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

Percentage Accurate: 27.5% → 32.0%

Time: 20.2s

Alternatives: 7

Speedup: 225.0×

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: 27.5% 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: 32.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, 2 \cdot z, z\right) \cdot 0.0625\right)}\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;x \cdot \left(\cos \left(t_1 \cdot \left(t_1 \cdot t_1\right)\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cbrt (* t (* (fma y (* 2.0 z) z) 0.0625)))))
   (if (<=
        (*
         (* x (cos (/ (* t (* z (+ 1.0 (* y 2.0)))) 16.0)))
         (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
        2e+306)
     (*
      x
      (* (cos (* t_1 (* t_1 t_1))) (cos (* (/ t 16.0) (fma (* 2.0 a) b b)))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cbrt((t * (fma(y, (2.0 * z), z) * 0.0625)));
	double tmp;
	if (((x * cos(((t * (z * (1.0 + (y * 2.0)))) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+306) {
		tmp = x * (cos((t_1 * (t_1 * t_1))) * cos(((t / 16.0) * fma((2.0 * a), b, b))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cbrt(Float64(t * Float64(fma(y, Float64(2.0 * z), z) * 0.0625)))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(y * 2.0)))) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+306)
		tmp = Float64(x * Float64(cos(Float64(t_1 * Float64(t_1 * t_1))) * cos(Float64(Float64(t / 16.0) * fma(Float64(2.0 * a), b, b)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[N[(t * N[(N[(y * N[(2.0 * z), $MachinePrecision] + z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(N[(x * N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+306], N[(x * N[(N[Cos[N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t / 16.0), $MachinePrecision] * N[(N[(2.0 * a), $MachinePrecision] * b + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 2.00000000000000003e306

   1. Initial program 48.8%

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

   3. Simplified 48.8%

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

      1. expm1-log1p-u 48.8%

         \[\leadsto x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(t \cdot \frac{\mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}\right)\right)\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      2. expm1-udef 48.8%

         \[\leadsto x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(t \cdot \frac{\mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}\right)\right)} - 1\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      3. associate-*r/ 48.8%

         \[\leadsto x \cdot \left(\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{t \cdot \mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}\right)}\right)} - 1\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

   5. Applied egg-rr 48.8%

      \[\leadsto x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\frac{t \cdot \mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}\right)\right)} - 1\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]
   6. Step-by-step derivation

      1. expm1-def 48.8%

         \[\leadsto x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{t \cdot \mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}\right)\right)\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      2. expm1-log1p 48.8%

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

      3. associate-/l* 49.1%

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

   7. Simplified 49.1%

      \[\leadsto x \cdot \left(\color{blue}{\cos \left(\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]
   8. Step-by-step derivation

      1. add-cube-cbrt 49.2%

         \[\leadsto x \cdot \left(\cos \color{blue}{\left(\left(\sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}} \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      2. div-inv 49.3%

         \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{\color{blue}{t \cdot \frac{1}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}} \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      3. clear-num 49.1%

         \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \color{blue}{\frac{\mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}}} \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      4. div-inv 49.1%

         \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \color{blue}{\left(\mathsf{fma}\left(y, 2 \cdot z, z\right) \cdot \frac{1}{16}\right)}} \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      5. *-commutative 49.1%

         \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, \color{blue}{z \cdot 2}, z\right) \cdot \frac{1}{16}\right)} \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      6. metadata-eval 49.1%

         \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot \color{blue}{0.0625}\right)} \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      7. div-inv 49.2%

         \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)} \cdot \sqrt[3]{\color{blue}{t \cdot \frac{1}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      8. clear-num 49.3%

         \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)} \cdot \sqrt[3]{t \cdot \color{blue}{\frac{\mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}}}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      9. div-inv 49.3%

         \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)} \cdot \sqrt[3]{t \cdot \color{blue}{\left(\mathsf{fma}\left(y, 2 \cdot z, z\right) \cdot \frac{1}{16}\right)}}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      10. *-commutative 49.3%

          \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)} \cdot \sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, \color{blue}{z \cdot 2}, z\right) \cdot \frac{1}{16}\right)}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      11. metadata-eval 49.3%

          \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)} \cdot \sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot \color{blue}{0.0625}\right)}\right) \cdot \sqrt[3]{\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      12. div-inv 49.1%

          \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)} \cdot \sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)}\right) \cdot \sqrt[3]{\color{blue}{t \cdot \frac{1}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      13. clear-num 49.3%

          \[\leadsto x \cdot \left(\cos \left(\left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)} \cdot \sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)}\right) \cdot \sqrt[3]{t \cdot \color{blue}{\frac{\mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}}}\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

   9. Applied egg-rr 49.3%

      \[\leadsto x \cdot \left(\cos \color{blue}{\left(\left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)} \cdot \sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)}\right) \cdot \sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, z \cdot 2, z\right) \cdot 0.0625\right)}\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

   if 2.00000000000000003e306 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 1.0%

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

   4. Taylor expanded in t around 0 5.0%

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

   5. Taylor expanded in t around 0 12.2%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;x \cdot \left(\cos \left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, 2 \cdot z, z\right) \cdot 0.0625\right)} \cdot \left(\sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, 2 \cdot z, z\right) \cdot 0.0625\right)} \cdot \sqrt[3]{t \cdot \left(\mathsf{fma}\left(y, 2 \cdot z, z\right) \cdot 0.0625\right)}\right)\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 32.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)}{16}\right)\right) \cdot t_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t_1 \cdot \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}\right)}^{3}}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (if (<= (* (* x (cos (/ (* t (* z (+ 1.0 (* y 2.0)))) 16.0))) t_1) 2e+306)
     (* t_1 (* x (cos (/ (pow (cbrt (* t (* z (fma 2.0 y 1.0)))) 3.0) 16.0))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x * cos(((t * (z * (1.0 + (y * 2.0)))) / 16.0))) * t_1) <= 2e+306) {
		tmp = t_1 * (x * cos((pow(cbrt((t * (z * fma(2.0, y, 1.0)))), 3.0) / 16.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(y * 2.0)))) / 16.0))) * t_1) <= 2e+306)
		tmp = Float64(t_1 * Float64(x * cos(Float64((cbrt(Float64(t * Float64(z * fma(2.0, y, 1.0)))) ^ 3.0) / 16.0))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(x * N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 2e+306], N[(t$95$1 * N[(x * N[Cos[N[(N[Power[N[Power[N[(t * N[(z * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 2.00000000000000003e306

   1. Initial program 48.8%

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

      1. add-cube-cbrt 49.1%

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t} \cdot \sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}\right) \cdot \sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      2. pow3 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}\right)}^{3}}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      3. *-commutative 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{\color{blue}{t \cdot \left(\left(y \cdot 2 + 1\right) \cdot z\right)}}\right)}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. *-commutative 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{t \cdot \color{blue}{\left(z \cdot \left(y \cdot 2 + 1\right)\right)}}\right)}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      5. *-commutative 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{t \cdot \left(z \cdot \left(\color{blue}{2 \cdot y} + 1\right)\right)}\right)}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      6. fma-def 49.2%

         \[\leadsto \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{t \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(2, y, 1\right)}\right)}\right)}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   3. Applied egg-rr 49.2%

      \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}\right)}^{3}}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   if 2.00000000000000003e306 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 1.0%

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

   4. Taylor expanded in t around 0 5.0%

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

   5. Taylor expanded in t around 0 12.2%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \cdot \left(x \cdot \cos \left(\frac{{\left(\sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)}\right)}^{3}}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 31.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right) \cdot \cos \left(\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* t (* z (+ 1.0 (* y 2.0)))) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      2e+306)
   (*
    x
    (*
     (cos (* (/ t 16.0) (fma (* 2.0 a) b b)))
     (cos (/ t (/ 16.0 (fma y (* 2.0 z) z))))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos(((t * (z * (1.0 + (y * 2.0)))) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+306) {
		tmp = x * (cos(((t / 16.0) * fma((2.0 * a), b, b))) * cos((t / (16.0 / fma(y, (2.0 * z), z)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(y * 2.0)))) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+306)
		tmp = Float64(x * Float64(cos(Float64(Float64(t / 16.0) * fma(Float64(2.0 * a), b, b))) * cos(Float64(t / Float64(16.0 / fma(y, Float64(2.0 * z), z))))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+306], N[(x * N[(N[Cos[N[(N[(t / 16.0), $MachinePrecision] * N[(N[(2.0 * a), $MachinePrecision] * b + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(t / N[(16.0 / N[(y * N[(2.0 * z), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 2.00000000000000003e306

   1. Initial program 48.8%

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

   3. Simplified 48.8%

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

      1. expm1-log1p-u 48.8%

         \[\leadsto x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(t \cdot \frac{\mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}\right)\right)\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      2. expm1-udef 48.8%

         \[\leadsto x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(t \cdot \frac{\mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}\right)\right)} - 1\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      3. associate-*r/ 48.8%

         \[\leadsto x \cdot \left(\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{t \cdot \mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}\right)}\right)} - 1\right) \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

   5. Applied egg-rr 48.8%

      \[\leadsto x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\frac{t \cdot \mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}\right)\right)} - 1\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]
   6. Step-by-step derivation

      1. expm1-def 48.8%

         \[\leadsto x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{t \cdot \mathsf{fma}\left(y, 2 \cdot z, z\right)}{16}\right)\right)\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]

      2. expm1-log1p 48.8%

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

      3. associate-/l* 49.1%

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

   7. Simplified 49.1%

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

   if 2.00000000000000003e306 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 1.0%

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

   4. Taylor expanded in t around 0 5.0%

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

   5. Taylor expanded in t around 0 12.2%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right) \cdot \cos \left(\frac{t}{\frac{16}{\mathsf{fma}\left(y, 2 \cdot z, z\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 32.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{t_1}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(x \cdot \cos \left(0.0625 \cdot t_1\right)\right) \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* z (+ 1.0 (* y 2.0))))))
   (if (<=
        (*
         (* x (cos (/ t_1 16.0)))
         (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
        2e+306)
     (* (* x (cos (* 0.0625 t_1))) (cos (/ (* b (fma a 2.0 1.0)) (/ 16.0 t))))
     x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (z * (1.0 + (y * 2.0)));
	double tmp;
	if (((x * cos((t_1 / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+306) {
		tmp = (x * cos((0.0625 * t_1))) * cos(((b * fma(a, 2.0, 1.0)) / (16.0 / t)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(z * Float64(1.0 + Float64(y * 2.0))))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(t_1 / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+306)
		tmp = Float64(Float64(x * cos(Float64(0.0625 * t_1))) * cos(Float64(Float64(b * fma(a, 2.0, 1.0)) / Float64(16.0 / t))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(z * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x * N[Cos[N[(t$95$1 / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(x * N[Cos[N[(0.0625 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 2.00000000000000003e306

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

      2. associate-*l* 48.8%

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

      3. cos-neg 48.8%

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

      4. distribute-frac-neg 48.8%

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

      5. distribute-lft-neg-in 48.8%

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

      6. distribute-rgt-neg-out 48.8%

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

      7. associate-*l* 48.8%

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

      8. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in z around inf 48.8%

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

   if 2.00000000000000003e306 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 1.0%

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

   4. Taylor expanded in t around 0 5.0%

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

   5. Taylor expanded in t around 0 12.2%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(x \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)\right)\right)\right) \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \cos \left(\frac{t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (*
          (* x (cos (/ (* t (* z (+ 1.0 (* y 2.0)))) 16.0)))
          (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))))
   (if (<= t_1 2e+306) t_1 x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * cos(((t * (z * (1.0 + (y * 2.0)))) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (t_1 <= 2e+306) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * cos(((t * (z * (1.0d0 + (y * 2.0d0)))) / 16.0d0))) * cos(((t * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))
    if (t_1 <= 2d+306) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.cos(((t * (z * (1.0 + (y * 2.0)))) / 16.0))) * Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (t_1 <= 2e+306) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.cos(((t * (z * (1.0 + (y * 2.0)))) / 16.0))) * math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))
	tmp = 0
	if t_1 <= 2e+306:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(y * 2.0)))) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0)))
	tmp = 0.0
	if (t_1 <= 2e+306)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * cos(((t * (z * (1.0 + (y * 2.0)))) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	tmp = 0.0;
	if (t_1 <= 2e+306)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+306], t$95$1, x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 2.00000000000000003e306

   1. Initial program 48.8%

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

   if 2.00000000000000003e306 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 0.0%

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

   3. Simplified 1.0%

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

   4. Taylor expanded in t around 0 5.0%

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

   5. Taylor expanded in t around 0 12.2%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{t \cdot \left(z \cdot \left(1 + y \cdot 2\right)\right)}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 29.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \left(z \cdot 0.0625\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 6.6e+104)
   (* x (* (cos (* t (* z 0.0625))) (cos (* (* t b) (+ 0.0625 (* a 0.125))))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 6.6e+104) {
		tmp = x * (cos((t * (z * 0.0625))) * cos(((t * b) * (0.0625 + (a * 0.125)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= 6.6d+104) then
        tmp = x * (cos((t * (z * 0.0625d0))) * cos(((t * b) * (0.0625d0 + (a * 0.125d0)))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 6.6e+104) {
		tmp = x * (Math.cos((t * (z * 0.0625))) * Math.cos(((t * b) * (0.0625 + (a * 0.125)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 6.6e+104:
		tmp = x * (math.cos((t * (z * 0.0625))) * math.cos(((t * b) * (0.0625 + (a * 0.125)))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 6.6e+104)
		tmp = Float64(x * Float64(cos(Float64(t * Float64(z * 0.0625))) * cos(Float64(Float64(t * b) * Float64(0.0625 + Float64(a * 0.125))))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 6.6e+104)
		tmp = x * (cos((t * (z * 0.0625))) * cos(((t * b) * (0.0625 + (a * 0.125)))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 6.6e+104], N[(x * N[(N[Cos[N[(t * N[(z * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t * b), $MachinePrecision] * N[(0.0625 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 6.59999999999999969e104

   1. Initial program 32.6%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in y around 0 34.1%

      \[\leadsto x \cdot \left(\color{blue}{\cos \left(0.0625 \cdot \left(t \cdot z\right)\right)} \cdot \cos \left(\frac{t}{16} \cdot \mathsf{fma}\left(2 \cdot a, b, b\right)\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r* 34.1%

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

      2. *-commutative 34.1%

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

      3. associate-*l* 34.1%

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

   6. Simplified 34.1%

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

   7. Taylor expanded in a around 0 34.7%

      \[\leadsto x \cdot \left(\cos \left(t \cdot \left(0.0625 \cdot z\right)\right) \cdot \cos \color{blue}{\left(0.125 \cdot \left(a \cdot \left(t \cdot b\right)\right) + 0.0625 \cdot \left(t \cdot b\right)\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 34.7%

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

      2. associate-*r* 34.8%

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

      3. distribute-rgt-out 34.8%

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

      4. *-commutative 34.8%

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

   9. Simplified 34.8%

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

   if 6.59999999999999969e104 < t

   1. Initial program 3.9%

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

   3. Simplified 3.9%

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

   4. Taylor expanded in t around 0 7.6%

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

   5. Taylor expanded in t around 0 14.9%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \left(z \cdot 0.0625\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 30.8% accurate, 225.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 27.4%

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

3. Simplified 27.9%

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

4. Taylor expanded in t around 0 28.1%

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

5. Taylor expanded in t around 0 30.2%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 30.2%

   \[\leadsto x \]

Developer target: 30.4% accurate, 1.0× 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 2023268 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

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