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

Percentage Accurate: 27.6% → 30.7%

Time: 6.4s

Alternatives: 4

Speedup: 27.0×

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: 27.6% 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: 30.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ \sin \left(\mathsf{fma}\left(z\_m, 0.0625 \cdot t\_m, \frac{\pi}{2}\right)\right) \cdot x \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (* (sin (fma z_m (* 0.0625 t_m) (/ PI 2.0))) x))

C

z_m = fabs(z);
t_m = fabs(t);
double code(double x, double y, double z_m, double t_m, double a, double b) {
	return sin(fma(z_m, (0.0625 * t_m), (((double) M_PI) / 2.0))) * x;
}

Julia

z_m = abs(z)
t_m = abs(t)
function code(x, y, z_m, t_m, a, b)
	return Float64(sin(fma(z_m, Float64(0.0625 * t_m), Float64(pi / 2.0))) * x)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z$95$m_, t$95$m_, a_, b_] := N[(N[Sin[N[(z$95$m * N[(0.0625 * t$95$m), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.6%

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

2. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. associate-*r* N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 28.7

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

4. Applied rewrites 28.7%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 29.7%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 29.7

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

9. Applied rewrites 29.7%

   \[\leadsto \sin \left(\mathsf{fma}\left(z, 0.0625 \cdot t, \frac{\pi}{2}\right)\right) \cdot x \]
10. Add Preprocessing

Alternative 2: 29.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ \sin \left(\mathsf{fma}\left(t\_m \cdot z\_m, 0.0625, 0.5 \cdot \pi\right)\right) \cdot x \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (* (sin (fma (* t_m z_m) 0.0625 (* 0.5 PI))) x))

C

z_m = fabs(z);
t_m = fabs(t);
double code(double x, double y, double z_m, double t_m, double a, double b) {
	return sin(fma((t_m * z_m), 0.0625, (0.5 * ((double) M_PI)))) * x;
}

Julia

z_m = abs(z)
t_m = abs(t)
function code(x, y, z_m, t_m, a, b)
	return Float64(sin(fma(Float64(t_m * z_m), 0.0625, Float64(0.5 * pi))) * x)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z$95$m_, t$95$m_, a_, b_] := N[(N[Sin[N[(N[(t$95$m * z$95$m), $MachinePrecision] * 0.0625 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.6%

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

2. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. associate-*r* N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 28.7

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

4. Applied rewrites 28.7%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 29.7%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 29.7

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

9. Applied rewrites 29.7%

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

10. Taylor expanded in y around 0

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

    1. *-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-PI.f64 29.7

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

12. Applied rewrites 29.7%

    \[\leadsto \sin \left(\mathsf{fma}\left(t \cdot z, 0.0625, 0.5 \cdot \pi\right)\right) \cdot x \]
13. Add Preprocessing

Alternative 3: 29.7% accurate, 3.4× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (* (cos (* (* 0.0625 t_m) z_m)) x))

C

z_m = fabs(z);
t_m = fabs(t);
double code(double x, double y, double z_m, double t_m, double a, double b) {
	return cos(((0.0625 * t_m) * z_m)) * x;
}

Fortran

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

Java

z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x, double y, double z_m, double t_m, double a, double b) {
	return Math.cos(((0.0625 * t_m) * z_m)) * x;
}

Python

z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x, y, z_m, t_m, a, b):
	return math.cos(((0.0625 * t_m) * z_m)) * x

Julia

z_m = abs(z)
t_m = abs(t)
function code(x, y, z_m, t_m, a, b)
	return Float64(cos(Float64(Float64(0.0625 * t_m) * z_m)) * x)
end

MATLAB

z_m = abs(z);
t_m = abs(t);
function tmp = code(x, y, z_m, t_m, a, b)
	tmp = cos(((0.0625 * t_m) * z_m)) * x;
end

Wolfram

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

Derivation

1. Initial program 27.6%

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

2. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. associate-*r* N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 28.7

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

4. Applied rewrites 28.7%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 29.7%

   \[\leadsto \cos \left(\left(0.0625 \cdot t\right) \cdot z\right) \cdot x \]
8. Add Preprocessing

Alternative 4: 29.7% accurate, 27.0× speedup

Math

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

FPCore

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

C

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

Fortran

z_m =     private
t_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_m, t_m, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z$95$m_, t$95$m_, a_, b_] := x

Derivation

1. Initial program 27.6%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 30.7%

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

Reproduce

herbie shell --seed 2025101 
(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))))