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

Percentage Accurate: 28.0% → 31.6%

Time: 18.1s

Alternatives: 5

Speedup: 269.0×

• could not determine a ground truth

  1. x = -4.857230178843377e-67
  2. y = -1.342997655949515e-281
  3. z = 7.751824977268224e+83
  4. t = 1.1118494972116e+304
  5. a = -94.20223596647142
  6. b = 1.8904283921683652e+148

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: 28.0% 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, 0.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := t\_m \cdot \left(0.0625 \cdot z\right)\\ t_2 := t\_1 \cdot \left(2 \cdot y\right)\\ \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-81}:\\ \;\;\;\;\left(x \cdot \cos \left(b \cdot \left(\mathsf{fma}\left(t\_m, a \cdot 2, t\_m\right) \cdot 0.0625\right)\right)\right) \cdot \left(\cos t\_2 \cdot \cos t\_1 - \sin t\_2 \cdot \sin t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (let* ((t_1 (* t_m (* 0.0625 z))) (t_2 (* t_1 (* 2.0 y))))
   (if (<= t_m 4.5e-81)
     (*
      (* x (cos (* b (* (fma t_m (* a 2.0) t_m) 0.0625))))
      (- (* (cos t_2) (cos t_1)) (* (sin t_2) (sin t_1))))
     x)))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double t_1 = t_m * (0.0625 * z);
	double t_2 = t_1 * (2.0 * y);
	double tmp;
	if (t_m <= 4.5e-81) {
		tmp = (x * cos((b * (fma(t_m, (a * 2.0), t_m) * 0.0625)))) * ((cos(t_2) * cos(t_1)) - (sin(t_2) * sin(t_1)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	t_1 = Float64(t_m * Float64(0.0625 * z))
	t_2 = Float64(t_1 * Float64(2.0 * y))
	tmp = 0.0
	if (t_m <= 4.5e-81)
		tmp = Float64(Float64(x * cos(Float64(b * Float64(fma(t_m, Float64(a * 2.0), t_m) * 0.0625)))) * Float64(Float64(cos(t_2) * cos(t_1)) - Float64(sin(t_2) * sin(t_1))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := Block[{t$95$1 = N[(t$95$m * N[(0.0625 * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$m, 4.5e-81], N[(N[(x * N[Cos[N[(b * N[(N[(t$95$m * N[(a * 2.0), $MachinePrecision] + t$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[t$95$2], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 2 regimes

2. if t < 4.5e-81

   1. Initial program 34.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. metadata-eval 34.4

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

   4. Applied egg-rr 34.4%

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

   6. Applied egg-rr 35.3%

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

      1. *-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. cos-sum N/A

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

      4. --lowering--.f64 N/A

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

   8. Applied egg-rr 35.7%

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

   if 4.5e-81 < t

   1. Initial program 6.5%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 10.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 14.8%

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

Alternative 2: 31.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-80}:\\ \;\;\;\;\left(x \cdot \cos \left(z \cdot \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(t\_m \cdot 0.0625\right)\right)\right)\right) \cdot \cos \left(b \cdot \left(0.0625 \cdot \left(t\_m \cdot \mathsf{fma}\left(2, a, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<= t_m 1.2e-80)
   (*
    (* x (cos (* z (* (fma 2.0 y 1.0) (* t_m 0.0625)))))
    (cos (* b (* 0.0625 (* t_m (fma 2.0 a 1.0))))))
   x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (t_m <= 1.2e-80) {
		tmp = (x * cos((z * (fma(2.0, y, 1.0) * (t_m * 0.0625))))) * cos((b * (0.0625 * (t_m * fma(2.0, a, 1.0)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (t_m <= 1.2e-80)
		tmp = Float64(Float64(x * cos(Float64(z * Float64(fma(2.0, y, 1.0) * Float64(t_m * 0.0625))))) * cos(Float64(b * Float64(0.0625 * Float64(t_m * fma(2.0, a, 1.0))))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[t$95$m, 1.2e-80], N[(N[(x * N[Cos[N[(z * N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(t$95$m * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(b * N[(0.0625 * N[(t$95$m * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 1.2e-80

   1. Initial program 34.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. metadata-eval 34.4

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

   4. Applied egg-rr 34.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 35.3

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

   6. Applied egg-rr 35.3%

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

   if 1.2e-80 < t

   1. Initial program 6.5%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 10.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 14.8%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-80}:\\ \;\;\;\;\left(x \cdot \cos \left(z \cdot \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(t \cdot 0.0625\right)\right)\right)\right) \cdot \cos \left(b \cdot \left(0.0625 \cdot \left(t \cdot \mathsf{fma}\left(2, a, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-80}:\\ \;\;\;\;\left(x \cdot \cos \left(b \cdot \left(\mathsf{fma}\left(t\_m, a \cdot 2, t\_m\right) \cdot 0.0625\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot \left(t\_m \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<= t_m 1e-80)
   (*
    (* x (cos (* b (* (fma t_m (* a 2.0) t_m) 0.0625))))
    (cos (* (fma 2.0 y 1.0) (* z (* t_m 0.0625)))))
   x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (t_m <= 1e-80) {
		tmp = (x * cos((b * (fma(t_m, (a * 2.0), t_m) * 0.0625)))) * cos((fma(2.0, y, 1.0) * (z * (t_m * 0.0625))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (t_m <= 1e-80)
		tmp = Float64(Float64(x * cos(Float64(b * Float64(fma(t_m, Float64(a * 2.0), t_m) * 0.0625)))) * cos(Float64(fma(2.0, y, 1.0) * Float64(z * Float64(t_m * 0.0625)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[t$95$m, 1e-80], N[(N[(x * N[Cos[N[(b * N[(N[(t$95$m * N[(a * 2.0), $MachinePrecision] + t$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(z * N[(t$95$m * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 9.99999999999999961e-81

   1. Initial program 34.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. metadata-eval 34.4

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

   4. Applied egg-rr 34.4%

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

   6. Applied egg-rr 35.3%

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

   if 9.99999999999999961e-81 < t

   1. Initial program 6.5%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 10.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 14.8%

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

Alternative 4: 31.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-101}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot \left(t\_m \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \left(b \cdot \left(0.125 \cdot \left(t\_m \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<= t_m 1.25e-101)
   (*
    (cos (* (fma 2.0 y 1.0) (* z (* t_m 0.0625))))
    (* x (cos (* b (* 0.125 (* t_m a))))))
   x))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (t_m <= 1.25e-101) {
		tmp = cos((fma(2.0, y, 1.0) * (z * (t_m * 0.0625)))) * (x * cos((b * (0.125 * (t_m * a)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (t_m <= 1.25e-101)
		tmp = Float64(cos(Float64(fma(2.0, y, 1.0) * Float64(z * Float64(t_m * 0.0625)))) * Float64(x * cos(Float64(b * Float64(0.125 * Float64(t_m * a))))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[t$95$m, 1.25e-101], N[(N[Cos[N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(z * N[(t$95$m * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[Cos[N[(b * N[(0.125 * N[(t$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 1.25e-101

   1. Initial program 33.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. metadata-eval 34.3

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

   4. Applied egg-rr 34.3%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \cos \left(b \cdot \color{blue}{\left(\frac{1}{8} \cdot \left(a \cdot t\right)\right)}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot \left(t \cdot \frac{1}{16}\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(b \cdot \color{blue}{\left(\frac{1}{8} \cdot \left(a \cdot t\right)\right)}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot \left(t \cdot \frac{1}{16}\right)\right)\right) \]

      2. *-lowering-*.f64 34.9

         \[\leadsto \left(x \cdot \cos \left(b \cdot \left(0.125 \cdot \color{blue}{\left(a \cdot t\right)}\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \]

   9. Simplified 34.9%

      \[\leadsto \left(x \cdot \cos \left(b \cdot \color{blue}{\left(0.125 \cdot \left(a \cdot t\right)\right)}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \]

   if 1.25e-101 < t

   1. Initial program 8.6%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 11.9%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 16.5%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-101}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \left(b \cdot \left(0.125 \cdot \left(t \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 31.1% accurate, 269.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in z around 0

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

5. Simplified 26.9%

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

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 29.1%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Developer Target 1: 30.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * cos(((b / 16.0d0) * (t / ((1.0d0 - (a * 2.0d0)) + ((a * 2.0d0) ** 2.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

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

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