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

Average Error: 46.1 → 44.0

Time: 17.5s

Precision: binary64

Cost: 64

Math

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

↓

\[x \]

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))))

↓

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

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	return 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 = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

↓

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
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));
}

↓

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

Python

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

↓

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

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

↓

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

↓

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

↓

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

Target

Original	46.1
Target	44.2
Herbie	44.0

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

Derivation

1. Initial program 46.1

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

2. Simplified 45.5

   \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + \frac{a}{8}\right)\right)\right)} \]
   Proof
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 1/16 (/.f64 y 8)))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite<= metadata-eval (*.f64 1/16 1)) (/.f64 y 8)))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 1 16)) 1) (/.f64 y 8)))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (*.f64 (/.f64 1 16) 1) (/.f64 y (Rewrite<= metadata-eval (/.f64 16 2)))))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (*.f64 (/.f64 1 16) 1) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y 2) 16))))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (*.f64 (/.f64 1 16) 1) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (*.f64 y 2))) 16)))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (*.f64 (/.f64 1 16) 1) (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 16 (*.f64 y 2))))))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 4 points increase in error, 1 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (*.f64 (/.f64 1 16) 1) (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 16) (*.f64 y 2)))))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 1 points increase in error, 4 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (Rewrite<= distribute-lft-in_binary64 (*.f64 (/.f64 1 16) (+.f64 1 (*.f64 y 2)))))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (*.f64 (/.f64 1 16) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 y 2) 1))))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 16 (+.f64 (*.f64 y 2) 1)))))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 4 points increase in error, 2 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1 (+.f64 (*.f64 y 2) 1)) 16)))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 2 points increase in error, 4 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (/.f64 (Rewrite=> *-lft-identity_binary64 (+.f64 (*.f64 y 2) 1)) 16))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (*.f64 z t) (+.f64 (*.f64 y 2) 1)) 16))) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 y 2) 1) (*.f64 z t))) 16)) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t)) 16)) (cos.f64 (*.f64 (*.f64 t b) (+.f64 1/16 (/.f64 a 8)))))): 15 points increase in error, 9 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 b t)) (+.f64 1/16 (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (+.f64 (Rewrite<= metadata-eval (*.f64 1/16 1)) (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (+.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 1 16)) 1) (/.f64 a 8)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (+.f64 (*.f64 (/.f64 1 16) 1) (/.f64 a (Rewrite<= metadata-eval (/.f64 16 2)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (+.f64 (*.f64 (/.f64 1 16) 1) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a 2) 16))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (+.f64 (*.f64 (/.f64 1 16) 1) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (*.f64 a 2))) 16)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (+.f64 (*.f64 (/.f64 1 16) 1) (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 16 (*.f64 a 2))))))))): 2 points increase in error, 3 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (+.f64 (*.f64 (/.f64 1 16) 1) (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 16) (*.f64 a 2)))))))): 3 points increase in error, 2 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (Rewrite<= distribute-lft-in_binary64 (*.f64 (/.f64 1 16) (+.f64 1 (*.f64 a 2)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (*.f64 (/.f64 1 16) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 a 2) 1))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 16 (+.f64 (*.f64 a 2) 1)))))))): 2 points increase in error, 2 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1 (+.f64 (*.f64 a 2) 1)) 16)))))): 2 points increase in error, 2 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 (*.f64 b t) (/.f64 (Rewrite=> *-lft-identity_binary64 (+.f64 (*.f64 a 2) 1)) 16))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (+.f64 (*.f64 a 2) 1) 16) (*.f64 b t)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (+.f64 (*.f64 a 2) 1) (*.f64 b t)) 16))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t)) 16)))): 12 points increase in error, 6 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))): 8 points increase in error, 2 points decrease in error

3. Taylor expanded in t around 0 45.1

   \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \color{blue}{1}\right) \]

4. Taylor expanded in z around 0 44.0

   \[\leadsto \color{blue}{x} \]

5. Final simplification 44.0

   \[\leadsto x \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

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