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

Percentage Accurate: 28.3% → 31.7%

Time: 6.5s

Alternatives: 5

Speedup: 269.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 31.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{+17}:\\ \;\;\;\;\left(\sin \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\_m\right) \cdot b\_m, 0.0625, \frac{\pi}{2}\right)\right) \cdot x\right) \cdot \sin \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot t\_m\right) \cdot 0.0625, -z, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<= t_m 2.3e+17)
   (*
    (* (sin (fma (* (* (fma a 2.0 1.0) t_m) b_m) 0.0625 (/ PI 2.0))) x)
    (sin (fma (* (* (fma 2.0 y 1.0) t_m) 0.0625) (- z) (/ PI 2.0))))
   x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (t_m <= 2.3e+17) {
		tmp = (sin(fma(((fma(a, 2.0, 1.0) * t_m) * b_m), 0.0625, (((double) M_PI) / 2.0))) * x) * sin(fma(((fma(2.0, y, 1.0) * t_m) * 0.0625), -z, (((double) M_PI) / 2.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (t_m <= 2.3e+17)
		tmp = Float64(Float64(sin(fma(Float64(Float64(fma(a, 2.0, 1.0) * t_m) * b_m), 0.0625, Float64(pi / 2.0))) * x) * sin(fma(Float64(Float64(fma(2.0, y, 1.0) * t_m) * 0.0625), Float64(-z), Float64(pi / 2.0))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[t$95$m, 2.3e+17], N[(N[(N[Sin[N[(N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * 0.0625 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Sin[N[(N[(N[(N[(2.0 * y + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * 0.0625), $MachinePrecision] * (-z) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 2.3e17

   1. Initial program 46.3%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 47.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-cos.f64 N/A

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

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

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 47.1%

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

   8. Applied rewrites 47.8%

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

   if 2.3e17 < t

   1. Initial program 7.7%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 13.1%

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

Alternative 2: 31.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{+43}:\\ \;\;\;\;\left(\cos \left(\left(t\_m \cdot z\right) \cdot 0.0625\right) \cdot \sin \left(\mathsf{fma}\left(0.0625 \cdot b\_m, \mathsf{fma}\left(a, 2, 1\right) \cdot t\_m, 0.5 \cdot \pi\right)\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<= t_m 1.35e+43)
   (*
    (*
     (cos (* (* t_m z) 0.0625))
     (sin (fma (* 0.0625 b_m) (* (fma a 2.0 1.0) t_m) (* 0.5 PI))))
    x)
   x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (t_m <= 1.35e+43) {
		tmp = (cos(((t_m * z) * 0.0625)) * sin(fma((0.0625 * b_m), (fma(a, 2.0, 1.0) * t_m), (0.5 * ((double) M_PI))))) * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (t_m <= 1.35e+43)
		tmp = Float64(Float64(cos(Float64(Float64(t_m * z) * 0.0625)) * sin(fma(Float64(0.0625 * b_m), Float64(fma(a, 2.0, 1.0) * t_m), Float64(0.5 * pi)))) * x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[t$95$m, 1.35e+43], N[(N[(N[Cos[N[(N[(t$95$m * z), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(0.0625 * b$95$m), $MachinePrecision] * N[(N[(a * 2.0 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 1.3500000000000001e43

   1. Initial program 44.3%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 45.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-cos.f64 N/A

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

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

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 45.0%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 45.6%

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

   if 1.3500000000000001e43 < t

   1. Initial program 6.8%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 12.6%

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

Alternative 3: 31.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{+17}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\_m\right) \cdot b\_m\right) \cdot 0.0625\right) \cdot x\right) \cdot \cos \left(\left(t\_m \cdot z\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<= t_m 2.3e+17)
   (*
    (* (cos (* (* (* (fma a 2.0 1.0) t_m) b_m) 0.0625)) x)
    (cos (* (* t_m z) 0.0625)))
   x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (t_m <= 2.3e+17) {
		tmp = (cos((((fma(a, 2.0, 1.0) * t_m) * b_m) * 0.0625)) * x) * cos(((t_m * z) * 0.0625));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (t_m <= 2.3e+17)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(fma(a, 2.0, 1.0) * t_m) * b_m) * 0.0625)) * x) * cos(Float64(Float64(t_m * z) * 0.0625)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[t$95$m, 2.3e+17], N[(N[(N[Cos[N[(N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * N[Cos[N[(N[(t$95$m * z), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 2.3e17

   1. Initial program 46.3%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 47.8%

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

   if 2.3e17 < t

   1. Initial program 7.7%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 13.1%

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

Alternative 4: 31.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{+44}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(0.0625 \cdot b\_m, \mathsf{fma}\left(a, 2, 1\right) \cdot t\_m, 0.5 \cdot \pi\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
b_m = (fabs.f64 b)
(FPCore (x y z t_m a b_m)
 :precision binary64
 (if (<= t_m 2.7e+44)
   (* (sin (fma (* 0.0625 b_m) (* (fma a 2.0 1.0) t_m) (* 0.5 PI))) x)
   x))

C

t_m = fabs(t);
b_m = fabs(b);
double code(double x, double y, double z, double t_m, double a, double b_m) {
	double tmp;
	if (t_m <= 2.7e+44) {
		tmp = sin(fma((0.0625 * b_m), (fma(a, 2.0, 1.0) * t_m), (0.5 * ((double) M_PI)))) * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

t_m = abs(t)
b_m = abs(b)
function code(x, y, z, t_m, a, b_m)
	tmp = 0.0
	if (t_m <= 2.7e+44)
		tmp = Float64(sin(fma(Float64(0.0625 * b_m), Float64(fma(a, 2.0, 1.0) * t_m), Float64(0.5 * pi))) * x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[x_, y_, z_, t$95$m_, a_, b$95$m_] := If[LessEqual[t$95$m, 2.7e+44], N[(N[Sin[N[(N[(0.0625 * b$95$m), $MachinePrecision] * N[(N[(a * 2.0 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 2.7e44

   1. Initial program 44.2%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 45.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-cos.f64 N/A

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

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

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 45.1%

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

   if 2.7e44 < t

   1. Initial program 6.7%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 12.5%

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

Alternative 5: 31.1% accurate, 269.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.3%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 31.1%

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

Developer Target 1: 30.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025106 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (cos (* (/ b 16) (/ t (+ (- 1 (* a 2)) (pow (* a 2) 2)))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))