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

Percentage Accurate: 27.4% → 28.8%

Time: 12.1s

Alternatives: 4

Speedup: 2.3×

• Could not converge on a ground truth

  1. x = 8.764474230034079e+102
  2. y = -8652374776987756.0
  3. z = 1.1228063791833419e+229
  4. t = 6.713240897337195e-61
  5. a = 6.183669332889124e-171
  6. b = 2.0685618314248252e-168

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

real(8) function code(x, y, z, t, a, b)
    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: 27.4% 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

real(8) function code(x, y, z, t, a, b)
    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: 28.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(\left(a \cdot t\right) \cdot b\right) \cdot 0.125\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (cos (* (* (* a t) b) 0.125)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return cos((((a * t) * b) * 0.125)) * x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 = cos((((a * t) * b) * 0.125d0)) * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return Math.cos((((a * t) * b) * 0.125)) * x;
}

Python

def code(x, y, z, t, a, b):
	return math.cos((((a * t) * b) * 0.125)) * x

Julia

function code(x, y, z, t, a, b)
	return Float64(cos(Float64(Float64(Float64(a * t) * b) * 0.125)) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = cos((((a * t) * b) * 0.125)) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[Cos[N[(N[(N[(a * t), $MachinePrecision] * b), $MachinePrecision] * 0.125), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.8%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 28.8

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

5. Applied rewrites 28.8%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 30.3

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

8. Applied rewrites 30.3%

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

9. Taylor expanded in a around inf

   \[\leadsto \cos \left(\frac{1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right)\right) \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 30.3%

    \[\leadsto \cos \left(\left(\left(b \cdot t\right) \cdot a\right) \cdot 0.125\right) \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 30.8%

    \[\leadsto \cos \left(\left(\left(a \cdot t\right) \cdot b\right) \cdot 0.125\right) \cdot x \]
14. Add Preprocessing

Alternative 2: 29.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (cos (* (* b t) 0.0625)) x))

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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 = cos(((b * t) * 0.0625d0)) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[Cos[N[(N[(b * t), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.8%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 28.8

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

5. Applied rewrites 28.8%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 30.3

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

8. Applied rewrites 30.3%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 30.6%

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

Alternative 3: 23.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(t \cdot t\right) \cdot \mathsf{fma}\left(-0.0078125, a, -0.001953125\right), b \cdot b, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* (fma (* (* t t) (fma -0.0078125 a -0.001953125)) (* b b) 1.0) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(((t * t) * fma(-0.0078125, a, -0.001953125)), (b * b), 1.0) * x;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(fma(Float64(Float64(t * t) * fma(-0.0078125, a, -0.001953125)), Float64(b * b), 1.0) * x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(t * t), $MachinePrecision] * N[(-0.0078125 * a + -0.001953125), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.8%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 28.8

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

5. Applied rewrites 28.8%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 30.3

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

8. Applied rewrites 30.3%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 27.9%

    \[\leadsto \mathsf{fma}\left(-0.125 \cdot a, \left(b \cdot t\right) \cdot \sin \left(\left(b \cdot t\right) \cdot 0.0625\right), \cos \left(\left(b \cdot t\right) \cdot 0.0625\right)\right) \cdot x \]

12. Taylor expanded in b around 0

    \[\leadsto \left(1 + {b}^{2} \cdot \left(\frac{-1}{128} \cdot \left(a \cdot {t}^{2}\right) + \frac{-1}{512} \cdot {t}^{2}\right)\right) \cdot x \]
13. Step-by-step derivation

14. Applied rewrites 26.0%

    \[\leadsto \mathsf{fma}\left(\left(t \cdot t\right) \cdot \mathsf{fma}\left(-0.0078125, a, -0.001953125\right), b \cdot b, 1\right) \cdot x \]
15. Add Preprocessing

Alternative 4: 23.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.0078125, a, -0.001953125\right), t \cdot t, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* (fma (* (* b b) (fma -0.0078125 a -0.001953125)) (* t t) 1.0) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(((b * b) * fma(-0.0078125, a, -0.001953125)), (t * t), 1.0) * x;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(fma(Float64(Float64(b * b) * fma(-0.0078125, a, -0.001953125)), Float64(t * t), 1.0) * x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(b * b), $MachinePrecision] * N[(-0.0078125 * a + -0.001953125), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.8%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 28.8

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

5. Applied rewrites 28.8%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 30.3

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

8. Applied rewrites 30.3%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 27.9%

    \[\leadsto \mathsf{fma}\left(-0.125 \cdot a, \left(b \cdot t\right) \cdot \sin \left(\left(b \cdot t\right) \cdot 0.0625\right), \cos \left(\left(b \cdot t\right) \cdot 0.0625\right)\right) \cdot x \]

12. Taylor expanded in t around 0

    \[\leadsto \left(1 + {t}^{2} \cdot \left(\frac{-1}{128} \cdot \left(a \cdot {b}^{2}\right) + \frac{-1}{512} \cdot {b}^{2}\right)\right) \cdot x \]
13. Step-by-step derivation

14. Applied rewrites 25.7%

    \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.0078125, a, -0.001953125\right), t \cdot t, 1\right) \cdot x \]
15. Add Preprocessing

Developer Target 1: 30.2% 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

real(8) function code(x, y, z, t, a, b)
    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 2024313 
(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))))