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

Percentage Accurate: 27.6% → 31.6%

Time: 24.6s

Alternatives: 5

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: 27.6% 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: 31.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+72}:\\ \;\;\;\;\cos \left(t \cdot \left(z \cdot \left({\left(\sqrt[3]{0.0625}\right)}^{3} \cdot \left(1 - y \cdot -2\right)\right)\right)\right) \cdot \left(x\_m \cdot \cos \left(t \cdot \left(\mathsf{fma}\left(2, a, 1\right) \cdot \frac{b}{16}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
        (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
       5e+72)
    (*
     (cos (* t (* z (* (pow (cbrt 0.0625) 3.0) (- 1.0 (* y -2.0))))))
     (* x_m (cos (* t (* (fma 2.0 a 1.0) (/ b 16.0))))))
    x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 5e+72) {
		tmp = cos((t * (z * (pow(cbrt(0.0625), 3.0) * (1.0 - (y * -2.0)))))) * (x_m * cos((t * (fma(2.0, a, 1.0) * (b / 16.0)))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * 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))) <= 5e+72)
		tmp = Float64(cos(Float64(t * Float64(z * Float64((cbrt(0.0625) ^ 3.0) * Float64(1.0 - Float64(y * -2.0)))))) * Float64(x_m * cos(Float64(t * Float64(fma(2.0, a, 1.0) * Float64(b / 16.0))))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 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], 5e+72], N[(N[Cos[N[(t * N[(z * N[(N[Power[N[Power[0.0625, 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[(1.0 - N[(y * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x$95$m * N[Cos[N[(t * N[(N[(2.0 * a + 1.0), $MachinePrecision] * N[(b / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 4.99999999999999992e72

   1. Initial program 47.6%

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

   3. Simplified 47.9%

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

      1. add-cube-cbrt 47.9%

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

      2. pow3 47.8%

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

      3. *-commutative 47.8%

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

      4. div-inv 47.8%

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

      5. metadata-eval 47.8%

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

      6. *-commutative 47.8%

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

      7. fma-define 47.8%

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

      8. *-commutative 47.8%

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

      9. fma-define 47.8%

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

      10. *-commutative 47.8%

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

      11. associate-*r* 47.8%

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

   6. Applied egg-rr 47.8%

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

   7. Taylor expanded in y around -inf 48.4%

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

   if 4.99999999999999992e72 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 13.4%

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

   3. Simplified 14.2%

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

   5. Taylor expanded in a around 0 15.4%

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

      1. *-commutative 15.4%

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

      2. *-commutative 15.4%

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

      3. associate-*r* 15.6%

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

      4. *-commutative 15.6%

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

   7. Simplified 15.6%

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

   8. Taylor expanded in t around 0 20.8%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+72}:\\ \;\;\;\;\cos \left(t \cdot \left(z \cdot \left({\left(\sqrt[3]{0.0625}\right)}^{3} \cdot \left(1 - y \cdot -2\right)\right)\right)\right) \cdot \left(x \cdot \cos \left(t \cdot \left(\mathsf{fma}\left(2, a, 1\right) \cdot \frac{b}{16}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+93}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x\_m \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(2, a, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
        (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
       5e+93)
    (*
     (cos (* (fma y 2.0 1.0) (* z (/ t 16.0))))
     (* x_m (cos (/ (* b (fma 2.0 a 1.0)) (/ 16.0 t)))))
    x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 5e+93) {
		tmp = cos((fma(y, 2.0, 1.0) * (z * (t / 16.0)))) * (x_m * cos(((b * fma(2.0, a, 1.0)) / (16.0 / t))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * 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))) <= 5e+93)
		tmp = Float64(cos(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t / 16.0)))) * Float64(x_m * cos(Float64(Float64(b * fma(2.0, a, 1.0)) / Float64(16.0 / t)))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 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], 5e+93], N[(N[Cos[N[(N[(y * 2.0 + 1.0), $MachinePrecision] * N[(z * N[(t / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x$95$m * N[Cos[N[(N[(b * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5.0000000000000001e93

   1. Initial program 47.7%

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

   3. Simplified 48.2%

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

      1. fma-undefine 48.2%

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

      2. *-commutative 48.2%

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

      3. *-commutative 48.2%

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

      4. div-inv 48.2%

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

      5. metadata-eval 48.2%

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

      6. associate-*r* 48.2%

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

      7. *-commutative 48.2%

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

      8. metadata-eval 48.2%

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

      9. div-inv 48.2%

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

      10. associate-*l* 47.9%

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

      11. clear-num 48.2%

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

      12. un-div-inv 48.3%

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

      13. *-commutative 48.3%

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

      14. *-commutative 48.3%

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

      15. fma-undefine 48.3%

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

   6. Applied egg-rr 48.3%

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

   if 5.0000000000000001e93 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 12.8%

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

   3. Simplified 13.6%

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

   5. Taylor expanded in a around 0 14.8%

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

      1. *-commutative 14.8%

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

      2. *-commutative 14.8%

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

      3. associate-*r* 15.0%

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

      4. *-commutative 15.0%

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

   7. Simplified 15.0%

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

   8. Taylor expanded in t around 0 20.1%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+93}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(2, a, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 1.8 \cdot 10^{+306}:\\ \;\;\;\;\cos \left(\left(t \cdot 0.125\right) \cdot \left(y \cdot z\right)\right) \cdot \left(x\_m \cdot \cos \left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot \frac{b}{16}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
        (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
       1.8e+306)
    (*
     (cos (* (* t 0.125) (* y z)))
     (* x_m (cos (* (fma 2.0 a 1.0) (* t (/ b 16.0))))))
    x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1.8e+306) {
		tmp = cos(((t * 0.125) * (y * z))) * (x_m * cos((fma(2.0, a, 1.0) * (t * (b / 16.0)))));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x_m * 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))) <= 1.8e+306)
		tmp = Float64(cos(Float64(Float64(t * 0.125) * Float64(y * z))) * Float64(x_m * cos(Float64(fma(2.0, a, 1.0) * Float64(t * Float64(b / 16.0))))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 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], 1.8e+306], N[(N[Cos[N[(N[(t * 0.125), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x$95$m * N[Cos[N[(N[(2.0 * a + 1.0), $MachinePrecision] * N[(t * N[(b / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.8000000000000001e306

   1. Initial program 49.7%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in y around inf 50.1%

      \[\leadsto \cos \color{blue}{\left(0.125 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)} \cdot \left(x \cdot \cos \left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot \frac{b}{16}\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* 50.1%

         \[\leadsto \cos \color{blue}{\left(\left(0.125 \cdot t\right) \cdot \left(y \cdot z\right)\right)} \cdot \left(x \cdot \cos \left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot \frac{b}{16}\right)\right)\right) \]

      2. *-commutative 50.1%

         \[\leadsto \cos \left(\left(0.125 \cdot t\right) \cdot \color{blue}{\left(z \cdot y\right)}\right) \cdot \left(x \cdot \cos \left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot \frac{b}{16}\right)\right)\right) \]

   7. Simplified 50.1%

      \[\leadsto \cos \color{blue}{\left(\left(0.125 \cdot t\right) \cdot \left(z \cdot y\right)\right)} \cdot \left(x \cdot \cos \left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot \frac{b}{16}\right)\right)\right) \]

   if 1.8000000000000001e306 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 1.0%

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

   3. Simplified 2.5%

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

   5. Taylor expanded in a around 0 4.0%

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

      1. *-commutative 4.0%

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

      2. *-commutative 4.0%

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

      3. associate-*r* 4.2%

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

      4. *-commutative 4.2%

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

   7. Simplified 4.2%

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

   8. Taylor expanded in t around 0 10.8%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 1.8 \cdot 10^{+306}:\\ \;\;\;\;\cos \left(\left(t \cdot 0.125\right) \cdot \left(y \cdot z\right)\right) \cdot \left(x \cdot \cos \left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot \frac{b}{16}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 31.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot t\_1 \leq 1.8 \cdot 10^{+306}:\\ \;\;\;\;t\_1 \cdot \left(x\_m \cdot \cos \left(\frac{2 \cdot \left(t \cdot \left(y \cdot z\right)\right)}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1)
         1.8e+306)
      (* t_1 (* x_m (cos (/ (* 2.0 (* t (* y z))) 16.0))))
      x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, 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_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1.8e+306) {
		tmp = t_1 * (x_m * cos(((2.0 * (t * (y * z))) / 16.0)));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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 = cos(((t * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))
    if (((x_m * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * t_1) <= 1.8d+306) then
        tmp = t_1 * (x_m * cos(((2.0d0 * (t * (y * z))) / 16.0d0)))
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x_m * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1.8e+306) {
		tmp = t_1 * (x_m * Math.cos(((2.0 * (t * (y * z))) / 16.0)));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))
	tmp = 0
	if ((x_m * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1.8e+306:
		tmp = t_1 * (x_m * math.cos(((2.0 * (t * (y * z))) / 16.0)))
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, 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_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1.8e+306)
		tmp = Float64(t_1 * Float64(x_m * cos(Float64(Float64(2.0 * Float64(t * Float64(y * z))) / 16.0))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	t_1 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	tmp = 0.0;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1.8e+306)
		tmp = t_1 * (x_m * cos(((2.0 * (t * (y * z))) / 16.0)));
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, 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]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1.8e+306], N[(t$95$1 * N[(x$95$m * N[Cos[N[(N[(2.0 * N[(t * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1.8000000000000001e306

   1. Initial program 49.7%

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

   3. Taylor expanded in y around inf 49.8%

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

      1. *-commutative 49.8%

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

   5. Simplified 49.8%

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

   if 1.8000000000000001e306 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 1.0%

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

   3. Simplified 2.5%

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

   5. Taylor expanded in a around 0 4.0%

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

      1. *-commutative 4.0%

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

      2. *-commutative 4.0%

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

      3. associate-*r* 4.2%

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

      4. *-commutative 4.2%

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

   7. Simplified 4.2%

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

   8. Taylor expanded in t around 0 10.8%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 1.8 \cdot 10^{+306}:\\ \;\;\;\;\cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \cdot \left(x \cdot \cos \left(\frac{2 \cdot \left(t \cdot \left(y \cdot z\right)\right)}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 30.7% accurate, 225.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * x_m;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t, a, b)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

3. Simplified 31.7%

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

5. Taylor expanded in a around 0 30.8%

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

   1. *-commutative 30.8%

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

   2. *-commutative 30.8%

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

   3. associate-*r* 30.9%

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

   4. *-commutative 30.9%

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

7. Simplified 30.9%

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

8. Taylor expanded in t around 0 32.8%

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

9. Final simplification 32.8%

   \[\leadsto x \]
10. Add Preprocessing

Developer target: 30.2% 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 2024066 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (* 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))))