SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 7.55% → 2.44%

Time: 16.3s

Precision: binary64

Cost: 19904

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (fma z (* y (- (tanh (/ t y)) (tanh (/ x y)))) 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) {
	return fma(z, (y * (tanh((t / y)) - tanh((x / y)))), x);
}

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)
	return fma(z, Float64(y * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))), x)
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_] := N[(z * N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Target

Original	7.55%
Target	3.12%
Herbie	2.44%

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

Derivation

1. Initial program 7.55

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

2. Simplified 2.44

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

   [Start] 7.55	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   +-commutative [=>] 7.55	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
   *-commutative [=>] 7.55	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
   associate-*l* [=>] 2.45	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
   fma-def [=>] 2.44	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

3. Final simplification 2.44

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

Alternatives

Alternative 1
Error	3.69%
Cost	41033

\[\begin{array}{l} t_1 := x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq \infty\right):\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	12.24%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;t \leq -1.62 \cdot 10^{-31} \lor \neg \left(t \leq 5.8 \cdot 10^{-63}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	27.94%
Cost	7769

\[\begin{array}{l} t_1 := x + z \cdot \left(t - x\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-269}:\\ \;\;\;\;y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-182}:\\ \;\;\;\;x + \frac{\frac{z \cdot y}{y}}{\frac{1}{t - x}}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-72} \lor \neg \left(x \leq 4 \cdot 10^{-31}\right):\\ \;\;\;\;x - z \cdot \left(y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	18.09%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-82}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-31}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \left(y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \end{array} \]

Alternative 5
Error	30.69%
Cost	7244

\[\begin{array}{l} t_1 := x + z \cdot \left(t - x\right)\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-269}:\\ \;\;\;\;y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-185}:\\ \;\;\;\;x + \frac{\frac{z \cdot y}{y}}{\frac{1}{t - x}}\\ \mathbf{elif}\;x \leq 10^{-74}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	30.79%
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-183} \lor \neg \left(x \leq 8 \cdot 10^{-72}\right) \land x \leq 9.8 \cdot 10^{-29}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	27.54%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+48} \lor \neg \left(y \leq 1.18 \cdot 10^{+48}\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	35.18%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.48 \cdot 10^{-269}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	35.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-269}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	34.96%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(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))))))