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

Percentage Accurate: 27.9% → 31.9%

Time: 25.5s

Alternatives: 6

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: 27.9% 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: 31.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t = |t|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \left(b \cdot 0.0625\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      2e+305)
   (*
    x
    (*
     (cos (* (/ (fma y 2.0 1.0) 16.0) (* z t)))
     (cos (expm1 (log1p (* t (* b 0.0625)))))))
   x))

C

t = abs(t);
b = abs(b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+305) {
		tmp = x * (cos(((fma(y, 2.0, 1.0) / 16.0) * (z * t))) * cos(expm1(log1p((t * (b * 0.0625))))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t = abs(t)
b = abs(b)
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+305)
		tmp = Float64(x * Float64(cos(Float64(Float64(fma(y, 2.0, 1.0) / 16.0) * Float64(z * t))) * cos(expm1(log1p(Float64(t * Float64(b * 0.0625)))))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+305], N[(x * N[(N[Cos[N[(N[(N[(y * 2.0 + 1.0), $MachinePrecision] / 16.0), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(Exp[N[Log[1 + N[(t * N[(b * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $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))) < 1.9999999999999999e305

   1. Initial program 49.0%

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

   3. Simplified 49.1%

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

      1. expm1-log1p-u 46.2%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{16} \cdot \left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right)\right)\right)\right)}\right) \]

      2. *-commutative 46.2%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right) \cdot \frac{t}{16}}\right)\right)\right)\right) \]

      3. *-commutative 46.2%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right)} \cdot \frac{t}{16}\right)\right)\right)\right) \]

      4. div-inv 46.2%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot \color{blue}{\left(t \cdot \frac{1}{16}\right)}\right)\right)\right)\right) \]

      5. metadata-eval 46.2%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot \left(t \cdot \color{blue}{0.0625}\right)\right)\right)\right)\right) \]

   5. Applied egg-rr 46.2%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b \cdot \mathsf{fma}\left(2, a, 1\right)\right) \cdot \left(t \cdot 0.0625\right)\right)\right)\right)}\right) \]

   6. Taylor expanded in a around 0 46.7%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\color{blue}{\log \left(1 + 0.0625 \cdot \left(b \cdot t\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. log1p-def 46.7%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(0.0625 \cdot \left(b \cdot t\right)\right)}\right)\right)\right) \]

      2. *-commutative 46.7%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(b \cdot t\right) \cdot 0.0625}\right)\right)\right)\right) \]

      3. *-commutative 46.7%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(t \cdot b\right)} \cdot 0.0625\right)\right)\right)\right) \]

      4. associate-*r* 46.7%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{t \cdot \left(b \cdot 0.0625\right)}\right)\right)\right)\right) \]

   8. Simplified 46.7%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(t \cdot \left(b \cdot 0.0625\right)\right)}\right)\right)\right) \]

   if 1.9999999999999999e305 < (*.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 2.1%

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

   4. Taylor expanded in t around 0 5.4%

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

   5. Taylor expanded in t around 0 12.6%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \left(b \cdot 0.0625\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 31.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t = |t|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+186}:\\ \;\;\;\;x \cdot \left(\cos \left(b \cdot \left(t \cdot 0.0625\right)\right) \cdot \cos \left(z \cdot \frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      4e+186)
   (*
    x
    (* (cos (* b (* t 0.0625))) (cos (* z (/ (fma y 2.0 1.0) (/ 16.0 t))))))
   x))

C

t = abs(t);
b = abs(b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 4e+186) {
		tmp = x * (cos((b * (t * 0.0625))) * cos((z * (fma(y, 2.0, 1.0) / (16.0 / t)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t = abs(t)
b = abs(b)
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 4e+186)
		tmp = Float64(x * Float64(cos(Float64(b * Float64(t * 0.0625))) * cos(Float64(z * Float64(fma(y, 2.0, 1.0) / Float64(16.0 / t))))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+186], N[(x * N[(N[Cos[N[(b * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(z * N[(N[(y * 2.0 + 1.0), $MachinePrecision] / N[(16.0 / t), $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))) < 3.99999999999999992e186

   1. Initial program 51.4%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in a around 0 51.9%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \color{blue}{\cos \left(0.0625 \cdot \left(b \cdot t\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 51.9%

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

      2. associate-*l* 51.9%

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

      3. *-commutative 51.9%

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

   6. Simplified 51.9%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \color{blue}{\cos \left(b \cdot \left(0.0625 \cdot t\right)\right)}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 51.6%

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

   8. Applied egg-rr 51.6%

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

   if 3.99999999999999992e186 < (*.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 5.2%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in t around 0 9.1%

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

   5. Taylor expanded in t around 0 15.9%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+186}:\\ \;\;\;\;x \cdot \left(\cos \left(b \cdot \left(t \cdot 0.0625\right)\right) \cdot \cos \left(z \cdot \frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 32.0% accurate, 0.4× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      2e+305)
   (*
    x
    (* (cos (/ (fma y 2.0 1.0) (/ (/ 16.0 t) z))) (cos (* b (* t 0.0625)))))
   x))

C

t = abs(t);
b = abs(b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+305) {
		tmp = x * (cos((fma(y, 2.0, 1.0) / ((16.0 / t) / z))) * cos((b * (t * 0.0625))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t = abs(t)
b = abs(b)
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+305)
		tmp = Float64(x * Float64(cos(Float64(fma(y, 2.0, 1.0) / Float64(Float64(16.0 / t) / z))) * cos(Float64(b * Float64(t * 0.0625)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+305], N[(x * N[(N[Cos[N[(N[(y * 2.0 + 1.0), $MachinePrecision] / N[(N[(16.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(b * N[(t * 0.0625), $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))) < 1.9999999999999999e305

   1. Initial program 49.0%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in a around 0 49.7%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \color{blue}{\cos \left(0.0625 \cdot \left(b \cdot t\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 49.7%

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

      2. associate-*l* 49.7%

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

      3. *-commutative 49.7%

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

   6. Simplified 49.7%

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

   if 1.9999999999999999e305 < (*.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 2.1%

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

   4. Taylor expanded in t around 0 5.4%

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

   5. Taylor expanded in t around 0 12.6%

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

4. Final simplification 35.8%

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

Alternative 4: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t = |t|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \left(\cos \left(b \cdot \left(t \cdot 0.0625\right)\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right) + 0.125 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      2e+305)
   (*
    x
    (*
     (cos (* b (* t 0.0625)))
     (cos (+ (* 0.0625 (* z t)) (* 0.125 (* t (* y z)))))))
   x))

C

t = abs(t);
b = abs(b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+305) {
		tmp = x * (cos((b * (t * 0.0625))) * cos(((0.0625 * (z * t)) + (0.125 * (t * (y * z))))));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
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 (((x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos(((t * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))) <= 2d+305) then
        tmp = x * (cos((b * (t * 0.0625d0))) * cos(((0.0625d0 * (z * t)) + (0.125d0 * (t * (y * z))))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

t = Math.abs(t);
b = Math.abs(b);
public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+305) {
		tmp = x * (Math.cos((b * (t * 0.0625))) * Math.cos(((0.0625 * (z * t)) + (0.125 * (t * (y * z))))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

t = abs(t)
b = abs(b)
def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+305:
		tmp = x * (math.cos((b * (t * 0.0625))) * math.cos(((0.0625 * (z * t)) + (0.125 * (t * (y * z))))))
	else:
		tmp = x
	return tmp

Julia

t = abs(t)
b = abs(b)
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+305)
		tmp = Float64(x * Float64(cos(Float64(b * Float64(t * 0.0625))) * cos(Float64(Float64(0.0625 * Float64(z * t)) + Float64(0.125 * Float64(t * Float64(y * z)))))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

t = abs(t)
b = abs(b)
function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+305)
		tmp = x * (cos((b * (t * 0.0625))) * cos(((0.0625 * (z * t)) + (0.125 * (t * (y * z))))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+305], N[(x * N[(N[Cos[N[(b * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[(t * N[(y * z), $MachinePrecision]), $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))) < 1.9999999999999999e305

   1. Initial program 49.0%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in a around 0 49.7%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \color{blue}{\cos \left(0.0625 \cdot \left(b \cdot t\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 49.7%

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

      2. associate-*l* 49.7%

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

      3. *-commutative 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in y around 0 49.2%

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

   if 1.9999999999999999e305 < (*.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 2.1%

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

   4. Taylor expanded in t around 0 5.4%

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

   5. Taylor expanded in t around 0 12.6%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \left(\cos \left(b \cdot \left(t \cdot 0.0625\right)\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right) + 0.125 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 31.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t = |t|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \left(\cos \left(b \cdot \left(t \cdot 0.0625\right)\right) \cdot \cos \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 3.2e-89)
   (* x (* (cos (* b (* t 0.0625))) (cos (* z (* t 0.0625)))))
   x))

C

t = abs(t);
b = abs(b);
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 3.2e-89) {
		tmp = x * (cos((b * (t * 0.0625))) * cos((z * (t * 0.0625))));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
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 <= 3.2d-89) then
        tmp = x * (cos((b * (t * 0.0625d0))) * cos((z * (t * 0.0625d0))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

t = Math.abs(t);
b = Math.abs(b);
public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 3.2e-89) {
		tmp = x * (Math.cos((b * (t * 0.0625))) * Math.cos((z * (t * 0.0625))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

t = abs(t)
b = abs(b)
def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 3.2e-89:
		tmp = x * (math.cos((b * (t * 0.0625))) * math.cos((z * (t * 0.0625))))
	else:
		tmp = x
	return tmp

Julia

t = abs(t)
b = abs(b)
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 3.2e-89)
		tmp = Float64(x * Float64(cos(Float64(b * Float64(t * 0.0625))) * cos(Float64(z * Float64(t * 0.0625)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

t = abs(t)
b = abs(b)
function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 3.2e-89)
		tmp = x * (cos((b * (t * 0.0625))) * cos((z * (t * 0.0625))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 3.2e-89], N[(x * N[(N[Cos[N[(b * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(z * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 3.19999999999999998e-89

   1. Initial program 39.3%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in a around 0 40.4%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \color{blue}{\cos \left(0.0625 \cdot \left(b \cdot t\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 40.4%

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

      2. associate-*l* 40.4%

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

      3. *-commutative 40.4%

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

   6. Simplified 40.4%

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

   7. Taylor expanded in y around 0 41.0%

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

      1. associate-*r* 41.0%

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

   9. Simplified 41.0%

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

   if 3.19999999999999998e-89 < t

   1. Initial program 12.3%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in t around 0 13.1%

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

   5. Taylor expanded in t around 0 17.6%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \left(\cos \left(b \cdot \left(t \cdot 0.0625\right)\right) \cdot \cos \left(z \cdot \left(t \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 31.0% accurate, 225.0× speedup

Math

\[\begin{array}{l} t = |t|\\ b = |b|\\ \\ x \end{array} \]

FPCore

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (x y z t a b) :precision binary64 x)

C

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

Fortran

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
NOTE: b should be positive before calling this function
code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 30.6%

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

3. Simplified 31.5%

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

4. Taylor expanded in t around 0 30.7%

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

5. Taylor expanded in t around 0 32.9%

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

6. Final simplification 32.9%

   \[\leadsto x \]

Developer target: 30.6% 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 2023308 
(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))))