SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 7.48% → 1.4%

Time: 15.9s

Precision: binary64

Cost: 41033

Math

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

↓

\[\begin{array}{l} t_1 := x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (* (- (tanh (/ t y)) (tanh (/ x y))) (* z y)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+306))) (* z (- t x)) t_1)))

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 t_1 = x + ((tanh((t / y)) - tanh((x / y))) * (z * y));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+306)) {
		tmp = z * (t - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = x + ((Math.tanh((t / y)) - Math.tanh((x / y))) * (z * y));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e+306)) {
		tmp = z * (t - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x + ((y * z) * (math.tanh((t / y)) - math.tanh((x / y))))

↓

def code(x, y, z, t):
	t_1 = x + ((math.tanh((t / y)) - math.tanh((x / y))) * (z * y))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e+306):
		tmp = z * (t - x)
	else:
		tmp = t_1
	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)
	t_1 = Float64(x + Float64(Float64(tanh(Float64(t / y)) - tanh(Float64(x / y))) * Float64(z * y)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+306))
		tmp = Float64(z * Float64(t - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y * z) * (tanh((t / y)) - tanh((x / y))));
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = x + ((tanh((t / y)) - tanh((x / y))) * (z * y));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e+306)))
		tmp = z * (t - x);
	else
		tmp = t_1;
	end
	tmp_2 = 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_] := Block[{t$95$1 = N[(x + N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+306]], $MachinePrecision]], N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision], t$95$1]]

Target

Original	7.48%
Target	3.28%
Herbie	1.4%

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < -inf.0 or 2.00000000000000003e306 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 97.93

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

   2. Simplified 23.65

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

      [Start] 97.93	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
      +-commutative [=>] 97.93	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
      *-commutative [=>] 97.93	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
      associate-*l* [=>] 23.65	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
      fma-def [=>] 23.65	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

   3. Taylor expanded in y around inf 5.49

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

   4. Taylor expanded in z around inf 6.29

      \[\leadsto \color{blue}{\left(t - x\right) \cdot z} \]

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 2.00000000000000003e306

   1. Initial program 1.05

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -\infty \lor \neg \left(x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.5%
Cost	19904

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

Alternative 2
Error	13.67%
Cost	13641

\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{+106} \lor \neg \left(y \leq 2.5 \cdot 10^{+98}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot t_1 - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_1 \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 3
Error	14.98%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-141} \lor \neg \left(t \leq 6500000\right):\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	15.06%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+116} \lor \neg \left(y \leq 1.12 \cdot 10^{+98}\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 5
Error	37.6%
Cost	849

\[\begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{+53}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{+21} \lor \neg \left(t \leq 2.8 \cdot 10^{-184}\right) \land t \leq 2.2 \cdot 10^{-150}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	23.96%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+100} \lor \neg \left(y \leq 2.65 \cdot 10^{-17}\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	28.54%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+100} \lor \neg \left(y \leq 1.75 \cdot 10^{+63}\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	35.62%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-235}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-187}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	35.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))))))