SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 7.5% → 2.34%

Time: 17.8s

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	7.5%
Target	3.27%
Herbie	2.34%

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

Derivation

1. Initial program 7.5

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

2. Simplified 2.34

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

   [Start] 7.5	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   +-commutative [=>] 7.5	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
   *-commutative [=>] 7.5	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
   associate-*l* [=>] 2.35	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
   fma-def [=>] 2.34	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

3. Final simplification 2.34

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

Alternatives

Alternative 1
Error	3.5%
Cost	27332

\[\begin{array}{l} t_1 := x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t_1 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 2
Error	11.59%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-48} \lor \neg \left(t \leq 88000000000000\right):\\ \;\;\;\;x + z \cdot \frac{1}{\frac{1}{y \cdot \tanh \left(\frac{t}{y}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	13.74%
Cost	13576

\[\begin{array}{l} t_1 := \tanh \left(\frac{x}{y}\right)\\ \mathbf{if}\;x \leq -100:\\ \;\;\;\;x - z \cdot \left(y \cdot t_1\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-67}:\\ \;\;\;\;x + z \cdot \frac{1}{\frac{1}{y \cdot \tanh \left(\frac{t}{y}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \left(-t_1\right), x\right)\\ \end{array} \]

Alternative 4
Error	13.74%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;x \leq -18 \lor \neg \left(x \leq 6.5 \cdot 10^{-68}\right):\\ \;\;\;\;x - z \cdot \left(y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{1}{\frac{1}{y \cdot \tanh \left(\frac{t}{y}\right)}}\\ \end{array} \]

Alternative 5
Error	15.68%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \leq -29 \lor \neg \left(x \leq 4.2 \cdot 10^{-67}\right):\\ \;\;\;\;x - z \cdot \left(y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{\frac{1}{\tanh \left(\frac{t}{y}\right)}}\\ \end{array} \]

Alternative 6
Error	15.58%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -32 \lor \neg \left(x \leq 6.2 \cdot 10^{-67}\right):\\ \;\;\;\;x - z \cdot \left(y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 7
Error	21.83%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+97}:\\ \;\;\;\;x + \frac{t + x}{\frac{\frac{t + x}{z}}{t - x}}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+124}:\\ \;\;\;\;x + \frac{z \cdot y}{\frac{t}{y} \cdot 0.3333333333333333 + \frac{y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x - z \cdot x\right)\\ \end{array} \]

Alternative 8
Error	18.09%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+98}:\\ \;\;\;\;x + \frac{t + x}{\frac{\frac{t + x}{z}}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 9
Error	20.93%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \left(1 - z\right) + z \cdot t\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+127}:\\ \;\;\;\;x + \frac{z \cdot y}{\frac{t}{y} \cdot 0.3333333333333333 + \frac{y}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 10
Error	21.84%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+97}:\\ \;\;\;\;x + \frac{t + x}{\frac{\frac{t + x}{z}}{t - x}}\\ \mathbf{elif}\;y \leq 7.9 \cdot 10^{+121}:\\ \;\;\;\;x + \frac{z \cdot y}{\frac{t}{y} \cdot 0.3333333333333333 + \frac{y}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 11
Error	22.46%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+62} \lor \neg \left(y \leq 220\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	22.41%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(1 - z\right) + z \cdot t\\ \mathbf{elif}\;y \leq 85:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 13
Error	27.32%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+64} \lor \neg \left(y \leq 82\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	34.43%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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))))))