SynthBasics:moogVCF from YampaSynth-0.2

Average Accuracy: 93.3% → 97.7%

Time: 17.4s

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	93.3%
Target	96.9%
Herbie	97.7%

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

Derivation

1. Initial program 94.6%

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

2. Simplified 98.0%

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

   [Start] 94.6	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   +-commutative [=>] 94.6	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
   *-commutative [=>] 94.6	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
   associate-*l* [=>] 98.0	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
   fma-def [=>] 98.0	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

3. Final simplification 98.0%

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

Alternatives

Alternative 1
Accuracy	96.9%
Cost	13632

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

Alternative 2
Accuracy	87.8%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;t \leq -4.15 \cdot 10^{-57} \lor \neg \left(t \leq 7.6 \cdot 10^{-46}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	85.6%
Cost	7892

\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ t_2 := x + t_1 \cdot \left(z \cdot y\right)\\ t_3 := z \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+141}:\\ \;\;\;\;x + t_3\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-202}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-168}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 31000000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+214}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \left(t_1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \frac{1}{y}\right) \cdot t_3\\ \end{array} \]

Alternative 4
Accuracy	84.9%
Cost	7504

\[\begin{array}{l} t_1 := x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ t_2 := z \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+142}:\\ \;\;\;\;x + t_2\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-165}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+191}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \frac{1}{y}\right) \cdot t_2\\ \end{array} \]

Alternative 5
Accuracy	86.5%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-56} \lor \neg \left(t \leq 1.85 \cdot 10^{-33}\right):\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	59.6%
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-284}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-210}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-19}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	78.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+30} \lor \neg \left(y \leq 2.4\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	57.9%
Cost	588

\[\begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+221}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+279}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]

Alternative 9
Accuracy	64.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+107} \lor \neg \left(z \leq 3.35 \cdot 10^{+16}\right):\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	70.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+28} \lor \neg \left(y \leq 2.4 \cdot 10^{-51}\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	60.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023160 
(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))))))