SynthBasics:moogVCF from YampaSynth-0.2

Average Accuracy: 92.5% → 98.2%

Time: 19.7s

Precision: binary64

Cost: 20036

Math

\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.1e+216)
   (fma z (* y (- (tanh (/ t y)) (tanh (/ x y)))) x)
   (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	return x + ((y * z) * (tanh((t / y)) - tanh((x / y))));
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.1e+216) {
		tmp = fma(z, (y * (tanh((t / y)) - tanh((x / y)))), x);
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y * z) * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.1e+216)
		tmp = fma(z, Float64(y * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))), x);
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y * z), $MachinePrecision] * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[y, 2.1e+216], N[(z * N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	92.5%
Target	97.0%
Herbie	98.2%

\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \]

Derivation

1. Split input into 2 regimes

2. if y < 2.10000000000000001e216

   1. Initial program 94.1%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]

   2. Simplified 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]
      Proof

      [Start] 94.1	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
      +-commutative [=>] 94.1	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
      *-commutative [=>] 94.1	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
      associate-*l* [=>] 98.3	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
      fma-def [=>] 98.3	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

   if 2.10000000000000001e216 < y

   1. Initial program 67.8%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]

   2. Simplified 86.2%

      \[\leadsto \color{blue}{x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} \]
      Proof

      [Start] 67.8	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
      associate-*l* [=>] 86.2	\[ x + \color{blue}{y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} \]

   3. Taylor expanded in y around inf 95.9%

      \[\leadsto x + \color{blue}{z \cdot \left(t - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	88.8%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;t \leq -800000 \lor \neg \left(t \leq 3 \cdot 10^{-89}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	97.6%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+216}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 3
Accuracy	86.6%
Cost	13644

\[\begin{array}{l} t_1 := \mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{if}\;t \leq -9.8 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-235}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-87}:\\ \;\;\;\;x + \left(z \cdot t - \tanh \left(\frac{x}{y}\right) \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	80.9%
Cost	7636

\[\begin{array}{l} t_1 := x + z \cdot \left(t - x\right)\\ t_2 := x + \tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{-147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.95 \cdot 10^{-290}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-269}:\\ \;\;\;\;x + \frac{z}{\frac{1}{t - x}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	84.1%
Cost	7628

\[\begin{array}{l} t_1 := x + \tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t \leq -5.1 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-236}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-93}:\\ \;\;\;\;x + \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	84.2%
Cost	7628

\[\begin{array}{l} t_1 := x + \tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t \leq -6 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-235}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-87}:\\ \;\;\;\;x + \left(z \cdot t - \tanh \left(\frac{x}{y}\right) \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	84.2%
Cost	7364

\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{-43}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(t_1 - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+91}:\\ \;\;\;\;x + t_1 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 8
Accuracy	74.3%
Cost	7244

\[\begin{array}{l} t_1 := x + z \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+20}:\\ \;\;\;\;\tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	76.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-39} \lor \neg \left(y \leq 1.86 \cdot 10^{+93}\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	67.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{-40} \lor \neg \left(y \leq 1.85 \cdot 10^{+93}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	70.7%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+57}:\\ \;\;\;\;x + z \cdot t\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 12
Accuracy	63.2%
Cost	520

\[\begin{array}{l} \mathbf{if}\;z \leq 2.6 \cdot 10^{+29}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+101}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Accuracy	64.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

  :herbie-target
  (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))

  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))