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

Percentage Accurate: 27.5% → 31.9%

Time: 10.4s

Alternatives: 8

Speedup: 2.6×

Specification

Math

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

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: 27.5% accurate, 1.0× speedup

Math

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

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: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \left|t\right|}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot \left|z\right|\right) \cdot \left|t\right|}{16}\right)\right) \cdot t\_1 \leq 5 \cdot 10^{+103}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(0.0625 \cdot \left|t\right|, \mathsf{fma}\left(y + y, \left|z\right|, \left|z\right|\right), \pi \cdot 0.5\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) (fabs t)) 16.0))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) (fabs z)) (fabs t)) 16.0)))
         t_1)
        5e+103)
     (*
      (*
       x
       (sin
        (fma (* 0.0625 (fabs t)) (fma (+ y y) (fabs z) (fabs z)) (* PI 0.5))))
      t_1)
     (* x (sin (* 0.5 PI))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos((((((a * 2.0) + 1.0) * b) * fabs(t)) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * fabs(z)) * fabs(t)) / 16.0))) * t_1) <= 5e+103) {
		tmp = (x * sin(fma((0.0625 * fabs(t)), fma((y + y), fabs(z), fabs(z)), (((double) M_PI) * 0.5)))) * t_1;
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * abs(t)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * abs(z)) * abs(t)) / 16.0))) * t_1) <= 5e+103)
		tmp = Float64(Float64(x * sin(fma(Float64(0.0625 * abs(t)), fma(Float64(y + y), abs(z), abs(z)), Float64(pi * 0.5)))) * t_1);
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 5e+103], N[(N[(x * N[Sin[N[(N[(0.0625 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(y + y), $MachinePrecision] * N[Abs[z], $MachinePrecision] + N[Abs[z], $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5e103

   1. Initial program 27.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 25.2%

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

   4. Applied rewrites 25.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 25.4%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 27.5%

      \[\leadsto \left(x \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.0625 \cdot t, \mathsf{fma}\left(y + y, z, z\right), \pi \cdot 0.5\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   if 5e103 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.1%

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

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \]
   6. Step-by-step derivation

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 29.7%

         \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]

   7. Applied rewrites 29.7%

      \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

         \[\leadsto x \cdot \cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \]

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

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. mult-flip N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 N/A

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \]

      16. metadata-eval 29.7%

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, \pi \cdot 0.5\right)\right) \]

   9. Applied rewrites 29.7%

      \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, \pi \cdot 0.5\right)\right) \]

   10. Taylor expanded in t around 0

       \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \pi\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-PI.f64 30.4%

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   12. Applied rewrites 30.4%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 31.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+294}:\\ \;\;\;\;t\_1 \cdot \cos \left(\frac{\left(b + 2 \cdot \left(a \cdot b\right)\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))))
   (if (<= (* t_1 (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))) 5e+294)
     (* t_1 (cos (/ (* (+ b (* 2.0 (* a b))) t) 16.0)))
     (* x (sin (* 0.5 PI))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+294) {
		tmp = t_1 * cos((((b + (2.0 * (a * b))) * t) / 16.0));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	double tmp;
	if ((t_1 * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+294) {
		tmp = t_1 * Math.cos((((b + (2.0 * (a * b))) * t) / 16.0));
	} else {
		tmp = x * Math.sin((0.5 * Math.PI));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))
	tmp = 0
	if (t_1 * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+294:
		tmp = t_1 * math.cos((((b + (2.0 * (a * b))) * t) / 16.0))
	else:
		tmp = x * math.sin((0.5 * math.pi))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0)))
	tmp = 0.0
	if (Float64(t_1 * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+294)
		tmp = Float64(t_1 * cos(Float64(Float64(Float64(b + Float64(2.0 * Float64(a * b))) * t) / 16.0)));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0));
	tmp = 0.0;
	if ((t_1 * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+294)
		tmp = t_1 * cos((((b + (2.0 * (a * b))) * t) / 16.0));
	else
		tmp = x * sin((0.5 * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[LessEqual[N[(t$95$1 * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+294], N[(t$95$1 * N[Cos[N[(N[(N[(b + N[(2.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 4.9999999999999999e294

   1. Initial program 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 27.5%

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

   4. Applied rewrites 27.5%

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

   if 4.9999999999999999e294 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.1%

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

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \]
   6. Step-by-step derivation

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 29.7%

         \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]

   7. Applied rewrites 29.7%

      \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

         \[\leadsto x \cdot \cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \]

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

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. mult-flip N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 N/A

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \]

      16. metadata-eval 29.7%

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, \pi \cdot 0.5\right)\right) \]

   9. Applied rewrites 29.7%

      \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, \pi \cdot 0.5\right)\right) \]

   10. Taylor expanded in t around 0

       \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \pi\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-PI.f64 30.4%

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   12. Applied rewrites 30.4%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 31.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\left(\cos \left(\left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t\right) \cdot -0.0625\right) \cdot x\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right) \cdot -0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      5e+294)
   (*
    (* (cos (* (* (* b (fma a 2.0 1.0)) t) -0.0625)) x)
    (cos (* (* t (* z (fma 2.0 y 1.0))) -0.0625)))
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+294) {
		tmp = (cos((((b * fma(a, 2.0, 1.0)) * t) * -0.0625)) * x) * cos(((t * (z * fma(2.0, y, 1.0))) * -0.0625));
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (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))) <= 5e+294)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(b * fma(a, 2.0, 1.0)) * t) * -0.0625)) * x) * cos(Float64(Float64(t * Float64(z * fma(2.0, y, 1.0))) * -0.0625)));
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 5e+294], N[(N[(N[Cos[N[(N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Cos[N[(N[(t * N[(z * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 4.9999999999999999e294

   1. Initial program 27.5%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 27.5%

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

   if 4.9999999999999999e294 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.1%

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

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \]
   6. Step-by-step derivation

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 29.7%

         \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]

   7. Applied rewrites 29.7%

      \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

         \[\leadsto x \cdot \cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \]

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

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. mult-flip N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 N/A

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \]

      16. metadata-eval 29.7%

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, \pi \cdot 0.5\right)\right) \]

   9. Applied rewrites 29.7%

      \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, \pi \cdot 0.5\right)\right) \]

   10. Taylor expanded in t around 0

       \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \pi\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-PI.f64 30.4%

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   12. Applied rewrites 30.4%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 31.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\left(\cos \left(\left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t\right) \cdot -0.0625\right) \cdot \cos \left(-0.125 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      5e+84)
   (*
    (*
     (cos (* (* (* b (fma a 2.0 1.0)) t) -0.0625))
     (cos (* -0.125 (* t (* y z)))))
    x)
   (* x (sin (* 0.5 PI)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+84) {
		tmp = (cos((((b * fma(a, 2.0, 1.0)) * t) * -0.0625)) * cos((-0.125 * (t * (y * z))))) * x;
	} else {
		tmp = x * sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (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))) <= 5e+84)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(b * fma(a, 2.0, 1.0)) * t) * -0.0625)) * cos(Float64(-0.125 * Float64(t * Float64(y * z))))) * x);
	else
		tmp = Float64(x * sin(Float64(0.5 * pi)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 5e+84], N[(N[(N[Cos[N[(N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.125 * N[(t * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5.0000000000000001e84

   1. Initial program 27.5%

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

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

   3. Applied rewrites 27.5%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 27.2%

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

   6. Applied rewrites 27.2%

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

   if 5.0000000000000001e84 < (*.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 27.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 28.1%

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

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \]
   6. Step-by-step derivation

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 29.7%

         \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]

   7. Applied rewrites 29.7%

      \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

         \[\leadsto x \cdot \cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \]

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

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. mult-flip N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 N/A

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \]

      16. metadata-eval 29.7%

          \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, \pi \cdot 0.5\right)\right) \]

   9. Applied rewrites 29.7%

      \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, \pi \cdot 0.5\right)\right) \]

   10. Taylor expanded in t around 0

       \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \pi\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-PI.f64 30.4%

          \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

   12. Applied rewrites 30.4%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 30.4% accurate, 2.6× speedup

Math

\[x \cdot \sin \left(0.5 \cdot \pi\right) \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* x (sin (* 0.5 PI))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * sin((0.5 * ((double) M_PI)));
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.sin((0.5 * Math.PI));
}

Python

def code(x, y, z, t, a, b):
	return x * math.sin((0.5 * math.pi))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * sin(Float64(0.5 * pi)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * sin((0.5 * pi));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

2. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 28.1%

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

4. Applied rewrites 28.1%

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

5. Taylor expanded in z around 0

   \[\leadsto x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \]
6. Step-by-step derivation

   1. lower-cos.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 29.7%

      \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]

7. Applied rewrites 29.7%

   \[\leadsto x \cdot \cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \]
8. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

      \[\leadsto x \cdot \cos \left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) \]

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

      \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16} \cdot \left(b \cdot t\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   6. distribute-lft-neg-in N/A

      \[\leadsto x \cdot \sin \left(\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   7. metadata-eval N/A

      \[\leadsto x \cdot \sin \left(\frac{-1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   8. metadata-eval N/A

      \[\leadsto x \cdot \sin \left(\frac{1}{-16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   9. metadata-eval N/A

      \[\leadsto x \cdot \sin \left(\frac{1}{\mathsf{neg}\left(16\right)} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(16\right)}, b \cdot t, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. mult-flip N/A

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

   14. lower-*.f64 N/A

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

   15. lower-PI.f64 N/A

       \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(\frac{-1}{16}, b \cdot t, \pi \cdot \frac{1}{2}\right)\right) \]

   16. metadata-eval 29.7%

       \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, \pi \cdot 0.5\right)\right) \]

9. Applied rewrites 29.7%

   \[\leadsto x \cdot \sin \left(\mathsf{fma}\left(-0.0625, b \cdot t, \pi \cdot 0.5\right)\right) \]

10. Taylor expanded in t around 0

    \[\leadsto x \cdot \sin \left(\frac{1}{2} \cdot \pi\right) \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

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

    2. lower-PI.f64 30.4%

       \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]

12. Applied rewrites 30.4%

    \[\leadsto x \cdot \sin \left(0.5 \cdot \pi\right) \]
13. Add Preprocessing

Alternative 6: 3.3% accurate, 4.8× speedup

Math

\[\left(\left(\left(-0.0078125 \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot z\right)\right) \cdot z\right) \cdot y\right) \cdot y \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* (* (* (* -0.0078125 (* (* (* t t) x) z)) z) y) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((-0.0078125 * (((t * t) * x) * z)) * z) * y) * y;
}

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 = ((((-0.0078125d0) * (((t * t) * x) * z)) * z) * y) * y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (((-0.0078125 * (((t * t) * x) * z)) * z) * y) * y;
}

Python

def code(x, y, z, t, a, b):
	return (((-0.0078125 * (((t * t) * x) * z)) * z) * y) * y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(Float64(-0.0078125 * Float64(Float64(Float64(t * t) * x) * z)) * z) * y) * y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (((-0.0078125 * (((t * t) * x) * z)) * z) * y) * y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(-0.0078125 * N[(N[(N[(t * t), $MachinePrecision] * x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

2. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

      \[\leadsto x + \color{blue}{{t}^{2} \cdot \left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]

   3. lower-pow.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \left(\color{blue}{x} \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \left(x \cdot \mathsf{fma}\left(\frac{-1}{512}, \color{blue}{{b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}}, \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right) \]

4. Applied rewrites 13.9%

   \[\leadsto \color{blue}{x + {t}^{2} \cdot \left(x \cdot \mathsf{fma}\left(-0.001953125, {b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}, -0.001953125 \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]

5. Taylor expanded in y around inf

   \[\leadsto \frac{-1}{128} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{\left({y}^{2} \cdot {z}^{2}\right)}\right)\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left(\color{blue}{{y}^{2}} \cdot {z}^{2}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot \color{blue}{{z}^{2}}\right)\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{\color{blue}{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right) \]

   7. lower-pow.f64 2.2%

      \[\leadsto -0.0078125 \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right) \]

7. Applied rewrites 2.2%

   \[\leadsto -0.0078125 \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{\left({y}^{2} \cdot {z}^{2}\right)}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot \color{blue}{{z}^{2}}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left({t}^{2} \cdot x\right) \cdot \left({y}^{2} \cdot \color{blue}{{z}^{2}}\right)\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left({t}^{2} \cdot x\right) \cdot \left({y}^{2} \cdot {z}^{\color{blue}{2}}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left({t}^{2} \cdot x\right) \cdot \left({z}^{2} \cdot {y}^{\color{blue}{2}}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(\left({t}^{2} \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{\color{blue}{2}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(\left({t}^{2} \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{\color{blue}{2}}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(\left({t}^{2} \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   9. lower-*.f64 2.2%

      \[\leadsto -0.0078125 \cdot \left(\left(\left({t}^{2} \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left({t}^{2} \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   12. lower-*.f64 2.2%

       \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   14. unpow2 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot {y}^{2}\right) \]

   15. lower-*.f64 2.2%

       \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot {y}^{2}\right) \]

   16. lift-pow.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot {y}^{2}\right) \]

   17. unpow2 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot \left(y \cdot y\right)\right) \]

   18. lower-*.f64 2.2%

       \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot \left(y \cdot y\right)\right) \]

9. Applied rewrites 2.2%

   \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot \left(y \cdot \color{blue}{y}\right)\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot \left(y \cdot \color{blue}{y}\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \left(\frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right)\right) \cdot \left(y \cdot \color{blue}{y}\right) \]

    4. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right)\right) \cdot \left(y \cdot y\right) \]

    5. associate-*r* N/A

       \[\leadsto \left(\left(\frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right)\right) \cdot y\right) \cdot y \]

    6. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right)\right) \cdot y\right) \cdot y \]

    7. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right)\right) \cdot y\right) \cdot y \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right)\right) \cdot y\right) \cdot y \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right)\right) \cdot y\right) \cdot y \]

    10. associate-*r* N/A

        \[\leadsto \left(\left(\frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot z\right) \cdot z\right)\right) \cdot y\right) \cdot y \]

    11. associate-*r* N/A

        \[\leadsto \left(\left(\left(\frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot z\right)\right) \cdot z\right) \cdot y\right) \cdot y \]

    12. lower-*.f64 N/A

        \[\leadsto \left(\left(\left(\frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot z\right)\right) \cdot z\right) \cdot y\right) \cdot y \]

    13. lower-*.f64 N/A

        \[\leadsto \left(\left(\left(\frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot z\right)\right) \cdot z\right) \cdot y\right) \cdot y \]

    14. lower-*.f64 3.3%

        \[\leadsto \left(\left(\left(-0.0078125 \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot z\right)\right) \cdot z\right) \cdot y\right) \cdot y \]

11. Applied rewrites 3.3%

    \[\leadsto \left(\left(\left(-0.0078125 \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot z\right)\right) \cdot z\right) \cdot y\right) \cdot y \]
12. Add Preprocessing

Alternative 7: 3.2% accurate, 4.8× speedup

Math

\[-0.0078125 \cdot \left(\left(\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot z\right) \cdot z\right) \cdot t\right) \cdot t\right) \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* -0.0078125 (* (* (* (* (* (* y y) x) z) z) t) t)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return -0.0078125 * ((((((y * y) * x) * z) * z) * t) * t);
}

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 = (-0.0078125d0) * ((((((y * y) * x) * z) * z) * t) * t)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return -0.0078125 * ((((((y * y) * x) * z) * z) * t) * t);
}

Python

def code(x, y, z, t, a, b):
	return -0.0078125 * ((((((y * y) * x) * z) * z) * t) * t)

Julia

function code(x, y, z, t, a, b)
	return Float64(-0.0078125 * Float64(Float64(Float64(Float64(Float64(Float64(y * y) * x) * z) * z) * t) * t))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = -0.0078125 * ((((((y * y) * x) * z) * z) * t) * t);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(-0.0078125 * N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

2. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

      \[\leadsto x + \color{blue}{{t}^{2} \cdot \left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]

   3. lower-pow.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \left(\color{blue}{x} \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \left(x \cdot \mathsf{fma}\left(\frac{-1}{512}, \color{blue}{{b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}}, \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right) \]

4. Applied rewrites 13.9%

   \[\leadsto \color{blue}{x + {t}^{2} \cdot \left(x \cdot \mathsf{fma}\left(-0.001953125, {b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}, -0.001953125 \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]

5. Taylor expanded in y around inf

   \[\leadsto \frac{-1}{128} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{\left({y}^{2} \cdot {z}^{2}\right)}\right)\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left(\color{blue}{{y}^{2}} \cdot {z}^{2}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot \color{blue}{{z}^{2}}\right)\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{\color{blue}{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right) \]

   7. lower-pow.f64 2.2%

      \[\leadsto -0.0078125 \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right) \]

7. Applied rewrites 2.2%

   \[\leadsto -0.0078125 \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{\left({y}^{2} \cdot {z}^{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right) \cdot {t}^{\color{blue}{2}}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right) \cdot {t}^{2}\right) \]

   4. unpow2 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right) \cdot \left(t \cdot t\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right) \cdot t\right) \cdot t\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right) \cdot t\right) \cdot t\right) \]

   7. lower-*.f64 2.7%

      \[\leadsto -0.0078125 \cdot \left(\left(\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right) \cdot t\right) \cdot t\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right) \cdot t\right) \cdot t\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right) \cdot t\right) \cdot t\right) \]

   10. associate-*r* N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(x \cdot {y}^{2}\right) \cdot {z}^{2}\right) \cdot t\right) \cdot t\right) \]

   11. lift-pow.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(x \cdot {y}^{2}\right) \cdot {z}^{2}\right) \cdot t\right) \cdot t\right) \]

   12. unpow2 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(x \cdot {y}^{2}\right) \cdot \left(z \cdot z\right)\right) \cdot t\right) \cdot t\right) \]

   13. associate-*r* N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(\left(x \cdot {y}^{2}\right) \cdot z\right) \cdot z\right) \cdot t\right) \cdot t\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(\left(x \cdot {y}^{2}\right) \cdot z\right) \cdot z\right) \cdot t\right) \cdot t\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(\left(x \cdot {y}^{2}\right) \cdot z\right) \cdot z\right) \cdot t\right) \cdot t\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(\left({y}^{2} \cdot x\right) \cdot z\right) \cdot z\right) \cdot t\right) \cdot t\right) \]

   17. lower-*.f64 3.2%

       \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(\left({y}^{2} \cdot x\right) \cdot z\right) \cdot z\right) \cdot t\right) \cdot t\right) \]

   18. lift-pow.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(\left({y}^{2} \cdot x\right) \cdot z\right) \cdot z\right) \cdot t\right) \cdot t\right) \]

   19. unpow2 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot z\right) \cdot z\right) \cdot t\right) \cdot t\right) \]

   20. lower-*.f64 3.2%

       \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot z\right) \cdot z\right) \cdot t\right) \cdot t\right) \]

9. Applied rewrites 3.2%

   \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot z\right) \cdot z\right) \cdot t\right) \cdot t\right) \]
10. Add Preprocessing

Alternative 8: 2.9% accurate, 4.8× speedup

Math

\[-0.0078125 \cdot \left(t \cdot \left(\left(t \cdot x\right) \cdot \left(\left(\left(z \cdot z\right) \cdot y\right) \cdot y\right)\right)\right) \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* -0.0078125 (* t (* (* t x) (* (* (* z z) y) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return -0.0078125 * (t * ((t * x) * (((z * z) * y) * y)));
}

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 = (-0.0078125d0) * (t * ((t * x) * (((z * z) * y) * y)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return -0.0078125 * (t * ((t * x) * (((z * z) * y) * y)));
}

Python

def code(x, y, z, t, a, b):
	return -0.0078125 * (t * ((t * x) * (((z * z) * y) * y)))

Julia

function code(x, y, z, t, a, b)
	return Float64(-0.0078125 * Float64(t * Float64(Float64(t * x) * Float64(Float64(Float64(z * z) * y) * y))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = -0.0078125 * (t * ((t * x) * (((z * z) * y) * y)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(-0.0078125 * N[(t * N[(N[(t * x), $MachinePrecision] * N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

2. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

      \[\leadsto x + \color{blue}{{t}^{2} \cdot \left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]

   3. lower-pow.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \left(\color{blue}{x} \cdot \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right) + \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto x + {t}^{2} \cdot \left(x \cdot \mathsf{fma}\left(\frac{-1}{512}, \color{blue}{{b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}}, \frac{-1}{512} \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right) \]

4. Applied rewrites 13.9%

   \[\leadsto \color{blue}{x + {t}^{2} \cdot \left(x \cdot \mathsf{fma}\left(-0.001953125, {b}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}, -0.001953125 \cdot \left({z}^{2} \cdot {\left(1 + 2 \cdot y\right)}^{2}\right)\right)\right)} \]

5. Taylor expanded in y around inf

   \[\leadsto \frac{-1}{128} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \color{blue}{\left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{\left({y}^{2} \cdot {z}^{2}\right)}\right)\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left(\color{blue}{{y}^{2}} \cdot {z}^{2}\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot \color{blue}{{z}^{2}}\right)\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{\color{blue}{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right) \]

   7. lower-pow.f64 2.2%

      \[\leadsto -0.0078125 \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right) \]

7. Applied rewrites 2.2%

   \[\leadsto -0.0078125 \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot {z}^{2}\right)\right)\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \color{blue}{\left({y}^{2} \cdot {z}^{2}\right)}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left({t}^{2} \cdot \left(x \cdot \left({y}^{2} \cdot \color{blue}{{z}^{2}}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left({t}^{2} \cdot x\right) \cdot \left({y}^{2} \cdot \color{blue}{{z}^{2}}\right)\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left({t}^{2} \cdot x\right) \cdot \left({y}^{2} \cdot {z}^{\color{blue}{2}}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left({t}^{2} \cdot x\right) \cdot \left({z}^{2} \cdot {y}^{\color{blue}{2}}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(\left({t}^{2} \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{\color{blue}{2}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(\left({t}^{2} \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{\color{blue}{2}}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{-1}{128} \cdot \left(\left(\left({t}^{2} \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   9. lower-*.f64 2.2%

      \[\leadsto -0.0078125 \cdot \left(\left(\left({t}^{2} \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left({t}^{2} \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   12. lower-*.f64 2.2%

       \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot {z}^{2}\right) \cdot {y}^{2}\right) \]

   14. unpow2 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot {y}^{2}\right) \]

   15. lower-*.f64 2.2%

       \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot {y}^{2}\right) \]

   16. lift-pow.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot {y}^{2}\right) \]

   17. unpow2 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot \left(y \cdot y\right)\right) \]

   18. lower-*.f64 2.2%

       \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot \left(y \cdot y\right)\right) \]

9. Applied rewrites 2.2%

   \[\leadsto -0.0078125 \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot \left(y \cdot \color{blue}{y}\right)\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot \left(y \cdot \color{blue}{y}\right)\right) \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(z \cdot z\right)\right) \cdot \left(y \cdot y\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(\left(z \cdot z\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(\left(z \cdot z\right) \cdot \left(\color{blue}{y} \cdot y\right)\right)\right) \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \left(\left(z \cdot z\right) \cdot \left(y \cdot y\right)\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \frac{-1}{128} \cdot \left(\left(t \cdot \left(t \cdot x\right)\right) \cdot \left(\left(z \cdot z\right) \cdot \left(\color{blue}{y} \cdot y\right)\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \frac{-1}{128} \cdot \left(t \cdot \left(\left(t \cdot x\right) \cdot \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot y\right)\right)}\right)\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(t \cdot \left(\left(t \cdot x\right) \cdot \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot y\right)\right)}\right)\right) \]

    9. lower-*.f64 N/A

       \[\leadsto \frac{-1}{128} \cdot \left(t \cdot \left(\left(t \cdot x\right) \cdot \left(\left(z \cdot z\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right) \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{-1}{128} \cdot \left(t \cdot \left(\left(t \cdot x\right) \cdot \left(\left(z \cdot z\right) \cdot \left(\color{blue}{y} \cdot y\right)\right)\right)\right) \]

    11. lift-*.f64 N/A

        \[\leadsto \frac{-1}{128} \cdot \left(t \cdot \left(\left(t \cdot x\right) \cdot \left(\left(z \cdot z\right) \cdot \left(y \cdot y\right)\right)\right)\right) \]

    12. associate-*r* N/A

        \[\leadsto \frac{-1}{128} \cdot \left(t \cdot \left(\left(t \cdot x\right) \cdot \left(\left(\left(z \cdot z\right) \cdot y\right) \cdot y\right)\right)\right) \]

    13. lower-*.f64 N/A

        \[\leadsto \frac{-1}{128} \cdot \left(t \cdot \left(\left(t \cdot x\right) \cdot \left(\left(\left(z \cdot z\right) \cdot y\right) \cdot y\right)\right)\right) \]

    14. lower-*.f64 2.9%

        \[\leadsto -0.0078125 \cdot \left(t \cdot \left(\left(t \cdot x\right) \cdot \left(\left(\left(z \cdot z\right) \cdot y\right) \cdot y\right)\right)\right) \]

11. Applied rewrites 2.9%

    \[\leadsto -0.0078125 \cdot \left(t \cdot \left(\left(t \cdot x\right) \cdot \color{blue}{\left(\left(\left(z \cdot z\right) \cdot y\right) \cdot y\right)}\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025202 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64
  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))