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

Percentage Accurate: 27.5% → 31.5%

Time: 20.0s

Alternatives: 8

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 31.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-142}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t\_m}{16}\right)\right) \cdot \left(x \cdot \cos \left({\left(\sqrt[3]{\left(t\_m \cdot \left(b \cdot 0.0625\right)\right) \cdot \mathsf{fma}\left(2, a, 1\right)}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(b \cdot \left(t\_m + \left(b \cdot -0.5\right) \cdot {t\_m}^{2}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (if (<= t_m 2.25e-142)
   (*
    (cos (* (fma y 2.0 1.0) (* z (/ t_m 16.0))))
    (* x (cos (pow (cbrt (* (* t_m (* b 0.0625)) (fma 2.0 a 1.0))) 3.0))))
   (*
    x
    (cos
     (* -0.125 (* a (expm1 (* b (+ t_m (* (* b -0.5) (pow t_m 2.0)))))))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	double tmp;
	if (t_m <= 2.25e-142) {
		tmp = cos((fma(y, 2.0, 1.0) * (z * (t_m / 16.0)))) * (x * cos(pow(cbrt(((t_m * (b * 0.0625)) * fma(2.0, a, 1.0))), 3.0)));
	} else {
		tmp = x * cos((-0.125 * (a * expm1((b * (t_m + ((b * -0.5) * pow(t_m, 2.0))))))));
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	tmp = 0.0
	if (t_m <= 2.25e-142)
		tmp = Float64(cos(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t_m / 16.0)))) * Float64(x * cos((cbrt(Float64(Float64(t_m * Float64(b * 0.0625)) * fma(2.0, a, 1.0))) ^ 3.0))));
	else
		tmp = Float64(x * cos(Float64(-0.125 * Float64(a * expm1(Float64(b * Float64(t_m + Float64(Float64(b * -0.5) * (t_m ^ 2.0)))))))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := If[LessEqual[t$95$m, 2.25e-142], N[(N[Cos[N[(N[(y * 2.0 + 1.0), $MachinePrecision] * N[(z * N[(t$95$m / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[Cos[N[Power[N[Power[N[(N[(t$95$m * N[(b * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[N[(-0.125 * N[(a * N[(Exp[N[(b * N[(t$95$m + N[(N[(b * -0.5), $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.25000000000000009e-142

   1. Initial program 33.8%

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

   3. Simplified 34.6%

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

      1. add-cube-cbrt 34.6%

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

      2. pow3 34.6%

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

      3. *-commutative 34.6%

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

      4. div-inv 34.6%

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

      5. metadata-eval 34.6%

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

   6. Applied egg-rr 34.6%

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

   if 2.25000000000000009e-142 < t

   1. Initial program 15.8%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in a around inf 16.3%

      \[\leadsto \cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \color{blue}{\left(-0.125 \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 16.3%

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

      2. *-commutative 16.3%

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

   7. Simplified 16.3%

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

   8. Taylor expanded in z around 0 17.1%

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

      1. expm1-log1p-u 15.7%

         \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot t\right)\right)}\right)\right) \]

      2. expm1-undefine 14.7%

         \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot t\right)} - 1\right)}\right)\right) \]

   10. Applied egg-rr 14.7%

       \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot t\right)} - 1\right)}\right)\right) \]
   11. Step-by-step derivation

       1. expm1-define 15.7%

          \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot t\right)\right)}\right)\right) \]

   12. Simplified 15.7%

       \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot t\right)\right)}\right)\right) \]

   13. Taylor expanded in b around 0 20.8%

       \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(\color{blue}{b \cdot \left(t + -0.5 \cdot \left(b \cdot {t}^{2}\right)\right)}\right)\right)\right) \]
   14. Step-by-step derivation

       1. associate-*r* 20.8%

          \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(b \cdot \left(t + \color{blue}{\left(-0.5 \cdot b\right) \cdot {t}^{2}}\right)\right)\right)\right) \]

   15. Simplified 20.8%

       \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(\color{blue}{b \cdot \left(t + \left(-0.5 \cdot b\right) \cdot {t}^{2}\right)}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-142}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x \cdot \cos \left({\left(\sqrt[3]{\left(t \cdot \left(b \cdot 0.0625\right)\right) \cdot \mathsf{fma}\left(2, a, 1\right)}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(b \cdot \left(t + \left(b \cdot -0.5\right) \cdot {t}^{2}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 30.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(t\_m \cdot \left(b + -0.5 \cdot \left(t\_m \cdot {b}^{2}\right)\right)\right)\right)\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (*
  x
  (cos (* -0.125 (* a (expm1 (* t_m (+ b (* -0.5 (* t_m (pow b 2.0)))))))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return x * cos((-0.125 * (a * expm1((t_m * (b + (-0.5 * (t_m * pow(b, 2.0)))))))));
}

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 * Math.cos((-0.125 * (a * Math.expm1((t_m * (b + (-0.5 * (t_m * Math.pow(b, 2.0)))))))));
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	return x * math.cos((-0.125 * (a * math.expm1((t_m * (b + (-0.5 * (t_m * math.pow(b, 2.0)))))))))

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return Float64(x * cos(Float64(-0.125 * Float64(a * expm1(Float64(t_m * Float64(b + Float64(-0.5 * Float64(t_m * (b ^ 2.0))))))))))
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := N[(x * N[Cos[N[(-0.125 * N[(a * N[(Exp[N[(t$95$m * N[(b + N[(-0.5 * N[(t$95$m * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.8%

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

3. Simplified 28.4%

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

5. Taylor expanded in a around inf 28.3%

   \[\leadsto \cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \color{blue}{\left(-0.125 \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r* 28.3%

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

   2. *-commutative 28.3%

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

7. Simplified 28.3%

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

8. Taylor expanded in z around 0 28.8%

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

   1. expm1-log1p-u 27.2%

      \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot t\right)\right)}\right)\right) \]

   2. expm1-undefine 26.5%

      \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot t\right)} - 1\right)}\right)\right) \]

10. Applied egg-rr 26.5%

    \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot t\right)} - 1\right)}\right)\right) \]
11. Step-by-step derivation

    1. expm1-define 27.2%

       \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot t\right)\right)}\right)\right) \]

12. Simplified 27.2%

    \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot t\right)\right)}\right)\right) \]

13. Taylor expanded in t around 0 30.6%

    \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(\color{blue}{t \cdot \left(b + -0.5 \cdot \left({b}^{2} \cdot t\right)\right)}\right)\right)\right) \]

14. Final simplification 30.6%

    \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(t \cdot \left(b + -0.5 \cdot \left(t \cdot {b}^{2}\right)\right)\right)\right)\right) \]
15. Add Preprocessing

Alternative 3: 31.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(b \cdot \left(t\_m + \left(b \cdot -0.5\right) \cdot {t\_m}^{2}\right)\right)\right)\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (*
  x
  (cos (* -0.125 (* a (expm1 (* b (+ t_m (* (* b -0.5) (pow t_m 2.0))))))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return x * cos((-0.125 * (a * expm1((b * (t_m + ((b * -0.5) * pow(t_m, 2.0))))))));
}

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 * Math.cos((-0.125 * (a * Math.expm1((b * (t_m + ((b * -0.5) * Math.pow(t_m, 2.0))))))));
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	return x * math.cos((-0.125 * (a * math.expm1((b * (t_m + ((b * -0.5) * math.pow(t_m, 2.0))))))))

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return Float64(x * cos(Float64(-0.125 * Float64(a * expm1(Float64(b * Float64(t_m + Float64(Float64(b * -0.5) * (t_m ^ 2.0)))))))))
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := N[(x * N[Cos[N[(-0.125 * N[(a * N[(Exp[N[(b * N[(t$95$m + N[(N[(b * -0.5), $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.8%

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

3. Simplified 28.4%

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

5. Taylor expanded in a around inf 28.3%

   \[\leadsto \cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)\right) \cdot \left(x \cdot \cos \color{blue}{\left(-0.125 \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r* 28.3%

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

   2. *-commutative 28.3%

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

7. Simplified 28.3%

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

8. Taylor expanded in z around 0 28.8%

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

   1. expm1-log1p-u 27.2%

      \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot t\right)\right)}\right)\right) \]

   2. expm1-undefine 26.5%

      \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot t\right)} - 1\right)}\right)\right) \]

10. Applied egg-rr 26.5%

    \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot t\right)} - 1\right)}\right)\right) \]
11. Step-by-step derivation

    1. expm1-define 27.2%

       \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot t\right)\right)}\right)\right) \]

12. Simplified 27.2%

    \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot t\right)\right)}\right)\right) \]

13. Taylor expanded in b around 0 30.8%

    \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(\color{blue}{b \cdot \left(t + -0.5 \cdot \left(b \cdot {t}^{2}\right)\right)}\right)\right)\right) \]
14. Step-by-step derivation

    1. associate-*r* 30.8%

       \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(b \cdot \left(t + \color{blue}{\left(-0.5 \cdot b\right) \cdot {t}^{2}}\right)\right)\right)\right) \]

15. Simplified 30.8%

    \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(\color{blue}{b \cdot \left(t + \left(-0.5 \cdot b\right) \cdot {t}^{2}\right)}\right)\right)\right) \]

16. Final simplification 30.8%

    \[\leadsto x \cdot \cos \left(-0.125 \cdot \left(a \cdot \mathsf{expm1}\left(b \cdot \left(t + \left(b \cdot -0.5\right) \cdot {t}^{2}\right)\right)\right)\right) \]
17. Add Preprocessing

Alternative 4: 29.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ x \cdot \left(\sqrt[3]{{\left(1 + \cos \left(t\_m \cdot \left(b \cdot -0.0625\right)\right)\right)}^{3}} + -1\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (* x (+ (cbrt (pow (+ 1.0 (cos (* t_m (* b -0.0625)))) 3.0)) -1.0)))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return x * (cbrt(pow((1.0 + cos((t_m * (b * -0.0625)))), 3.0)) + -1.0);
}

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 * (Math.cbrt(Math.pow((1.0 + Math.cos((t_m * (b * -0.0625)))), 3.0)) + -1.0);
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return Float64(x * Float64(cbrt((Float64(1.0 + cos(Float64(t_m * Float64(b * -0.0625)))) ^ 3.0)) + -1.0))
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := N[(x * N[(N[Power[N[Power[N[(1.0 + N[Cos[N[(t$95$m * N[(b * -0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.8%

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

3. Simplified 28.4%

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

5. Taylor expanded in z around 0 29.1%

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

6. Taylor expanded in a around 0 30.2%

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

   1. *-commutative 30.2%

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

   2. associate-*r* 30.2%

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

8. Simplified 30.2%

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

   1. expm1-log1p-u 30.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)\right)\right)} \cdot x \]

   2. expm1-undefine 30.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)\right)} - 1\right)} \cdot x \]

   3. associate-*l* 30.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(-0.0625 \cdot \left(b \cdot t\right)\right)}\right)} - 1\right) \cdot x \]

10. Applied egg-rr 30.2%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right)} - 1\right)} \cdot x \]
11. Step-by-step derivation

    1. add-cbrt-cube 30.2%

       \[\leadsto \left(\color{blue}{\sqrt[3]{\left(e^{\mathsf{log1p}\left(\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right)}\right) \cdot e^{\mathsf{log1p}\left(\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right)}}} - 1\right) \cdot x \]

    2. pow3 30.2%

       \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(e^{\mathsf{log1p}\left(\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right)}\right)}^{3}}} - 1\right) \cdot x \]

    3. log1p-undefine 30.2%

       \[\leadsto \left(\sqrt[3]{{\left(e^{\color{blue}{\log \left(1 + \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right)}}\right)}^{3}} - 1\right) \cdot x \]

    4. rem-exp-log 30.2%

       \[\leadsto \left(\sqrt[3]{{\color{blue}{\left(1 + \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right)}}^{3}} - 1\right) \cdot x \]

    5. associate-*r* 30.2%

       \[\leadsto \left(\sqrt[3]{{\left(1 + \cos \color{blue}{\left(\left(-0.0625 \cdot b\right) \cdot t\right)}\right)}^{3}} - 1\right) \cdot x \]

12. Applied egg-rr 30.2%

    \[\leadsto \left(\color{blue}{\sqrt[3]{{\left(1 + \cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)\right)}^{3}}} - 1\right) \cdot x \]

13. Final simplification 30.2%

    \[\leadsto x \cdot \left(\sqrt[3]{{\left(1 + \cos \left(t \cdot \left(b \cdot -0.0625\right)\right)\right)}^{3}} + -1\right) \]
14. Add Preprocessing

Alternative 5: 29.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ x \cdot {\left(\sqrt[3]{\cos \left(-0.0625 \cdot \left(t\_m \cdot b\right)\right)}\right)}^{3} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m a b)
 :precision binary64
 (* x (pow (cbrt (cos (* -0.0625 (* t_m b)))) 3.0)))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m, double a, double b) {
	return x * pow(cbrt(cos((-0.0625 * (t_m * b)))), 3.0);
}

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 * Math.pow(Math.cbrt(Math.cos((-0.0625 * (t_m * b)))), 3.0);
}

Julia

t_m = abs(t)
function code(x, y, z, t_m, a, b)
	return Float64(x * (cbrt(cos(Float64(-0.0625 * Float64(t_m * b)))) ^ 3.0))
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b_] := N[(x * N[Power[N[Power[N[Cos[N[(-0.0625 * N[(t$95$m * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.8%

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

3. Simplified 28.4%

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

5. Taylor expanded in z around 0 29.1%

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

6. Taylor expanded in a around 0 30.2%

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

   1. *-commutative 30.2%

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

   2. associate-*r* 30.2%

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

8. Simplified 30.2%

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

   1. add-cube-cbrt 30.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)} \cdot \sqrt[3]{\cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)}\right) \cdot \sqrt[3]{\cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)}\right)} \cdot x \]

   2. pow3 30.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)}\right)}^{3}} \cdot x \]

   3. associate-*l* 30.2%

      \[\leadsto {\left(\sqrt[3]{\cos \color{blue}{\left(-0.0625 \cdot \left(b \cdot t\right)\right)}}\right)}^{3} \cdot x \]

10. Applied egg-rr 30.2%

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)}\right)}^{3}} \cdot x \]

11. Final simplification 30.2%

    \[\leadsto x \cdot {\left(\sqrt[3]{\cos \left(-0.0625 \cdot \left(t \cdot b\right)\right)}\right)}^{3} \]
12. Add Preprocessing

Alternative 6: 29.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ x \cdot \left(\left(1 + \cos \left(t\_m \cdot \left(b \cdot -0.0625\right)\right)\right) + -1\right) \end{array} \]

FPCore

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

C

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

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 * ((1.0d0 + cos((t_m * (b * (-0.0625d0))))) + (-1.0d0))
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 * ((1.0 + Math.cos((t_m * (b * -0.0625)))) + -1.0);
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m, a, b):
	return x * ((1.0 + math.cos((t_m * (b * -0.0625)))) + -1.0)

Julia

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

MATLAB

t_m = abs(t);
function tmp = code(x, y, z, t_m, a, b)
	tmp = x * ((1.0 + cos((t_m * (b * -0.0625)))) + -1.0);
end

Wolfram

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

Derivation

1. Initial program 27.8%

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

3. Simplified 28.4%

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

5. Taylor expanded in z around 0 29.1%

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

6. Taylor expanded in a around 0 30.2%

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

   1. *-commutative 30.2%

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

   2. associate-*r* 30.2%

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

8. Simplified 30.2%

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

   1. expm1-log1p-u 30.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)\right)\right)} \cdot x \]

   2. expm1-undefine 30.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)\right)} - 1\right)} \cdot x \]

   3. associate-*l* 30.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(-0.0625 \cdot \left(b \cdot t\right)\right)}\right)} - 1\right) \cdot x \]

10. Applied egg-rr 30.2%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right)} - 1\right)} \cdot x \]
11. Step-by-step derivation

    1. log1p-undefine 30.2%

       \[\leadsto \left(e^{\color{blue}{\log \left(1 + \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right)}} - 1\right) \cdot x \]

    2. rem-exp-log 30.2%

       \[\leadsto \left(\color{blue}{\left(1 + \cos \left(-0.0625 \cdot \left(b \cdot t\right)\right)\right)} - 1\right) \cdot x \]

    3. associate-*r* 30.2%

       \[\leadsto \left(\left(1 + \cos \color{blue}{\left(\left(-0.0625 \cdot b\right) \cdot t\right)}\right) - 1\right) \cdot x \]

12. Applied egg-rr 30.2%

    \[\leadsto \left(\color{blue}{\left(1 + \cos \left(\left(-0.0625 \cdot b\right) \cdot t\right)\right)} - 1\right) \cdot x \]

13. Final simplification 30.2%

    \[\leadsto x \cdot \left(\left(1 + \cos \left(t \cdot \left(b \cdot -0.0625\right)\right)\right) + -1\right) \]
14. Add Preprocessing

Alternative 7: 29.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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 * cos((t_m * (b * (-0.0625d0))))
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 * Math.cos((t_m * (b * -0.0625)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.8%

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

3. Simplified 28.4%

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

5. Taylor expanded in z around 0 29.1%

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

6. Taylor expanded in a around 0 30.2%

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

   1. *-commutative 30.2%

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

   2. associate-*r* 30.2%

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

8. Simplified 30.2%

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

9. Final simplification 30.2%

   \[\leadsto x \cdot \cos \left(t \cdot \left(b \cdot -0.0625\right)\right) \]
10. Add Preprocessing

Alternative 8: 31.0% accurate, 225.0× 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 27.8%

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

   1. associate-*l* 27.8%

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

   2. *-commutative 27.8%

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

   3. *-commutative 27.8%

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

   4. associate-/l* 27.8%

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

   5. fma-define 27.8%

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

   6. associate-/l* 27.8%

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

   7. fma-define 27.8%

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

3. Simplified 27.8%

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

5. Taylor expanded in t around 0 29.6%

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

Developer Target 1: 30.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024128 
(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))))