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

Average Error: 46.2 → 43.3

Time: 22.3s

Precision: binary64

Cost: 53956

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+285}:\\ \;\;\;\;x \cdot \left(\log \left(e^{\cos \left(t \cdot \left(z \cdot \mathsf{fma}\left(0.125, y, 0.0625\right)\right)\right)}\right) \cdot \cos \left(t \cdot \left(b \cdot \mathsf{fma}\left(a, 0.125, 0.0625\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      2e+285)
   (*
    x
    (*
     (log (exp (cos (* t (* z (fma 0.125 y 0.0625))))))
     (cos (* t (* b (fma a 0.125 0.0625))))))
   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) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 2e+285) {
		tmp = x * (log(exp(cos((t * (z * fma(0.125, y, 0.0625)))))) * cos((t * (b * fma(a, 0.125, 0.0625)))));
	} else {
		tmp = x;
	}
	return tmp;
}

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)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+285)
		tmp = Float64(x * Float64(log(exp(cos(Float64(t * Float64(z * fma(0.125, y, 0.0625)))))) * cos(Float64(t * Float64(b * fma(a, 0.125, 0.0625))))));
	else
		tmp = x;
	end
	return tmp
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_] := If[LessEqual[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[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+285], N[(x * N[(N[Log[N[Exp[N[Cos[N[(t * N[(z * N[(0.125 * y + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(t * N[(b * N[(a * 0.125 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Target

Original	46.2
Target	44.5
Herbie	43.3

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

Derivation

1. Split input into 2 regimes

2. if (*.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))) < 2e285

   1. Initial program 33.5

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

   2. Simplified 33.5

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \mathsf{fma}\left(y, 0.125, 0.0625\right)\right) \cdot \cos \left(t \cdot \left(b \cdot \mathsf{fma}\left(a, 0.125, 0.0625\right)\right)\right)\right)} \]
      Proof
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (fma.f64 y 1/8 1/16))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (fma.f64 y (Rewrite<= metadata-eval (/.f64 2 16)) 1/16))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (fma.f64 y (/.f64 2 16) (Rewrite<= metadata-eval (/.f64 1 16))))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (/.f64 2 16)) (/.f64 1 16))))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 2 16) y)) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 16 y))) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 1 points increase in error, 3 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 y) 16)) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 3 points increase in error, 1 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y 2)) 16) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (*.f64 y 2))) 16) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 16 (*.f64 y 2)))) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 1 points increase in error, 3 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 16) (*.f64 y 2))) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 3 points increase in error, 1 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 y 2) (/.f64 1 16))) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (*.f64 y 2) 1) (/.f64 1 16))))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 16) (+.f64 (*.f64 y 2) 1))))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 1 points increase in error, 3 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 3 points increase in error, 1 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 8 points increase in error, 10 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a (Rewrite<= metadata-eval (/.f64 2 16)) 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 (*.f64 t (*.f64 b (fma.f64 a (/.f64 2 16) (Rewrite<= metadata-eval (/.f64 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 (*.f64 t (*.f64 b (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (/.f64 2 16)) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 2 16) a)) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 a (/.f64 2 16))) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (*.f64 a (Rewrite=> metadata-eval 1/8)) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (*.f64 a (Rewrite<= metadata-eval (*.f64 2 1/16))) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (*.f64 a (*.f64 2 (Rewrite<= metadata-eval (/.f64 1 16)))) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a 2) (/.f64 1 16))) (/.f64 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 (*.f64 t (*.f64 b (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (*.f64 a 2) 1) (/.f64 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 (*.f64 t (*.f64 b (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 16) (+.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 t (*.f64 b (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 16 (+.f64 (*.f64 a 2) 1))))))))): 2 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 t (*.f64 b (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1 (+.f64 (*.f64 a 2) 1)) 16))))))): 0 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 t (*.f64 b (/.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 (*.f64 t (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (+.f64 (*.f64 a 2) 1) 16) b)))))): 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 t (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) 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=> associate-*r/_binary64 (/.f64 (*.f64 t (*.f64 (+.f64 (*.f64 a 2) 1) b)) 16))))): 1 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<= *-commutative_binary64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t)) 16)))): 0 points increase in error, 0 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)))): 4 points increase in error, 3 points decrease in error

   3. Applied egg-rr 33.5

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

   if 2e285 < (*.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)))

   1. Initial program 63.2

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

   2. Simplified 62.6

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \mathsf{fma}\left(y, 0.125, 0.0625\right)\right) \cdot \cos \left(t \cdot \left(b \cdot \mathsf{fma}\left(a, 0.125, 0.0625\right)\right)\right)\right)} \]
      Proof
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (fma.f64 y 1/8 1/16))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (fma.f64 y (Rewrite<= metadata-eval (/.f64 2 16)) 1/16))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (fma.f64 y (/.f64 2 16) (Rewrite<= metadata-eval (/.f64 1 16))))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (/.f64 2 16)) (/.f64 1 16))))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 2 16) y)) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 16 y))) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 1 points increase in error, 3 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 y) 16)) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 3 points increase in error, 1 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y 2)) 16) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (*.f64 y 2))) 16) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 16 (*.f64 y 2)))) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 1 points increase in error, 3 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 16) (*.f64 y 2))) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 3 points increase in error, 1 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 y 2) (/.f64 1 16))) (/.f64 1 16)))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (*.f64 y 2) 1) (/.f64 1 16))))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (*.f64 (*.f64 z t) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 16) (+.f64 (*.f64 y 2) 1))))) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 1 points increase in error, 3 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 3 points increase in error, 1 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 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 t (*.f64 b (fma.f64 a 1/8 1/16)))))): 8 points increase in error, 10 points decrease in error
      (*.f64 x (*.f64 (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16)) (cos.f64 (*.f64 t (*.f64 b (fma.f64 a (Rewrite<= metadata-eval (/.f64 2 16)) 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 (*.f64 t (*.f64 b (fma.f64 a (/.f64 2 16) (Rewrite<= metadata-eval (/.f64 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 (*.f64 t (*.f64 b (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (/.f64 2 16)) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 2 16) a)) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 a (/.f64 2 16))) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (*.f64 a (Rewrite=> metadata-eval 1/8)) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (*.f64 a (Rewrite<= metadata-eval (*.f64 2 1/16))) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (*.f64 a (*.f64 2 (Rewrite<= metadata-eval (/.f64 1 16)))) (/.f64 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 (*.f64 t (*.f64 b (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a 2) (/.f64 1 16))) (/.f64 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 (*.f64 t (*.f64 b (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (*.f64 a 2) 1) (/.f64 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 (*.f64 t (*.f64 b (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 16) (+.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 t (*.f64 b (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 16 (+.f64 (*.f64 a 2) 1))))))))): 2 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 t (*.f64 b (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1 (+.f64 (*.f64 a 2) 1)) 16))))))): 0 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 t (*.f64 b (/.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 (*.f64 t (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (+.f64 (*.f64 a 2) 1) 16) b)))))): 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 t (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) 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=> associate-*r/_binary64 (/.f64 (*.f64 t (*.f64 (+.f64 (*.f64 a 2) 1) b)) 16))))): 1 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<= *-commutative_binary64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t)) 16)))): 0 points increase in error, 0 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)))): 4 points increase in error, 3 points decrease in error

   3. Taylor expanded in t around 0 59.9

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

   4. Taylor expanded in z around 0 56.4

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

4. Final simplification 43.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+285}:\\ \;\;\;\;x \cdot \left(\log \left(e^{\cos \left(t \cdot \left(z \cdot \mathsf{fma}\left(0.125, y, 0.0625\right)\right)\right)}\right) \cdot \cos \left(t \cdot \left(b \cdot \mathsf{fma}\left(a, 0.125, 0.0625\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Error	43.3
Cost	28868

\[\begin{array}{l} t_1 := \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	44.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022300 
(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))))