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

Percentage Accurate: 27.2% → 31.7%

Time: 7.9s

Alternatives: 4

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: 27.2% 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, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\left(\sin \left(\left(-0.0625\right) \cdot \left(\left(b\_m \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x\_m\right) \cdot \cos \left(\left(0.0625 \cdot t\right) \cdot \left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b_m)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t) 16.0)))
       2e+299)
    (*
     (*
      (sin (+ (* (- 0.0625) (* (* b_m (fma a 2.0 1.0)) t)) (/ (PI) 2.0)))
      x_m)
     (cos (* (* 0.0625 t) (* (fma 2.0 y 1.0) z))))
    x_m)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.0000000000000001e299

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 44.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      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-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-sin.f64 N/A

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

      14. lower-+.f64 N/A

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

   7. Applied rewrites 43.8%

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

   if 2.0000000000000001e299 < (*.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 0.4%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 11.8%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\left(\sin \left(\left(-0.0625\right) \cdot \left(\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x\right) \cdot \cos \left(\left(0.0625 \cdot t\right) \cdot \left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 31.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t}{16}\right) \leq 10^{+181}:\\ \;\;\;\;\left(\cos \left(\left(0.0625 \cdot t\right) \cdot z\right) \cdot x\_m\right) \cdot \sin \left(\mathsf{fma}\left(-0.0625, \left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\_m, 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b_m)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t) 16.0)))
       1e+181)
    (*
     (* (cos (* (* 0.0625 t) z)) x_m)
     (sin (fma -0.0625 (* (* (fma a 2.0 1.0) t) b_m) (* 0.5 (PI)))))
    x_m)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 9.9999999999999992e180

   1. Initial program 43.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\left(\frac{t \cdot z}{y} \cdot \frac{1}{16} + \frac{1}{8} \cdot \left(t \cdot z\right)\right) \cdot y\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 42.9

         \[\leadsto \left(x \cdot \cos \left(\mathsf{fma}\left(\frac{t \cdot z}{y}, 0.0625, 0.125 \cdot \left(t \cdot z\right)\right) \cdot y\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Applied rewrites 42.9%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\frac{t \cdot z}{y}, 0.0625, 0.125 \cdot \left(t \cdot z\right)\right) \cdot y\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-neg-rev N/A

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

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

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

      9. lower-sin.f64 N/A

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

      10. lower-+.f64 N/A

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

   7. Applied rewrites 42.7%

      \[\leadsto \left(x \cdot \cos \left(\mathsf{fma}\left(\frac{t \cdot z}{y}, 0.0625, 0.125 \cdot \left(t \cdot z\right)\right) \cdot y\right)\right) \cdot \color{blue}{\sin \left(\left(-\frac{\left(b \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot t}{16}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 44.1%

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

   if 9.9999999999999992e180 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 5.8%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 14.7%

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

Alternative 3: 31.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\_m\right) \cdot t}{16}\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\cos \left(\left(\left(b\_m \cdot t\right) \cdot a\right) \cdot 0.125\right) \cdot x\_m\right) \cdot \cos \left(\left(0.0625 \cdot t\right) \cdot \left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b_m)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (* x_m (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
        (cos (/ (* (* (+ (* a 2.0) 1.0) b_m) t) 16.0)))
       2e+306)
    (*
     (* (cos (* (* (* b_m t) a) 0.125)) x_m)
     (cos (* (* 0.0625 t) (* (fma 2.0 y 1.0) z))))
    x_m)))

C

b_m = fabs(b);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b_m) {
	double tmp;
	if (((x_m * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b_m) * t) / 16.0))) <= 2e+306) {
		tmp = (cos((((b_m * t) * a) * 0.125)) * x_m) * cos(((0.0625 * t) * (fma(2.0, y, 1.0) * z)));
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Julia

b_m = abs(b)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b_m)
	tmp = 0.0
	if (Float64(Float64(x_m * 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_m) * t) / 16.0))) <= 2e+306)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(b_m * t) * a) * 0.125)) * x_m) * cos(Float64(Float64(0.0625 * t) * Float64(fma(2.0, y, 1.0) * z))));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b$95$m_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 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$95$m), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(N[Cos[N[(N[(N[(b$95$m * t), $MachinePrecision] * a), $MachinePrecision] * 0.125), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision] * N[Cos[N[(N[(0.0625 * t), $MachinePrecision] * N[(N[(2.0 * y + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 2.00000000000000003e306

   1. Initial program 43.6%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 43.5

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

   8. Applied rewrites 43.5%

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

   if 2.00000000000000003e306 < (*.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 0.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 11.9%

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

Alternative 4: 30.5% accurate, 269.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

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

C

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

Fortran

b_m =     private
x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z, t, a, b_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    code = x_s * x_m
end function

Java

b_m = Math.abs(b);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b_m) {
	return x_s * x_m;
}

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 24.5%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 28.1%

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

Developer Target 1: 29.9% 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 2025051 
(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))))