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

Percentage Accurate: 27.7% → 32.1%

Time: 6.8s

Alternatives: 8

Speedup: 100.6×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot \left(t\_m \cdot z\right)}{16}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\_m, \frac{t\_m}{16}, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      2e+290)
   (*
    (* x (cos (/ (* (fma 2.0 y 1.0) (* t_m z)) 16.0)))
    (sin (fma (* (fma a 2.0 1.0) b_m) (/ t_m 16.0) (/ PI 2.0))))
   x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+290) {
		tmp = (x * cos(((fma(2.0, y, 1.0) * (t_m * z)) / 16.0))) * sin(fma((fma(a, 2.0, 1.0) * b_m), (t_m / 16.0), (((double) M_PI) / 2.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+290)
		tmp = Float64(Float64(x * cos(Float64(Float64(fma(2.0, y, 1.0) * Float64(t_m * z)) / 16.0))) * sin(fma(Float64(fma(a, 2.0, 1.0) * b_m), Float64(t_m / 16.0), Float64(pi / 2.0))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(N[(x * N[Cos[N[(N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(t$95$m * z), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * N[(t$95$m / 16.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

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)))) < 2.00000000000000012e290

   1. Initial program 47.1%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 47.1

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

   3. Applied rewrites 47.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. sin-+PI/2-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

   5. Applied rewrites 47.0%

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

   if 2.00000000000000012e290 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 1.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 11.4%

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

Alternative 2: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;t\_1 \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot \left(b\_m \cdot t\_m\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (let* ((t_1 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))))
   (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0))) 2e+290)
     (* t_1 (cos (/ (* (fma a 2.0 1.0) (* b_m t_m)) 16.0)))
     x)))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double t_1 = x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0));
	double tmp;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+290) {
		tmp = t_1 * cos(((fma(a, 2.0, 1.0) * (b_m * t_m)) / 16.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	t_1 = Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+290)
		tmp = Float64(t_1 * cos(Float64(Float64(fma(a, 2.0, 1.0) * Float64(b_m * t_m)) / 16.0)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := Block[{t$95$1 = N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(t$95$1 * N[Cos[N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * N[(b$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]]

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)))) < 2.00000000000000012e290

   1. Initial program 47.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 47.1

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

   3. Applied rewrites 47.1%

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

   if 2.00000000000000012e290 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 1.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 11.4%

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

Alternative 3: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \left(\cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \frac{t\_m \cdot z}{16}\right) \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\_m\right) \cdot \frac{t\_m}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      2e+290)
   (*
    x
    (*
     (cos (* (fma 2.0 y 1.0) (/ (* t_m z) 16.0)))
     (cos (* (* (fma a 2.0 1.0) b_m) (/ t_m 16.0)))))
   x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+290) {
		tmp = x * (cos((fma(2.0, y, 1.0) * ((t_m * z) / 16.0))) * cos(((fma(a, 2.0, 1.0) * b_m) * (t_m / 16.0))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+290)
		tmp = Float64(x * Float64(cos(Float64(fma(2.0, y, 1.0) * Float64(Float64(t_m * z) / 16.0))) * cos(Float64(Float64(fma(a, 2.0, 1.0) * b_m) * Float64(t_m / 16.0)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(x * N[(N[Cos[N[(N[(2.0 * y + 1.0), $MachinePrecision] * N[(N[(t$95$m * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * N[(t$95$m / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

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)))) < 2.00000000000000012e290

   1. Initial program 47.1%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 47.1

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

   3. Applied rewrites 47.1%

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

   5. Applied rewrites 47.1%

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

   if 2.00000000000000012e290 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 1.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 11.4%

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

Alternative 4: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \left(\cos \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot \frac{t\_m}{16}\right) \cdot \cos \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\_m\right) \cdot \frac{t\_m}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      2e+290)
   (*
    x
    (*
     (cos (* (* (fma 2.0 y 1.0) z) (/ t_m 16.0)))
     (cos (* (* (fma a 2.0 1.0) b_m) (/ t_m 16.0)))))
   x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+290) {
		tmp = x * (cos(((fma(2.0, y, 1.0) * z) * (t_m / 16.0))) * cos(((fma(a, 2.0, 1.0) * b_m) * (t_m / 16.0))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+290)
		tmp = Float64(x * Float64(cos(Float64(Float64(fma(2.0, y, 1.0) * z) * Float64(t_m / 16.0))) * cos(Float64(Float64(fma(a, 2.0, 1.0) * b_m) * Float64(t_m / 16.0)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(x * N[(N[Cos[N[(N[(N[(2.0 * y + 1.0), $MachinePrecision] * z), $MachinePrecision] * N[(t$95$m / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * N[(t$95$m / 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

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)))) < 2.00000000000000012e290

   1. Initial program 47.1%

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

   3. Applied rewrites 47.1%

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

   if 2.00000000000000012e290 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 1.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 11.4%

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

Alternative 5: 31.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 10^{+210}:\\ \;\;\;\;\left(\cos \left(\left(t\_m \cdot z\right) \cdot 0.0625\right) \cdot x\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot \left(b\_m \cdot t\_m\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      1e+210)
   (*
    (* (cos (* (* t_m z) 0.0625)) x)
    (cos (/ (* (fma a 2.0 1.0) (* b_m t_m)) 16.0)))
   x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+210) {
		tmp = (cos(((t_m * z) * 0.0625)) * x) * cos(((fma(a, 2.0, 1.0) * (b_m * t_m)) / 16.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+210)
		tmp = Float64(Float64(cos(Float64(Float64(t_m * z) * 0.0625)) * x) * cos(Float64(Float64(fma(a, 2.0, 1.0) * Float64(b_m * t_m)) / 16.0)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+210], N[(N[(N[Cos[N[(N[(t$95$m * z), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Cos[N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * N[(b$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

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)))) < 9.99999999999999927e209

   1. Initial program 47.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 47.0

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

   4. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-*.f64 46.9

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

   6. Applied rewrites 46.9%

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

   if 9.99999999999999927e209 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 4.4%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 13.7%

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

Alternative 6: 31.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 10^{+210}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\_m\right) \cdot b\_m\right) \cdot 0.0625\right) \cdot x\right) \cdot \cos \left(\left(t\_m \cdot z\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      1e+210)
   (*
    (* (cos (* (* (* (fma a 2.0 1.0) t_m) b_m) 0.0625)) x)
    (cos (* (* t_m z) 0.0625)))
   x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+210) {
		tmp = (cos((((fma(a, 2.0, 1.0) * t_m) * b_m) * 0.0625)) * x) * cos(((t_m * z) * 0.0625));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 1e+210)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(fma(a, 2.0, 1.0) * t_m) * b_m) * 0.0625)) * x) * cos(Float64(Float64(t_m * z) * 0.0625)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+210], N[(N[(N[Cos[N[(N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Cos[N[(N[(t$95$m * z), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

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)))) < 9.99999999999999927e209

   1. Initial program 47.5%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 46.5%

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

   if 9.99999999999999927e209 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 4.4%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 13.7%

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

Alternative 7: 31.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t\_m}{16}\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(b\_m \cdot t\_m\right) \cdot 0.0625\right)\right) \cdot \cos \left(\left(0.0625 \cdot t\_m\right) \cdot \left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t_m) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t_m) 16.0)))
      2e+290)
   (*
    (* x (cos (* (* b_m t_m) 0.0625)))
    (cos (* (* 0.0625 t_m) (* (fma 2.0 y 1.0) z))))
   x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+290) {
		tmp = (x * cos(((b_m * t_m) * 0.0625))) * cos(((0.0625 * t_m) * (fma(2.0, y, 1.0) * z)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t_m) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b_m) * t_m) / 16.0))) <= 2e+290)
		tmp = Float64(Float64(x * cos(Float64(Float64(b_m * t_m) * 0.0625))) * cos(Float64(Float64(0.0625 * t_m) * Float64(fma(2.0, y, 1.0) * z))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+290], N[(N[(x * N[Cos[N[(N[(b$95$m * t$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(0.0625 * t$95$m), $MachinePrecision] * N[(N[(2.0 * y + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

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)))) < 2.00000000000000012e290

   1. Initial program 47.1%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 46.4%

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

   if 2.00000000000000012e290 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 1.0%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 11.4%

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

Alternative 8: 30.6% accurate, 100.6× speedup

Math

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

FPCore

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

C

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

Fortran

t_m =     private
b_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t_m, a, b_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.7%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 30.6%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64
  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))