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

Percentage Accurate: 29.0% → 33.4%

Time: 7.9s

Alternatives: 7

Speedup: 100.6×

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: 29.0% 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: 33.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot \left|z\right|\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\left(\cos \left(\left(\left(b \cdot \left(a + \left(a - -1\right)\right)\right) \cdot t\right) \cdot -0.0625\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\mathsf{fma}\left(y + y, \left|z\right|, \left|z\right|\right) \cdot t\right) \cdot 0.0625\right)\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) (fabs z)) t) 16.0)))
       (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0)))
      5e+290)
   (*
    (*
     (cos (* (* (* b (+ a (- a -1.0))) t) -0.0625))
     (sin (fma 0.5 PI (* (* (fma (+ y y) (fabs z) (fabs z)) t) 0.0625))))
    x)
   x))

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) * fabs(z)) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+290) {
		tmp = (cos((((b * (a + (a - -1.0))) * t) * -0.0625)) * sin(fma(0.5, ((double) M_PI), ((fma((y + y), fabs(z), fabs(z)) * t) * 0.0625)))) * x;
	} else {
		tmp = x;
	}
	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) * abs(z)) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+290)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(b * Float64(a + Float64(a - -1.0))) * t) * -0.0625)) * sin(fma(0.5, pi, Float64(Float64(fma(Float64(y + y), abs(z), abs(z)) * t) * 0.0625)))) * x);
	else
		tmp = x;
	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] * N[Abs[z], $MachinePrecision]), $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+290], N[(N[(N[Cos[N[(N[(N[(b * N[(a + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * Pi + N[(N[(N[(N[(y + y), $MachinePrecision] * N[Abs[z], $MachinePrecision] + N[Abs[z], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 4.9999999999999998e290

   1. Initial program 29.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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 29.0%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. mult-flip N/A

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

      19. lift-/.f64 N/A

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

   5. Applied rewrites 29.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. count-2-rev N/A

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

      5. metadata-eval N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 29.0%

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

   7. Applied rewrites 29.0%

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

   if 4.9999999999999998e290 < (*.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 29.0%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. metadata-eval 29.0%

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

   3. Applied rewrites 29.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 32.1%

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

   6. Applied rewrites 32.1%

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

   7. Evaluated real constant 32.1%

      \[\leadsto x \cdot 1 \]

   8. Taylor expanded in x around 0

      \[\leadsto x \]
   9. Step-by-step derivation

   10. Applied rewrites 32.1%

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

Alternative 2: 33.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+279}:\\ \;\;\;\;t\_1 \cdot \cos \left(\frac{1}{\frac{16}{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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+279)
     (* t_1 (cos (/ 1.0 (/ 16.0 (* (* b (fma a 2.0 1.0)) t)))))
     x)))

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+279) {
		tmp = t_1 * cos((1.0 / (16.0 / ((b * fma(a, 2.0, 1.0)) * t))));
	} else {
		tmp = x;
	}
	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+279)
		tmp = Float64(t_1 * cos(Float64(1.0 / Float64(16.0 / Float64(Float64(b * fma(a, 2.0, 1.0)) * t)))));
	else
		tmp = x;
	end
	return 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+279], N[(t$95$1 * N[Cos[N[(1.0 / N[(16.0 / N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5.0000000000000002e279

   1. Initial program 29.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 28.9%

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 28.9%

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 28.9%

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

   3. Applied rewrites 28.9%

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

   if 5.0000000000000002e279 < (*.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 29.0%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. metadata-eval 29.0%

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

   3. Applied rewrites 29.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 32.1%

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

   6. Applied rewrites 32.1%

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

   7. Evaluated real constant 32.1%

      \[\leadsto x \cdot 1 \]

   8. Taylor expanded in x around 0

      \[\leadsto x \]
   9. Step-by-step derivation

   10. Applied rewrites 32.1%

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

Alternative 3: 33.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{\frac{1}{\cos \left(-0.0625 \cdot \left(\mathsf{fma}\left(a + a, t, t\right) \cdot b\right)\right) \cdot \cos \left(\left(\mathsf{fma}\left(z, y + y, z\right) \cdot t\right) \cdot -0.0625\right)}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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+279)
   (/
    x
    (/
     1.0
     (*
      (cos (* -0.0625 (* (fma (+ a a) t t) b)))
      (cos (* (* (fma z (+ y y) z) t) -0.0625)))))
   x))

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+279) {
		tmp = x / (1.0 / (cos((-0.0625 * (fma((a + a), t, t) * b))) * cos(((fma(z, (y + y), z) * t) * -0.0625))));
	} else {
		tmp = x;
	}
	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+279)
		tmp = Float64(x / Float64(1.0 / Float64(cos(Float64(-0.0625 * Float64(fma(Float64(a + a), t, t) * b))) * cos(Float64(Float64(fma(z, Float64(y + y), z) * t) * -0.0625)))));
	else
		tmp = x;
	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+279], N[(x / N[(1.0 / N[(N[Cos[N[(-0.0625 * N[(N[(N[(a + a), $MachinePrecision] * t + t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(z * N[(y + y), $MachinePrecision] + z), $MachinePrecision] * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5.0000000000000002e279

   1. Initial program 29.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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 29.0%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. mult-flip N/A

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

      19. lift-/.f64 N/A

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

   5. Applied rewrites 29.0%

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

   7. Applied rewrites 29.2%

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

   if 5.0000000000000002e279 < (*.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 29.0%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. metadata-eval 29.0%

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

   3. Applied rewrites 29.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 32.1%

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

   6. Applied rewrites 32.1%

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

   7. Evaluated real constant 32.1%

      \[\leadsto x \cdot 1 \]

   8. Taylor expanded in x around 0

      \[\leadsto x \]
   9. Step-by-step derivation

   10. Applied rewrites 32.1%

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

Alternative 4: 33.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+279}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot \left(t \cdot \left(1 - -2 \cdot a\right)\right)\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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+279)
   (*
    x
    (*
     (cos (* 0.0625 (* b (* t (- 1.0 (* -2.0 a))))))
     (cos (* 0.0625 (* t (* z (+ 1.0 (* 2.0 y))))))))
   x))

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+279) {
		tmp = x * (cos((0.0625 * (b * (t * (1.0 - (-2.0 * a)))))) * cos((0.0625 * (t * (z * (1.0 + (2.0 * y)))))));
	} else {
		tmp = x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))) <= 5d+279) then
        tmp = x * (cos((0.0625d0 * (b * (t * (1.0d0 - ((-2.0d0) * a)))))) * cos((0.0625d0 * (t * (z * (1.0d0 + (2.0d0 * y)))))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+279) {
		tmp = x * (Math.cos((0.0625 * (b * (t * (1.0 - (-2.0 * a)))))) * Math.cos((0.0625 * (t * (z * (1.0 + (2.0 * y)))))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+279:
		tmp = x * (math.cos((0.0625 * (b * (t * (1.0 - (-2.0 * a)))))) * math.cos((0.0625 * (t * (z * (1.0 + (2.0 * y)))))))
	else:
		tmp = x
	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+279)
		tmp = Float64(x * Float64(cos(Float64(0.0625 * Float64(b * Float64(t * Float64(1.0 - Float64(-2.0 * a)))))) * cos(Float64(0.0625 * Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y))))))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))) <= 5e+279)
		tmp = x * (cos((0.0625 * (b * (t * (1.0 - (-2.0 * a)))))) * cos((0.0625 * (t * (z * (1.0 + (2.0 * y)))))));
	else
		tmp = x;
	end
	tmp_2 = 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+279], N[(x * N[(N[Cos[N[(0.0625 * N[(b * N[(t * N[(1.0 - N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * N[(t * N[(z * N[(1.0 + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5.0000000000000002e279

   1. Initial program 29.0%

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

   2. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

   4. Applied rewrites 29.2%

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

   if 5.0000000000000002e279 < (*.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 29.0%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. metadata-eval 29.0%

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

   3. Applied rewrites 29.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 32.1%

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

   6. Applied rewrites 32.1%

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

   7. Evaluated real constant 32.1%

      \[\leadsto x \cdot 1 \]

   8. Taylor expanded in x around 0

      \[\leadsto x \]
   9. Step-by-step derivation

   10. Applied rewrites 32.1%

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

Alternative 5: 33.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\left(\cos \left(\left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t\right) \cdot -0.0625\right) \cdot x\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right) \cdot -0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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+279)
   (*
    (* (cos (* (* (* b (fma a 2.0 1.0)) t) -0.0625)) x)
    (cos (* (* t (* z (fma 2.0 y 1.0))) -0.0625)))
   x))

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+279) {
		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;
	}
	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+279)
		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 = x;
	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+279], 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], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5.0000000000000002e279

   1. Initial program 29.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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 29.0%

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

   if 5.0000000000000002e279 < (*.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 29.0%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. metadata-eval 29.0%

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

   3. Applied rewrites 29.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 32.1%

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

   6. Applied rewrites 32.1%

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

   7. Evaluated real constant 32.1%

      \[\leadsto x \cdot 1 \]

   8. Taylor expanded in x around 0

      \[\leadsto x \]
   9. Step-by-step derivation

   10. Applied rewrites 32.1%

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

Alternative 6: 33.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\left(\cos \left(\left(b \cdot t\right) \cdot -0.0625\right) \cdot \cos \left(\left(t \cdot \left(z \cdot \mathsf{fma}\left(2, y, 1\right)\right)\right) \cdot -0.0625\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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+279)
   (*
    (* (cos (* (* b t) -0.0625)) (cos (* (* t (* z (fma 2.0 y 1.0))) -0.0625)))
    x)
   x))

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+279) {
		tmp = (cos(((b * t) * -0.0625)) * cos(((t * (z * fma(2.0, y, 1.0))) * -0.0625))) * x;
	} else {
		tmp = x;
	}
	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+279)
		tmp = Float64(Float64(cos(Float64(Float64(b * t) * -0.0625)) * cos(Float64(Float64(t * Float64(z * fma(2.0, y, 1.0))) * -0.0625))) * x);
	else
		tmp = x;
	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+279], N[(N[(N[Cos[N[(N[(b * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t * N[(z * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 5.0000000000000002e279

   1. Initial program 29.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. 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 29.0%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 29.7%

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

   if 5.0000000000000002e279 < (*.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 29.0%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mult-flip N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. metadata-eval 29.0%

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

   3. Applied rewrites 29.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 32.1%

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

   6. Applied rewrites 32.1%

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

   7. Evaluated real constant 32.1%

      \[\leadsto x \cdot 1 \]

   8. Taylor expanded in x around 0

      \[\leadsto x \]
   9. Step-by-step derivation

   10. Applied rewrites 32.1%

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

Alternative 7: 32.1% accurate, 100.6× speedup

Math

\[x \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 29.0%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. mult-flip N/A

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

   16. lower-*.f64 N/A

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

   17. lower-PI.f64 N/A

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

   18. metadata-eval 29.0%

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

3. Applied rewrites 29.0%

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

4. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 32.1%

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

6. Applied rewrites 32.1%

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

7. Evaluated real constant 32.1%

   \[\leadsto x \cdot 1 \]

8. Taylor expanded in x around 0

   \[\leadsto x \]
9. Step-by-step derivation

10. Applied rewrites 32.1%

    \[\leadsto x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(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))))