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

Percentage Accurate: 27.5% → 31.9%

Time: 19.2s

Alternatives: 11

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: 31.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x\_m \cdot \cos \left({\left(\sqrt[3]{\frac{t \cdot b}{16} \cdot \mathsf{fma}\left(2, a, 1\right)}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(e^{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625} + -1\right)\\ \end{array} \end{array} \]

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 (/ (* (* z (+ 1.0 (* y 2.0))) t) 16.0)))
        (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
       4e+298)
    (*
     (cos (* (fma y 2.0 1.0) (* z (/ t 16.0))))
     (* x_m (cos (pow (cbrt (* (/ (* t b) 16.0) (fma 2.0 a 1.0))) 3.0))))
    (* x_m (+ (exp (+ (log 2.0) (* (pow (* t b) 2.0) -0.0009765625))) -1.0)))))

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((((z * (1.0 + (y * 2.0))) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 4e+298) {
		tmp = cos((fma(y, 2.0, 1.0) * (z * (t / 16.0)))) * (x_m * cos(pow(cbrt((((t * b) / 16.0) * fma(2.0, a, 1.0))), 3.0)));
	} else {
		tmp = x_m * (exp((log(2.0) + (pow((t * b), 2.0) * -0.0009765625))) + -1.0);
	}
	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(z * Float64(1.0 + Float64(y * 2.0))) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 4e+298)
		tmp = Float64(cos(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t / 16.0)))) * Float64(x_m * cos((cbrt(Float64(Float64(Float64(t * b) / 16.0) * fma(2.0, a, 1.0))) ^ 3.0))));
	else
		tmp = Float64(x_m * Float64(exp(Float64(log(2.0) + Float64((Float64(t * b) ^ 2.0) * -0.0009765625))) + -1.0));
	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[(z * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+298], 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[Power[N[Power[N[(N[(N[(t * b), $MachinePrecision] / 16.0), $MachinePrecision] * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[Power[N[(t * b), $MachinePrecision], 2.0], $MachinePrecision] * -0.0009765625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $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)))) < 3.9999999999999998e298

   1. Initial program 45.5%

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

   3. Simplified 46.0%

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

      1. add-cube-cbrt 46.6%

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

      2. pow3 46.8%

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

      3. *-commutative 46.8%

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

      4. associate-*r/ 46.8%

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

   6. Applied egg-rr 46.8%

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

   if 3.9999999999999998e298 < (*.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 0.2%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in z around 0 5.0%

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

   6. Taylor expanded in a around 0 6.6%

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

      1. expm1-log1p-u 6.6%

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

      2. expm1-undefine 6.6%

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

      3. *-commutative 6.6%

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

      4. *-commutative 6.6%

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

   8. Applied egg-rr 6.6%

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

   9. Taylor expanded in t around 0 8.9%

      \[\leadsto x \cdot \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left({b}^{2} \cdot {t}^{2}\right)}} - 1\right) \]
   10. Step-by-step derivation

       1. *-commutative 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left({b}^{2} \cdot {t}^{2}\right) \cdot -0.0009765625}} - 1\right) \]

       2. *-commutative 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left({t}^{2} \cdot {b}^{2}\right)} \cdot -0.0009765625} - 1\right) \]

       3. unpow2 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \left(\color{blue}{\left(t \cdot t\right)} \cdot {b}^{2}\right) \cdot -0.0009765625} - 1\right) \]

       4. unpow2 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \left(\left(t \cdot t\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -0.0009765625} - 1\right) \]

       5. swap-sqr 10.8%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left(\left(t \cdot b\right) \cdot \left(t \cdot b\right)\right)} \cdot -0.0009765625} - 1\right) \]

       6. unpow2 10.8%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{{\left(t \cdot b\right)}^{2}} \cdot -0.0009765625} - 1\right) \]

   11. Simplified 10.8%

       \[\leadsto x \cdot \left(e^{\color{blue}{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625}} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x \cdot \cos \left({\left(\sqrt[3]{\frac{t \cdot b}{16} \cdot \mathsf{fma}\left(2, a, 1\right)}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x\_m \cdot \cos \left({\left(\sqrt[3]{t \cdot b} \cdot \sqrt[3]{0.0625}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(e^{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625} + -1\right)\\ \end{array} \end{array} \]

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 (/ (* (* z (+ 1.0 (* y 2.0))) t) 16.0)))
        (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
       4e+298)
    (*
     (cos (* (fma y 2.0 1.0) (* z (/ t 16.0))))
     (* x_m (cos (pow (* (cbrt (* t b)) (cbrt 0.0625)) 3.0))))
    (* x_m (+ (exp (+ (log 2.0) (* (pow (* t b) 2.0) -0.0009765625))) -1.0)))))

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((((z * (1.0 + (y * 2.0))) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 4e+298) {
		tmp = cos((fma(y, 2.0, 1.0) * (z * (t / 16.0)))) * (x_m * cos(pow((cbrt((t * b)) * cbrt(0.0625)), 3.0)));
	} else {
		tmp = x_m * (exp((log(2.0) + (pow((t * b), 2.0) * -0.0009765625))) + -1.0);
	}
	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(z * Float64(1.0 + Float64(y * 2.0))) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 4e+298)
		tmp = Float64(cos(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t / 16.0)))) * Float64(x_m * cos((Float64(cbrt(Float64(t * b)) * cbrt(0.0625)) ^ 3.0))));
	else
		tmp = Float64(x_m * Float64(exp(Float64(log(2.0) + Float64((Float64(t * b) ^ 2.0) * -0.0009765625))) + -1.0));
	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[(z * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+298], 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[Power[N[(N[Power[N[(t * b), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.0625, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[Power[N[(t * b), $MachinePrecision], 2.0], $MachinePrecision] * -0.0009765625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $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)))) < 3.9999999999999998e298

   1. Initial program 45.5%

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

   3. Simplified 46.0%

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

      1. add-cube-cbrt 46.6%

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

      2. pow3 46.8%

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

      3. *-commutative 46.8%

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

      4. associate-*r/ 46.8%

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

   6. Applied egg-rr 46.8%

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

   7. Taylor expanded in a around 0 46.6%

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

   if 3.9999999999999998e298 < (*.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 0.2%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in z around 0 5.0%

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

   6. Taylor expanded in a around 0 6.6%

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

      1. expm1-log1p-u 6.6%

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

      2. expm1-undefine 6.6%

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

      3. *-commutative 6.6%

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

      4. *-commutative 6.6%

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

   8. Applied egg-rr 6.6%

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

   9. Taylor expanded in t around 0 8.9%

      \[\leadsto x \cdot \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left({b}^{2} \cdot {t}^{2}\right)}} - 1\right) \]
   10. Step-by-step derivation

       1. *-commutative 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left({b}^{2} \cdot {t}^{2}\right) \cdot -0.0009765625}} - 1\right) \]

       2. *-commutative 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left({t}^{2} \cdot {b}^{2}\right)} \cdot -0.0009765625} - 1\right) \]

       3. unpow2 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \left(\color{blue}{\left(t \cdot t\right)} \cdot {b}^{2}\right) \cdot -0.0009765625} - 1\right) \]

       4. unpow2 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \left(\left(t \cdot t\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -0.0009765625} - 1\right) \]

       5. swap-sqr 10.8%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left(\left(t \cdot b\right) \cdot \left(t \cdot b\right)\right)} \cdot -0.0009765625} - 1\right) \]

       6. unpow2 10.8%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{{\left(t \cdot b\right)}^{2}} \cdot -0.0009765625} - 1\right) \]

   11. Simplified 10.8%

       \[\leadsto x \cdot \left(e^{\color{blue}{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625}} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x \cdot \cos \left({\left(\sqrt[3]{t \cdot b} \cdot \sqrt[3]{0.0625}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x\_m \cdot \cos \left(b \cdot \left(t \cdot {\left(\sqrt[3]{0.0625}\right)}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(e^{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625} + -1\right)\\ \end{array} \end{array} \]

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 (/ (* (* z (+ 1.0 (* y 2.0))) t) 16.0)))
        (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
       4e+298)
    (*
     (cos (* (fma y 2.0 1.0) (* z (/ t 16.0))))
     (* x_m (cos (* b (* t (pow (cbrt 0.0625) 3.0))))))
    (* x_m (+ (exp (+ (log 2.0) (* (pow (* t b) 2.0) -0.0009765625))) -1.0)))))

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((((z * (1.0 + (y * 2.0))) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 4e+298) {
		tmp = cos((fma(y, 2.0, 1.0) * (z * (t / 16.0)))) * (x_m * cos((b * (t * pow(cbrt(0.0625), 3.0)))));
	} else {
		tmp = x_m * (exp((log(2.0) + (pow((t * b), 2.0) * -0.0009765625))) + -1.0);
	}
	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(z * Float64(1.0 + Float64(y * 2.0))) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 4e+298)
		tmp = Float64(cos(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t / 16.0)))) * Float64(x_m * cos(Float64(b * Float64(t * (cbrt(0.0625) ^ 3.0))))));
	else
		tmp = Float64(x_m * Float64(exp(Float64(log(2.0) + Float64((Float64(t * b) ^ 2.0) * -0.0009765625))) + -1.0));
	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[(z * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+298], 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[(b * N[(t * N[Power[N[Power[0.0625, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[Power[N[(t * b), $MachinePrecision], 2.0], $MachinePrecision] * -0.0009765625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $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)))) < 3.9999999999999998e298

   1. Initial program 45.5%

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

   3. Simplified 46.0%

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

      1. add-cube-cbrt 46.6%

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

      2. pow3 46.8%

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

      3. *-commutative 46.8%

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

      4. associate-*r/ 46.8%

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

   6. Applied egg-rr 46.8%

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

   7. Taylor expanded in a around 0 46.2%

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

   if 3.9999999999999998e298 < (*.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 0.2%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in z around 0 5.0%

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

   6. Taylor expanded in a around 0 6.6%

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

      1. expm1-log1p-u 6.6%

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

      2. expm1-undefine 6.6%

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

      3. *-commutative 6.6%

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

      4. *-commutative 6.6%

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

   8. Applied egg-rr 6.6%

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

   9. Taylor expanded in t around 0 8.9%

      \[\leadsto x \cdot \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left({b}^{2} \cdot {t}^{2}\right)}} - 1\right) \]
   10. Step-by-step derivation

       1. *-commutative 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left({b}^{2} \cdot {t}^{2}\right) \cdot -0.0009765625}} - 1\right) \]

       2. *-commutative 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left({t}^{2} \cdot {b}^{2}\right)} \cdot -0.0009765625} - 1\right) \]

       3. unpow2 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \left(\color{blue}{\left(t \cdot t\right)} \cdot {b}^{2}\right) \cdot -0.0009765625} - 1\right) \]

       4. unpow2 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \left(\left(t \cdot t\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -0.0009765625} - 1\right) \]

       5. swap-sqr 10.8%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left(\left(t \cdot b\right) \cdot \left(t \cdot b\right)\right)} \cdot -0.0009765625} - 1\right) \]

       6. unpow2 10.8%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{{\left(t \cdot b\right)}^{2}} \cdot -0.0009765625} - 1\right) \]

   11. Simplified 10.8%

       \[\leadsto x \cdot \left(e^{\color{blue}{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625}} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x \cdot \cos \left(b \cdot \left(t \cdot {\left(\sqrt[3]{0.0625}\right)}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 31.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x\_m \cdot \cos \left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot \frac{b}{16}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(e^{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625} + -1\right)\\ \end{array} \end{array} \]

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 (/ (* (* z (+ 1.0 (* y 2.0))) t) 16.0)))
        (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
       2e+259)
    (*
     (cos (* (fma y 2.0 1.0) (* z (/ t 16.0))))
     (* x_m (cos (* (fma 2.0 a 1.0) (* t (/ b 16.0))))))
    (* x_m (+ (exp (+ (log 2.0) (* (pow (* t b) 2.0) -0.0009765625))) -1.0)))))

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((((z * (1.0 + (y * 2.0))) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+259) {
		tmp = cos((fma(y, 2.0, 1.0) * (z * (t / 16.0)))) * (x_m * cos((fma(2.0, a, 1.0) * (t * (b / 16.0)))));
	} else {
		tmp = x_m * (exp((log(2.0) + (pow((t * b), 2.0) * -0.0009765625))) + -1.0);
	}
	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(z * Float64(1.0 + Float64(y * 2.0))) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+259)
		tmp = Float64(cos(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t / 16.0)))) * Float64(x_m * cos(Float64(fma(2.0, a, 1.0) * Float64(t * Float64(b / 16.0))))));
	else
		tmp = Float64(x_m * Float64(exp(Float64(log(2.0) + Float64((Float64(t * b) ^ 2.0) * -0.0009765625))) + -1.0));
	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[(z * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+259], 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[(2.0 * a + 1.0), $MachinePrecision] * N[(t * N[(b / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[Power[N[(t * b), $MachinePrecision], 2.0], $MachinePrecision] * -0.0009765625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $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)))) < 2e259

   1. Initial program 44.6%

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

   3. Simplified 45.0%

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

   if 2e259 < (*.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 4.0%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in z around 0 7.8%

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

   6. Taylor expanded in a around 0 9.4%

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

      1. expm1-log1p-u 9.4%

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

      2. expm1-undefine 9.4%

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

      3. *-commutative 9.4%

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

      4. *-commutative 9.4%

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

   8. Applied egg-rr 9.4%

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

   9. Taylor expanded in t around 0 11.6%

      \[\leadsto x \cdot \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left({b}^{2} \cdot {t}^{2}\right)}} - 1\right) \]
   10. Step-by-step derivation

       1. *-commutative 11.6%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left({b}^{2} \cdot {t}^{2}\right) \cdot -0.0009765625}} - 1\right) \]

       2. *-commutative 11.6%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left({t}^{2} \cdot {b}^{2}\right)} \cdot -0.0009765625} - 1\right) \]

       3. unpow2 11.6%

          \[\leadsto x \cdot \left(e^{\log 2 + \left(\color{blue}{\left(t \cdot t\right)} \cdot {b}^{2}\right) \cdot -0.0009765625} - 1\right) \]

       4. unpow2 11.6%

          \[\leadsto x \cdot \left(e^{\log 2 + \left(\left(t \cdot t\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -0.0009765625} - 1\right) \]

       5. swap-sqr 13.3%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left(\left(t \cdot b\right) \cdot \left(t \cdot b\right)\right)} \cdot -0.0009765625} - 1\right) \]

       6. unpow2 13.3%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{{\left(t \cdot b\right)}^{2}} \cdot -0.0009765625} - 1\right) \]

   11. Simplified 13.3%

       \[\leadsto x \cdot \left(e^{\color{blue}{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625}} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x \cdot \cos \left(\mathsf{fma}\left(2, a, 1\right) \cdot \left(t \cdot \frac{b}{16}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+298}:\\ \;\;\;\;t\_1 \cdot \cos \left(\frac{1}{\frac{\frac{16}{t}}{b \cdot \mathsf{fma}\left(2, a, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(e^{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625} + -1\right)\\ \end{array} \end{array} \end{array} \]

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 (* x_m (cos (/ (* (* z (+ 1.0 (* y 2.0))) t) 16.0)))))
   (*
    x_s
    (if (<= (* t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))) 4e+298)
      (* t_1 (cos (/ 1.0 (/ (/ 16.0 t) (* b (fma 2.0 a 1.0))))))
      (*
       x_m
       (+ (exp (+ (log 2.0) (* (pow (* t b) 2.0) -0.0009765625))) -1.0))))))

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 = x_m * cos((((z * (1.0 + (y * 2.0))) * t) / 16.0));
	double tmp;
	if ((t_1 * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 4e+298) {
		tmp = t_1 * cos((1.0 / ((16.0 / t) / (b * fma(2.0, a, 1.0)))));
	} else {
		tmp = x_m * (exp((log(2.0) + (pow((t * b), 2.0) * -0.0009765625))) + -1.0);
	}
	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 = Float64(x_m * cos(Float64(Float64(Float64(z * Float64(1.0 + Float64(y * 2.0))) * t) / 16.0)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 4e+298)
		tmp = Float64(t_1 * cos(Float64(1.0 / Float64(Float64(16.0 / t) / Float64(b * fma(2.0, a, 1.0))))));
	else
		tmp = Float64(x_m * Float64(exp(Float64(log(2.0) + Float64((Float64(t * b) ^ 2.0) * -0.0009765625))) + -1.0));
	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_] := Block[{t$95$1 = N[(x$95$m * N[Cos[N[(N[(N[(z * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(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]), $MachinePrecision], 4e+298], N[(t$95$1 * N[Cos[N[(1.0 / N[(N[(16.0 / t), $MachinePrecision] / N[(b * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[Power[N[(t * b), $MachinePrecision], 2.0], $MachinePrecision] * -0.0009765625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $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)))) < 3.9999999999999998e298

   1. Initial program 45.5%

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

      1. add-cube-cbrt 45.4%

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

      2. pow3 45.6%

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

      3. *-commutative 45.6%

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

   4. Applied egg-rr 45.6%

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

      1. clear-num 45.6%

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

      2. inv-pow 45.6%

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

      3. *-commutative 45.6%

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

      4. *-commutative 45.6%

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

      5. unpow3 45.4%

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

      6. add-cube-cbrt 45.6%

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

      7. fma-undefine 45.6%

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

   6. Applied egg-rr 45.6%

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

      1. unpow-1 45.6%

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

      2. associate-/r* 45.9%

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

   8. Simplified 45.9%

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

   if 3.9999999999999998e298 < (*.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 0.2%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in z around 0 5.0%

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

   6. Taylor expanded in a around 0 6.6%

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

      1. expm1-log1p-u 6.6%

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

      2. expm1-undefine 6.6%

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

      3. *-commutative 6.6%

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

      4. *-commutative 6.6%

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

   8. Applied egg-rr 6.6%

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

   9. Taylor expanded in t around 0 8.9%

      \[\leadsto x \cdot \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left({b}^{2} \cdot {t}^{2}\right)}} - 1\right) \]
   10. Step-by-step derivation

       1. *-commutative 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left({b}^{2} \cdot {t}^{2}\right) \cdot -0.0009765625}} - 1\right) \]

       2. *-commutative 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left({t}^{2} \cdot {b}^{2}\right)} \cdot -0.0009765625} - 1\right) \]

       3. unpow2 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \left(\color{blue}{\left(t \cdot t\right)} \cdot {b}^{2}\right) \cdot -0.0009765625} - 1\right) \]

       4. unpow2 8.9%

          \[\leadsto x \cdot \left(e^{\log 2 + \left(\left(t \cdot t\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -0.0009765625} - 1\right) \]

       5. swap-sqr 10.8%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{\left(\left(t \cdot b\right) \cdot \left(t \cdot b\right)\right)} \cdot -0.0009765625} - 1\right) \]

       6. unpow2 10.8%

          \[\leadsto x \cdot \left(e^{\log 2 + \color{blue}{{\left(t \cdot b\right)}^{2}} \cdot -0.0009765625} - 1\right) \]

   11. Simplified 10.8%

       \[\leadsto x \cdot \left(e^{\color{blue}{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625}} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(z \cdot \left(1 + y \cdot 2\right)\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{1}{\frac{\frac{16}{t}}{b \cdot \mathsf{fma}\left(2, a, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + {\left(t \cdot b\right)}^{2} \cdot -0.0009765625} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-37}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x\_m \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(1 - a \cdot -2\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (<= t 1.7e-37)
    (*
     (cos (* (fma y 2.0 1.0) (* z (/ t 16.0))))
     (* x_m (cos (* 0.0625 (* b (* t (- 1.0 (* a -2.0))))))))
    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 (t <= 1.7e-37) {
		tmp = cos((fma(y, 2.0, 1.0) * (z * (t / 16.0)))) * (x_m * cos((0.0625 * (b * (t * (1.0 - (a * -2.0)))))));
	} 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 (t <= 1.7e-37)
		tmp = Float64(cos(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t / 16.0)))) * Float64(x_m * cos(Float64(0.0625 * Float64(b * Float64(t * Float64(1.0 - Float64(a * -2.0))))))));
	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[t, 1.7e-37], 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[(0.0625 * N[(b * N[(t * N[(1.0 - N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.70000000000000009e-37

   1. Initial program 35.5%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in a around -inf 36.9%

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

   if 1.70000000000000009e-37 < t

   1. Initial program 11.5%

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

   3. Simplified 11.5%

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

   5. Taylor expanded in t around 0 15.7%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-37}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \frac{t}{16}\right)\right) \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(1 - a \cdot -2\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-37}:\\ \;\;\;\;\left(x\_m \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(1 - a \cdot -2\right)\right)\right)\right)\right) \cdot \cos \left(\left(z \cdot \frac{t}{16}\right) \cdot \left(y \cdot \left(2 + \frac{1}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (<= t 1.85e-37)
    (*
     (* x_m (cos (* 0.0625 (* b (* t (- 1.0 (* a -2.0)))))))
     (cos (* (* z (/ t 16.0)) (* y (+ 2.0 (/ 1.0 y))))))
    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 (t <= 1.85e-37) {
		tmp = (x_m * cos((0.0625 * (b * (t * (1.0 - (a * -2.0))))))) * cos(((z * (t / 16.0)) * (y * (2.0 + (1.0 / y)))));
	} 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) :: tmp
    if (t <= 1.85d-37) then
        tmp = (x_m * cos((0.0625d0 * (b * (t * (1.0d0 - (a * (-2.0d0)))))))) * cos(((z * (t / 16.0d0)) * (y * (2.0d0 + (1.0d0 / y)))))
    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 tmp;
	if (t <= 1.85e-37) {
		tmp = (x_m * Math.cos((0.0625 * (b * (t * (1.0 - (a * -2.0))))))) * Math.cos(((z * (t / 16.0)) * (y * (2.0 + (1.0 / y)))));
	} 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):
	tmp = 0
	if t <= 1.85e-37:
		tmp = (x_m * math.cos((0.0625 * (b * (t * (1.0 - (a * -2.0))))))) * math.cos(((z * (t / 16.0)) * (y * (2.0 + (1.0 / y)))))
	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 (t <= 1.85e-37)
		tmp = Float64(Float64(x_m * cos(Float64(0.0625 * Float64(b * Float64(t * Float64(1.0 - Float64(a * -2.0))))))) * cos(Float64(Float64(z * Float64(t / 16.0)) * Float64(y * Float64(2.0 + Float64(1.0 / y))))));
	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)
	tmp = 0.0;
	if (t <= 1.85e-37)
		tmp = (x_m * cos((0.0625 * (b * (t * (1.0 - (a * -2.0))))))) * cos(((z * (t / 16.0)) * (y * (2.0 + (1.0 / y)))));
	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_] := N[(x$95$s * If[LessEqual[t, 1.85e-37], N[(N[(x$95$m * N[Cos[N[(0.0625 * N[(b * N[(t * N[(1.0 - N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(z * N[(t / 16.0), $MachinePrecision]), $MachinePrecision] * N[(y * N[(2.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.85e-37

   1. Initial program 35.5%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in a around -inf 36.9%

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

   6. Taylor expanded in y around inf 36.9%

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

   if 1.85e-37 < t

   1. Initial program 11.5%

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

   3. Simplified 11.5%

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

   5. Taylor expanded in t around 0 15.7%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-37}:\\ \;\;\;\;\left(x \cdot \cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(1 - a \cdot -2\right)\right)\right)\right)\right) \cdot \cos \left(\left(z \cdot \frac{t}{16}\right) \cdot \left(y \cdot \left(2 + \frac{1}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{+59}:\\ \;\;\;\;x\_m \cdot \left(\cos \left(0.0625 \cdot \left(\left(z \cdot t\right) \cdot \left(1 + y \cdot 2\right)\right)\right) \cdot \cos \left(b \cdot \left(t \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (<= t 1.55e+59)
    (*
     x_m
     (*
      (cos (* 0.0625 (* (* z t) (+ 1.0 (* y 2.0)))))
      (cos (* b (* t 0.0625)))))
    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 (t <= 1.55e+59) {
		tmp = x_m * (cos((0.0625 * ((z * t) * (1.0 + (y * 2.0))))) * cos((b * (t * 0.0625))));
	} 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) :: tmp
    if (t <= 1.55d+59) then
        tmp = x_m * (cos((0.0625d0 * ((z * t) * (1.0d0 + (y * 2.0d0))))) * cos((b * (t * 0.0625d0))))
    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 tmp;
	if (t <= 1.55e+59) {
		tmp = x_m * (Math.cos((0.0625 * ((z * t) * (1.0 + (y * 2.0))))) * Math.cos((b * (t * 0.0625))));
	} 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):
	tmp = 0
	if t <= 1.55e+59:
		tmp = x_m * (math.cos((0.0625 * ((z * t) * (1.0 + (y * 2.0))))) * math.cos((b * (t * 0.0625))))
	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 (t <= 1.55e+59)
		tmp = Float64(x_m * Float64(cos(Float64(0.0625 * Float64(Float64(z * t) * Float64(1.0 + Float64(y * 2.0))))) * cos(Float64(b * Float64(t * 0.0625)))));
	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)
	tmp = 0.0;
	if (t <= 1.55e+59)
		tmp = x_m * (cos((0.0625 * ((z * t) * (1.0 + (y * 2.0))))) * cos((b * (t * 0.0625))));
	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_] := N[(x$95$s * If[LessEqual[t, 1.55e+59], N[(x$95$m * N[(N[Cos[N[(0.0625 * N[(N[(z * t), $MachinePrecision] * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(b * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.55000000000000007e59

   1. Initial program 33.7%

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

   3. Simplified 35.0%

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

      1. add-cube-cbrt 35.1%

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

      2. pow3 35.0%

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

      3. *-commutative 35.0%

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

      4. associate-*r/ 35.0%

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

   6. Applied egg-rr 35.0%

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

   7. Taylor expanded in a around 0 34.9%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right) \cdot \cos \left(b \cdot \left(t \cdot {\left(\sqrt[3]{0.0625}\right)}^{3}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. metadata-eval 34.9%

         \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + \color{blue}{\left(--2\right)} \cdot y\right)\right)\right)\right) \cdot \cos \left(b \cdot \left(t \cdot {\left(\sqrt[3]{0.0625}\right)}^{3}\right)\right)\right) \]

      2. cancel-sign-sub-inv 34.9%

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

      3. cancel-sign-sub-inv 34.9%

         \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \color{blue}{\left(1 + \left(--2\right) \cdot y\right)}\right)\right)\right) \cdot \cos \left(b \cdot \left(t \cdot {\left(\sqrt[3]{0.0625}\right)}^{3}\right)\right)\right) \]

      4. metadata-eval 34.9%

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

      5. associate-*r* 35.5%

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

      6. rem-cube-cbrt 34.8%

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

   9. Simplified 34.8%

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

   if 1.55000000000000007e59 < t

   1. Initial program 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in t around 0 14.9%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(\left(z \cdot t\right) \cdot \left(1 + y \cdot 2\right)\right)\right) \cdot \cos \left(b \cdot \left(t \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-37}:\\ \;\;\;\;x\_m \cdot \cos \left(0.0625 \cdot \left(z \cdot \left(t \cdot \mathsf{fma}\left(y, 2, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (<= t 1.8e-37)
    (* x_m (cos (* 0.0625 (* z (* t (fma y 2.0 1.0))))))
    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 (t <= 1.8e-37) {
		tmp = x_m * cos((0.0625 * (z * (t * fma(y, 2.0, 1.0)))));
	} 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 (t <= 1.8e-37)
		tmp = Float64(x_m * cos(Float64(0.0625 * Float64(z * Float64(t * fma(y, 2.0, 1.0))))));
	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[t, 1.8e-37], N[(x$95$m * N[Cos[N[(0.0625 * N[(z * N[(t * N[(y * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000004e-37

   1. Initial program 35.5%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in a around -inf 36.9%

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

   6. Taylor expanded in b around 0 36.2%

      \[\leadsto \color{blue}{x \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]

   8. Simplified 36.8%

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

   if 1.80000000000000004e-37 < t

   1. Initial program 11.5%

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

   3. Simplified 11.5%

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

   5. Taylor expanded in t around 0 15.7%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \cos \left(0.0625 \cdot \left(z \cdot \left(t \cdot \mathsf{fma}\left(y, 2, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-37}:\\ \;\;\;\;x\_m \cdot \cos \left(0.0625 \cdot \left(\left(z \cdot t\right) \cdot \left(1 + y \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (<= t 3e-37)
    (* x_m (cos (* 0.0625 (* (* z t) (+ 1.0 (* y 2.0))))))
    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 (t <= 3e-37) {
		tmp = x_m * cos((0.0625 * ((z * t) * (1.0 + (y * 2.0)))));
	} 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) :: tmp
    if (t <= 3d-37) then
        tmp = x_m * cos((0.0625d0 * ((z * t) * (1.0d0 + (y * 2.0d0)))))
    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 tmp;
	if (t <= 3e-37) {
		tmp = x_m * Math.cos((0.0625 * ((z * t) * (1.0 + (y * 2.0)))));
	} 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):
	tmp = 0
	if t <= 3e-37:
		tmp = x_m * math.cos((0.0625 * ((z * t) * (1.0 + (y * 2.0)))))
	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 (t <= 3e-37)
		tmp = Float64(x_m * cos(Float64(0.0625 * Float64(Float64(z * t) * Float64(1.0 + Float64(y * 2.0))))));
	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)
	tmp = 0.0;
	if (t <= 3e-37)
		tmp = x_m * cos((0.0625 * ((z * t) * (1.0 + (y * 2.0)))));
	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_] := N[(x$95$s * If[LessEqual[t, 3e-37], N[(x$95$m * N[Cos[N[(0.0625 * N[(N[(z * t), $MachinePrecision] * N[(1.0 + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3e-37

   1. Initial program 35.5%

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

   3. Simplified 36.9%

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

      1. add-cube-cbrt 37.1%

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

      2. pow3 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-*r/ 36.9%

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

   6. Applied egg-rr 36.9%

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

   7. Taylor expanded in b around 0 36.2%

      \[\leadsto \color{blue}{x \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. metadata-eval 36.2%

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

      2. cancel-sign-sub-inv 36.2%

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

      3. cancel-sign-sub-inv 36.2%

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

      4. metadata-eval 36.2%

         \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + \color{blue}{2} \cdot y\right)\right)\right)\right) \]

      5. associate-*r* 36.8%

         \[\leadsto x \cdot \cos \left(0.0625 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot \left(1 + 2 \cdot y\right)\right)}\right) \]

   9. Simplified 36.8%

      \[\leadsto \color{blue}{x \cdot \cos \left(0.0625 \cdot \left(\left(t \cdot z\right) \cdot \left(1 + 2 \cdot y\right)\right)\right)} \]

   if 3e-37 < t

   1. Initial program 11.5%

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

   3. Simplified 11.5%

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

   5. Taylor expanded in t around 0 15.7%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \cos \left(0.0625 \cdot \left(\left(z \cdot t\right) \cdot \left(1 + y \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.8% 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 29.2%

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

3. Simplified 29.2%

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

5. Taylor expanded in t around 0 30.9%

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

Developer Target 1: 30.3% 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 2024185 
(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))))