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

Percentage Accurate: 27.7% → 32.0%

Time: 16.4s

Alternatives: 10

Speedup: 24.5×

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: 27.7% 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: 32.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \left(\left(2 \cdot a\right) \cdot b\right) \cdot \left(t \cdot 0.0625\right)\\ t_2 := x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ t_3 := t \cdot \left(b \cdot 0.0625\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+89}:\\ \;\;\;\;t\_2 \cdot \left(\cos t\_1 \cdot \cos t\_3 - \sin t\_1 \cdot \sin t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot 1\right) \cdot 1\\ \end{array} \end{array} \end{array} \]

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 (* (* (* 2.0 a) b) (* t 0.0625)))
        (t_2 (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))))
        (t_3 (* t (* b 0.0625))))
   (*
    x_s
    (if (<= (* t_2 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))) 1e+89)
      (* t_2 (- (* (cos t_1) (cos t_3)) (* (sin t_1) (sin t_3))))
      (* (* x_m 1.0) 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 = ((2.0 * a) * b) * (t * 0.0625);
	double t_2 = x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double t_3 = t * (b * 0.0625);
	double tmp;
	if ((t_2 * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+89) {
		tmp = t_2 * ((cos(t_1) * cos(t_3)) - (sin(t_1) * sin(t_3)));
	} else {
		tmp = (x_m * 1.0) * 1.0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = ((2.0d0 * a) * b) * (t * 0.0625d0)
    t_2 = x_m * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))
    t_3 = t * (b * 0.0625d0)
    if ((t_2 * cos(((t * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))) <= 1d+89) then
        tmp = t_2 * ((cos(t_1) * cos(t_3)) - (sin(t_1) * sin(t_3)))
    else
        tmp = (x_m * 1.0d0) * 1.0d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = ((2.0 * a) * b) * (t * 0.0625);
	double t_2 = x_m * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double t_3 = t * (b * 0.0625);
	double tmp;
	if ((t_2 * Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+89) {
		tmp = t_2 * ((Math.cos(t_1) * Math.cos(t_3)) - (Math.sin(t_1) * Math.sin(t_3)));
	} else {
		tmp = (x_m * 1.0) * 1.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = ((2.0 * a) * b) * (t * 0.0625)
	t_2 = x_m * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))
	t_3 = t * (b * 0.0625)
	tmp = 0
	if (t_2 * math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+89:
		tmp = t_2 * ((math.cos(t_1) * math.cos(t_3)) - (math.sin(t_1) * math.sin(t_3)))
	else:
		tmp = (x_m * 1.0) * 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(Float64(Float64(2.0 * a) * b) * Float64(t * 0.0625))
	t_2 = Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0)))
	t_3 = Float64(t * Float64(b * 0.0625))
	tmp = 0.0
	if (Float64(t_2 * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 1e+89)
		tmp = Float64(t_2 * Float64(Float64(cos(t_1) * cos(t_3)) - Float64(sin(t_1) * sin(t_3))));
	else
		tmp = Float64(Float64(x_m * 1.0) * 1.0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	t_1 = ((2.0 * a) * b) * (t * 0.0625);
	t_2 = x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	t_3 = t * (b * 0.0625);
	tmp = 0.0;
	if ((t_2 * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+89)
		tmp = t_2 * ((cos(t_1) * cos(t_3)) - (sin(t_1) * sin(t_3)));
	else
		tmp = (x_m * 1.0) * 1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(2.0 * a), $MachinePrecision] * b), $MachinePrecision] * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[(b * 0.0625), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(t$95$2 * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+89], N[(t$95$2 * N[(N[(N[Cos[t$95$1], $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$1], $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 1.0), $MachinePrecision] * 1.0), $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)))) < 9.99999999999999995e88

   1. Initial program 41.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. cos-sum N/A

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

      14. lower--.f64 N/A

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

   4. Applied rewrites 41.7%

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

   if 9.99999999999999995e88 < (*.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 10.2%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 18.8%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+89}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \left(\cos \left(\left(\left(2 \cdot a\right) \cdot b\right) \cdot \left(t \cdot 0.0625\right)\right) \cdot \cos \left(t \cdot \left(b \cdot 0.0625\right)\right) - \sin \left(\left(\left(2 \cdot a\right) \cdot b\right) \cdot \left(t \cdot 0.0625\right)\right) \cdot \sin \left(t \cdot \left(b \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 32.2% accurate, 0.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1)
         1e+306)
      (*
       t_1
       (*
        x_m
        (cos
         (/
          (*
           t
           (*
            z
            (* (fma y 2.0 1.0) (* (fma y 2.0 -1.0) (/ 1.0 (fma y 2.0 -1.0))))))
          16.0))))
      (* (* x_m 1.0) 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 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306) {
		tmp = t_1 * (x_m * cos(((t * (z * (fma(y, 2.0, 1.0) * (fma(y, 2.0, -1.0) * (1.0 / fma(y, 2.0, -1.0)))))) / 16.0)));
	} else {
		tmp = (x_m * 1.0) * 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 = cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306)
		tmp = Float64(t_1 * Float64(x_m * cos(Float64(Float64(t * Float64(z * Float64(fma(y, 2.0, 1.0) * Float64(fma(y, 2.0, -1.0) * Float64(1.0 / fma(y, 2.0, -1.0)))))) / 16.0))));
	else
		tmp = Float64(Float64(x_m * 1.0) * 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[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1e+306], N[(t$95$1 * N[(x$95$m * N[Cos[N[(N[(t * N[(z * N[(N[(y * 2.0 + 1.0), $MachinePrecision] * N[(N[(y * 2.0 + -1.0), $MachinePrecision] * N[(1.0 / N[(y * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 1.0), $MachinePrecision] * 1.0), $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)))) < 1.00000000000000002e306

   1. Initial program 43.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. difference-of-sqr-1 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. sub-neg N/A

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

      19. lift-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 43.9

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

   4. Applied rewrites 43.9%

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

   if 1.00000000000000002e306 < (*.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 0.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.6%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

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

Alternative 3: 32.1% accurate, 0.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1)
         1e+306)
      (* t_1 (* x_m (cos (/ (* z t) (* 16.0 (/ 1.0 (fma y 2.0 1.0)))))))
      (* (* x_m 1.0) 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 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306) {
		tmp = t_1 * (x_m * cos(((z * t) / (16.0 * (1.0 / fma(y, 2.0, 1.0))))));
	} else {
		tmp = (x_m * 1.0) * 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 = cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306)
		tmp = Float64(t_1 * Float64(x_m * cos(Float64(Float64(z * t) / Float64(16.0 * Float64(1.0 / fma(y, 2.0, 1.0)))))));
	else
		tmp = Float64(Float64(x_m * 1.0) * 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[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1e+306], N[(t$95$1 * N[(x$95$m * N[Cos[N[(N[(z * t), $MachinePrecision] / N[(16.0 * N[(1.0 / N[(y * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 1.0), $MachinePrecision] * 1.0), $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)))) < 1.00000000000000002e306

   1. Initial program 43.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lift-+.f64 N/A

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

      7. flip3-+ N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 43.9%

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

   if 1.00000000000000002e306 < (*.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 0.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.6%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

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

Alternative 4: 32.2% accurate, 0.5× speedup

Math

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

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 (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
          (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))))
   (* x_s (if (<= t_1 1e+306) t_1 (* (* x_m 1.0) 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((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (t_1 <= 1e+306) {
		tmp = t_1;
	} else {
		tmp = (x_m * 1.0) * 1.0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos(((t * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))
    if (t_1 <= 1d+306) then
        tmp = t_1
    else
        tmp = (x_m * 1.0d0) * 1.0d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = (x_m * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (t_1 <= 1e+306) {
		tmp = t_1;
	} else {
		tmp = (x_m * 1.0) * 1.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = (x_m * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))
	tmp = 0
	if t_1 <= 1e+306:
		tmp = t_1
	else:
		tmp = (x_m * 1.0) * 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(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0)))
	tmp = 0.0
	if (t_1 <= 1e+306)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m * 1.0) * 1.0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	t_1 = (x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	tmp = 0.0;
	if (t_1 <= 1e+306)
		tmp = t_1;
	else
		tmp = (x_m * 1.0) * 1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 1e+306], t$95$1, N[(N[(x$95$m * 1.0), $MachinePrecision] * 1.0), $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)))) < 1.00000000000000002e306

   1. Initial program 43.8%

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

   if 1.00000000000000002e306 < (*.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 0.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.6%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

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

Alternative 5: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+89}:\\ \;\;\;\;t\_1 \cdot \cos \left(\left(t \cdot b\right) \cdot \mathsf{fma}\left(0.125, a, 0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot 1\right) \cdot 1\\ \end{array} \end{array} \end{array} \]

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 (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (*
    x_s
    (if (<= (* t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))) 1e+89)
      (* t_1 (cos (* (* t b) (fma 0.125 a 0.0625))))
      (* (* x_m 1.0) 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((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_1 * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+89) {
		tmp = t_1 * cos(((t * b) * fma(0.125, a, 0.0625)));
	} else {
		tmp = (x_m * 1.0) * 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(Float64(Float64(y * 2.0) + 1.0) * z) * 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))) <= 1e+89)
		tmp = Float64(t_1 * cos(Float64(Float64(t * b) * fma(0.125, a, 0.0625))));
	else
		tmp = Float64(Float64(x_m * 1.0) * 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[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $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], 1e+89], N[(t$95$1 * N[Cos[N[(N[(t * b), $MachinePrecision] * N[(0.125 * a + 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 1.0), $MachinePrecision] * 1.0), $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)))) < 9.99999999999999995e88

   1. Initial program 41.5%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 41.5

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(0.125, a, 0.0625\right)}\right) \]

   5. Applied rewrites 41.5%

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(t \cdot b\right) \cdot \mathsf{fma}\left(0.125, a, 0.0625\right)\right)} \]

   if 9.99999999999999995e88 < (*.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 10.2%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 18.8%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+89}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \mathsf{fma}\left(0.125, a, 0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot t\_1 \leq 10^{+306}:\\ \;\;\;\;t\_1 \cdot \left(x\_m \cdot \cos \left(\left(z \cdot t\right) \cdot \mathsf{fma}\left(0.125, y, 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot 1\right) \cdot 1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1)
         1e+306)
      (* t_1 (* x_m (cos (* (* z t) (fma 0.125 y 0.0625)))))
      (* (* x_m 1.0) 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 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306) {
		tmp = t_1 * (x_m * cos(((z * t) * fma(0.125, y, 0.0625))));
	} else {
		tmp = (x_m * 1.0) * 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 = cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306)
		tmp = Float64(t_1 * Float64(x_m * cos(Float64(Float64(z * t) * fma(0.125, y, 0.0625)))));
	else
		tmp = Float64(Float64(x_m * 1.0) * 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[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1e+306], N[(t$95$1 * N[(x$95$m * N[Cos[N[(N[(z * t), $MachinePrecision] * N[(0.125 * y + 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 1.0), $MachinePrecision] * 1.0), $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)))) < 1.00000000000000002e306

   1. Initial program 43.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 43.4

          \[\leadsto \left(x \cdot \cos \left(\left(t \cdot z\right) \cdot \color{blue}{\mathsf{fma}\left(0.125, y, 0.0625\right)}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Applied rewrites 43.4%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(t \cdot z\right) \cdot \mathsf{fma}\left(0.125, y, 0.0625\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   if 1.00000000000000002e306 < (*.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 0.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.6%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+306}:\\ \;\;\;\;\cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \cdot \left(x \cdot \cos \left(\left(z \cdot t\right) \cdot \mathsf{fma}\left(0.125, y, 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot t\_1 \leq 10^{+306}:\\ \;\;\;\;t\_1 \cdot \left(x\_m \cdot \cos \left(0.125 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot 1\right) \cdot 1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1)
         1e+306)
      (* t_1 (* x_m (cos (* 0.125 (* t (* y z))))))
      (* (* x_m 1.0) 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 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306) {
		tmp = t_1 * (x_m * cos((0.125 * (t * (y * z)))));
	} else {
		tmp = (x_m * 1.0) * 1.0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(((t * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))
    if (((x_m * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * t_1) <= 1d+306) then
        tmp = t_1 * (x_m * cos((0.125d0 * (t * (y * z)))))
    else
        tmp = (x_m * 1.0d0) * 1.0d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x_m * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306) {
		tmp = t_1 * (x_m * Math.cos((0.125 * (t * (y * z)))));
	} else {
		tmp = (x_m * 1.0) * 1.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))
	tmp = 0
	if ((x_m * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306:
		tmp = t_1 * (x_m * math.cos((0.125 * (t * (y * z)))))
	else:
		tmp = (x_m * 1.0) * 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 = cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306)
		tmp = Float64(t_1 * Float64(x_m * cos(Float64(0.125 * Float64(t * Float64(y * z))))));
	else
		tmp = Float64(Float64(x_m * 1.0) * 1.0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	t_1 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	tmp = 0.0;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306)
		tmp = t_1 * (x_m * cos((0.125 * (t * (y * z)))));
	else
		tmp = (x_m * 1.0) * 1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1e+306], N[(t$95$1 * N[(x$95$m * N[Cos[N[(0.125 * N[(t * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 1.0), $MachinePrecision] * 1.0), $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)))) < 1.00000000000000002e306

   1. Initial program 43.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 42.7

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

   5. Applied rewrites 42.7%

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

   if 1.00000000000000002e306 < (*.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 0.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.6%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

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

Alternative 8: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot t\_1 \leq 10^{+306}:\\ \;\;\;\;t\_1 \cdot \left(x\_m \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot 1\right) \cdot 1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) t_1)
         1e+306)
      (* t_1 (* x_m (cos (* 0.0625 (* z t)))))
      (* (* x_m 1.0) 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 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306) {
		tmp = t_1 * (x_m * cos((0.0625 * (z * t))));
	} else {
		tmp = (x_m * 1.0) * 1.0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(((t * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))
    if (((x_m * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * t_1) <= 1d+306) then
        tmp = t_1 * (x_m * cos((0.0625d0 * (z * t))))
    else
        tmp = (x_m * 1.0d0) * 1.0d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x_m * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306) {
		tmp = t_1 * (x_m * Math.cos((0.0625 * (z * t))));
	} else {
		tmp = (x_m * 1.0) * 1.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))
	tmp = 0
	if ((x_m * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306:
		tmp = t_1 * (x_m * math.cos((0.0625 * (z * t))))
	else:
		tmp = (x_m * 1.0) * 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 = cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x_m * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306)
		tmp = Float64(t_1 * Float64(x_m * cos(Float64(0.0625 * Float64(z * t)))));
	else
		tmp = Float64(Float64(x_m * 1.0) * 1.0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	t_1 = cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0));
	tmp = 0.0;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * t_1) <= 1e+306)
		tmp = t_1 * (x_m * cos((0.0625 * (z * t))));
	else
		tmp = (x_m * 1.0) * 1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1e+306], N[(t$95$1 * N[(x$95$m * N[Cos[N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 1.0), $MachinePrecision] * 1.0), $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)))) < 1.00000000000000002e306

   1. Initial program 43.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 42.6

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

   5. Applied rewrites 42.6%

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

   if 1.00000000000000002e306 < (*.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 0.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.6%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+306}:\\ \;\;\;\;\cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \cdot \left(x \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-94}:\\ \;\;\;\;x\_m \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot \mathsf{fma}\left(t, 2 \cdot a, t\right)\right)\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot 1\right) \cdot 1\\ \end{array} \end{array} \]

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 4e-94)
    (*
     x_m
     (* (cos (* 0.0625 (* b (fma t (* 2.0 a) t)))) (cos (* 0.0625 (* z t)))))
    (* (* x_m 1.0) 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 (t <= 4e-94) {
		tmp = x_m * (cos((0.0625 * (b * fma(t, (2.0 * a), t)))) * cos((0.0625 * (z * t))));
	} else {
		tmp = (x_m * 1.0) * 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 (t <= 4e-94)
		tmp = Float64(x_m * Float64(cos(Float64(0.0625 * Float64(b * fma(t, Float64(2.0 * a), t)))) * cos(Float64(0.0625 * Float64(z * t)))));
	else
		tmp = Float64(Float64(x_m * 1.0) * 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[t, 4e-94], N[(x$95$m * N[(N[Cos[N[(0.0625 * N[(b * N[(t * N[(2.0 * a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.9999999999999998e-94

   1. Initial program 28.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 29.9

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

   8. Applied rewrites 29.9%

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

   if 3.9999999999999998e-94 < t

   1. Initial program 15.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 16.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 20.2%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-94}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot \mathsf{fma}\left(t, 2 \cdot a, t\right)\right)\right) \cdot \cos \left(0.0625 \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.9% accurate, 24.5× speedup

Math

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

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 1.0) 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) {
	return x_s * ((x_m * 1.0) * 1.0);
}

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 * 1.0d0) * 1.0d0)
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 * 1.0) * 1.0);
}

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 * 1.0) * 1.0)

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 * Float64(Float64(x_m * 1.0) * 1.0))
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 * 1.0) * 1.0);
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 * N[(N[(x$95$m * 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.0%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 24.9%

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

6. Taylor expanded in b around 0

   \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 26.2%

   \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 30.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(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))))