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

Percentage Accurate: 27.7% → 32.0%

Time: 9.1s

Alternatives: 10

Speedup: 2.4×

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: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot t\_1 \leq 3 \cdot 10^{+73}:\\ \;\;\;\;\left(x\_m \cdot \sin \left(\frac{t\_m \cdot \left(z\_m \cdot \left(\left(\frac{1}{y} + 2\right) \cdot y\right)\right)}{-16} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0))))
   (*
    x_s
    (if (<=
         (* (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0))) t_1)
         3e+73)
      (*
       (*
        x_m
        (sin
         (+ (/ (* t_m (* z_m (* (+ (/ 1.0 y) 2.0) y))) -16.0) (/ (PI) 2.0))))
       t_1)
      (* (sin (* 0.5 (PI))) x_m)))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 3.00000000000000011e73

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 47.2

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

   7. Applied rewrites 47.2%

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

   if 3.00000000000000011e73 < (*.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 5.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 7.1%

      \[\leadsto \color{blue}{\left(\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right)\right)\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      5. lower-PI.f64 14.6

         \[\leadsto \sin \left(0.5 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot x \]

   10. Applied rewrites 14.6%

       \[\leadsto \color{blue}{\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 32.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 3 \cdot 10^{+73}:\\ \;\;\;\;\left(\cos \left(\frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t\_m}{-16}\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, y, 1\right) \cdot z\_m}{-16}, t\_m, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       3e+73)
    (*
     (*
      (cos (/ (* (* b (fma a 2.0 1.0)) t_m) -16.0))
      (sin (fma (/ (* (fma 2.0 y 1.0) z_m) -16.0) t_m (/ (PI) 2.0))))
     x_m)
    (* (sin (* 0.5 (PI))) x_m))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 3.00000000000000011e73

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 47.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-sin.f64 47.2

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 47.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 47.2

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

   6. Applied rewrites 47.2%

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

   if 3.00000000000000011e73 < (*.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 5.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 7.1%

      \[\leadsto \color{blue}{\left(\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right)\right)\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      5. lower-PI.f64 14.6

         \[\leadsto \sin \left(0.5 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot x \]

   10. Applied rewrites 14.6%

       \[\leadsto \color{blue}{\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 3 \cdot 10^{+73}:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(a, 2, 1\right) \cdot \left(\left(b \cdot t\_m\right) \cdot -0.0625\right)\right) \cdot x\_m\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\_m\right) \cdot t\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       3e+73)
    (*
     (* (cos (* (fma a 2.0 1.0) (* (* b t_m) -0.0625))) x_m)
     (sin (fma (PI) 0.5 (* -0.0625 (* (* (fma 2.0 y 1.0) z_m) t_m)))))
    (* (sin (* 0.5 (PI))) x_m))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 3.00000000000000011e73

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 46.9%

      \[\leadsto \color{blue}{\left(\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right)\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 46.9%

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(a, 2, 1\right) \cdot \left(\left(b \cdot t\right) \cdot -0.0625\right)\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right)\right)\right) \]

   if 3.00000000000000011e73 < (*.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 5.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 7.1%

      \[\leadsto \color{blue}{\left(\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right)\right)\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      5. lower-PI.f64 14.6

         \[\leadsto \sin \left(0.5 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot x \]

   10. Applied rewrites 14.6%

       \[\leadsto \color{blue}{\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t\_m}{16}\right) \leq 3 \cdot 10^{+227}:\\ \;\;\;\;\left(\sin \left(\mathsf{fma}\left(0.0625, \left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t\_m, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x\_m\right) \cdot \cos \left(-0.0625 \cdot \left(t\_m \cdot z\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b) t_m) 16.0)))
       3e+227)
    (*
     (* (sin (fma 0.0625 (* (* (fma a 2.0 1.0) b) t_m) (/ (PI) 2.0))) x_m)
     (cos (* -0.0625 (* t_m z_m))))
    (* (sin (* 0.5 (PI))) x_m))))

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)))) < 2.99999999999999986e227

   1. Initial program 44.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.7%

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

   7. Applied rewrites 45.1%

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

   if 2.99999999999999986e227 < (*.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

   4. Applied rewrites 1.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 2.2%

      \[\leadsto \color{blue}{\left(\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right)\right)\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

      5. lower-PI.f64 11.8

         \[\leadsto \sin \left(0.5 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot x \]

   10. Applied rewrites 11.8%

       \[\leadsto \color{blue}{\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 30.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\cos \left(\left(t\_m \cdot z\_m\right) \cdot 0.0625\right) \cdot x\_m\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (* x_s (* (cos (* (* t_m z_m) 0.0625)) x_m)))

C

t_m = fabs(t);
z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	return x_s * (cos(((t_m * z_m) * 0.0625)) * x_m);
}

Fortran

t_m =     private
z_m =     private
x\_m =     private
x\_s =     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_s, x_m, y, z_m, t_m, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x_s * (cos(((t_m * z_m) * 0.0625d0)) * x_m)
end function

Java

t_m = Math.abs(t);
z_m = Math.abs(z);
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_m, double t_m, double a, double b) {
	return x_s * (Math.cos(((t_m * z_m) * 0.0625)) * x_m);
}

Python

t_m = math.fabs(t)
z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z_m, t_m, a, b):
	return x_s * (math.cos(((t_m * z_m) * 0.0625)) * x_m)

Julia

t_m = abs(t)
z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	return Float64(x_s * Float64(cos(Float64(Float64(t_m * z_m) * 0.0625)) * x_m))
end

MATLAB

t_m = abs(t);
z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = x_s * (cos(((t_m * z_m) * 0.0625)) * x_m);
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, t$95$m_, a_, b_] := N[(x$95$s * N[(N[Cos[N[(N[(t$95$m * z$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.3%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 27.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 29.1%

   \[\leadsto \cos \left(\left(t \cdot z\right) \cdot 0.0625\right) \cdot \color{blue}{x} \]
9. Add Preprocessing

Alternative 6: 31.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot x\_m\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (* x_s (* (sin (* 0.5 (PI))) x_m)))

Derivation

1. Initial program 25.3%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. metadata-eval N/A

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

4. Applied rewrites 25.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 25.8%

   \[\leadsto \color{blue}{\left(\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right)\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -0.0625 \cdot \left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right)\right)\right)} \]

8. Taylor expanded in t around 0

   \[\leadsto \color{blue}{x \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]

   3. lower-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

   4. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot x \]

   5. lower-PI.f64 29.0

      \[\leadsto \sin \left(0.5 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot x \]

10. Applied rewrites 29.0%

    \[\leadsto \color{blue}{\sin \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot x} \]
11. Add Preprocessing

Alternative 7: 26.9% accurate, 10.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \mathsf{fma}\left(\left(\left(\left(t\_m \cdot t\_m\right) \cdot -0.001953125\right) \cdot z\_m\right) \cdot z\_m, x\_m, x\_m\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (* x_s (fma (* (* (* (* t_m t_m) -0.001953125) z_m) z_m) x_m x_m)))

C

t_m = fabs(t);
z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	return x_s * fma(((((t_m * t_m) * -0.001953125) * z_m) * z_m), x_m, x_m);
}

Julia

t_m = abs(t)
z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	return Float64(x_s * fma(Float64(Float64(Float64(Float64(t_m * t_m) * -0.001953125) * z_m) * z_m), x_m, x_m))
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, t$95$m_, a_, b_] := N[(x$95$s * N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.001953125), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.3%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 27.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 29.1%

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

9. Taylor expanded in z around 0

   \[\leadsto x + \frac{-1}{512} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot {z}^{2}\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 23.1%

    \[\leadsto \mathsf{fma}\left(-0.001953125 \cdot \left(t \cdot t\right), \left(z \cdot z\right) \cdot \color{blue}{x}, x\right) \]
12. Step-by-step derivation

13. Applied rewrites 25.1%

    \[\leadsto \mathsf{fma}\left(\left(\left(\left(t \cdot t\right) \cdot -0.001953125\right) \cdot z\right) \cdot z, x, x\right) \]
14. Add Preprocessing

Alternative 8: 25.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \mathsf{fma}\left(\left(\left(t\_m \cdot t\_m\right) \cdot -0.001953125\right) \cdot z\_m, z\_m \cdot x\_m, x\_m\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (* x_s (fma (* (* (* t_m t_m) -0.001953125) z_m) (* z_m x_m) x_m)))

C

t_m = fabs(t);
z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	return x_s * fma((((t_m * t_m) * -0.001953125) * z_m), (z_m * x_m), x_m);
}

Julia

t_m = abs(t)
z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	return Float64(x_s * fma(Float64(Float64(Float64(t_m * t_m) * -0.001953125) * z_m), Float64(z_m * x_m), x_m))
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, t$95$m_, a_, b_] := N[(x$95$s * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.001953125), $MachinePrecision] * z$95$m), $MachinePrecision] * N[(z$95$m * x$95$m), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.3%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 27.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 29.1%

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

9. Taylor expanded in z around 0

   \[\leadsto x + \frac{-1}{512} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot {z}^{2}\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 23.1%

    \[\leadsto \mathsf{fma}\left(-0.001953125 \cdot \left(t \cdot t\right), \left(z \cdot z\right) \cdot \color{blue}{x}, x\right) \]
12. Step-by-step derivation

13. Applied rewrites 24.7%

    \[\leadsto \mathsf{fma}\left(\left(\left(t \cdot t\right) \cdot -0.001953125\right) \cdot z, z \cdot x, x\right) \]
14. Add Preprocessing

Alternative 9: 25.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \mathsf{fma}\left(z\_m, \left(z\_m \cdot x\_m\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot -0.001953125\right), x\_m\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (* x_s (fma z_m (* (* z_m x_m) (* (* t_m t_m) -0.001953125)) x_m)))

C

t_m = fabs(t);
z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	return x_s * fma(z_m, ((z_m * x_m) * ((t_m * t_m) * -0.001953125)), x_m);
}

Julia

t_m = abs(t)
z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	return Float64(x_s * fma(z_m, Float64(Float64(z_m * x_m) * Float64(Float64(t_m * t_m) * -0.001953125)), x_m))
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, t$95$m_, a_, b_] := N[(x$95$s * N[(z$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.001953125), $MachinePrecision]), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.3%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 27.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 29.1%

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

9. Taylor expanded in z around 0

   \[\leadsto x + \frac{-1}{512} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot {z}^{2}\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 23.1%

    \[\leadsto \mathsf{fma}\left(-0.001953125 \cdot \left(t \cdot t\right), \left(z \cdot z\right) \cdot \color{blue}{x}, x\right) \]
12. Step-by-step derivation

13. Applied rewrites 24.7%

    \[\leadsto \mathsf{fma}\left(z, \left(z \cdot x\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{-0.001953125}\right), x\right) \]
14. Add Preprocessing

Alternative 10: 2.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(\left(t\_m \cdot t\_m\right) \cdot -0.001953125\right) \cdot \left(\left(z\_m \cdot z\_m\right) \cdot x\_m\right)\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z_m t_m a b)
 :precision binary64
 (* x_s (* (* (* t_m t_m) -0.001953125) (* (* z_m z_m) x_m))))

C

t_m = fabs(t);
z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z_m, double t_m, double a, double b) {
	return x_s * (((t_m * t_m) * -0.001953125) * ((z_m * z_m) * x_m));
}

Fortran

t_m =     private
z_m =     private
x\_m =     private
x\_s =     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_s, x_m, y, z_m, t_m, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x_s * (((t_m * t_m) * (-0.001953125d0)) * ((z_m * z_m) * x_m))
end function

Java

t_m = Math.abs(t);
z_m = Math.abs(z);
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_m, double t_m, double a, double b) {
	return x_s * (((t_m * t_m) * -0.001953125) * ((z_m * z_m) * x_m));
}

Python

t_m = math.fabs(t)
z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z_m, t_m, a, b):
	return x_s * (((t_m * t_m) * -0.001953125) * ((z_m * z_m) * x_m))

Julia

t_m = abs(t)
z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z_m, t_m, a, b)
	return Float64(x_s * Float64(Float64(Float64(t_m * t_m) * -0.001953125) * Float64(Float64(z_m * z_m) * x_m)))
end

MATLAB

t_m = abs(t);
z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z_m, t_m, a, b)
	tmp = x_s * (((t_m * t_m) * -0.001953125) * ((z_m * z_m) * x_m));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, t$95$m_, a_, b_] := N[(x$95$s * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.001953125), $MachinePrecision] * N[(N[(z$95$m * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.3%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 27.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 29.1%

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

9. Taylor expanded in z around 0

   \[\leadsto x + \frac{-1}{512} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot {z}^{2}\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 23.1%

    \[\leadsto \mathsf{fma}\left(-0.001953125 \cdot \left(t \cdot t\right), \left(z \cdot z\right) \cdot \color{blue}{x}, x\right) \]

12. Taylor expanded in z around inf

    \[\leadsto \frac{-1}{512} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{{z}^{2}}\right)\right) \]
13. Step-by-step derivation

14. Applied rewrites 2.6%

    \[\leadsto \left(\left(t \cdot t\right) \cdot -0.001953125\right) \cdot \left(\left(z \cdot z\right) \cdot x\right) \]
15. Add Preprocessing

Developer Target 1: 30.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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(((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 2025017 
(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))))