - Float64(a * 2.0)) + (Float64(a * 2.0) ^ 2.0))))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + ((a * 2.0) ^ 2.0)))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Cos[N[(N[(b / 16.0), $MachinePrecision] * N[(t / N[(N[(1.0 - N[(a * 2.0), $MachinePrecision]), $MachinePrecision] + N[Power[N[(a * 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024244 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (cos (* (/ b 16) (/ t (+ (- 1 (* a 2)) (pow (* a 2) 2)))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))