SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.5 → 0.8

Time: 16.6s

Precision: binary64

Cost: 41032

Math

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

↓

\[\begin{array}{l} t_1 := x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot t - z \cdot x\\ \end{array} \]

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 (<= t_1 (- INFINITY))
     (* z (- t x))
     (if (<= t_1 5e+306) t_1 (- (* z t) (* z 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 t_1 = x + ((tanh((t / y)) - tanh((x / y))) * (z * y));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z * (t - x);
	} else if (t_1 <= 5e+306) {
		tmp = t_1;
	} else {
		tmp = (z * t) - (z * x);
	}
	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) {
		tmp = z * (t - x);
	} else if (t_1 <= 5e+306) {
		tmp = t_1;
	} else {
		tmp = (z * t) - (z * x);
	}
	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:
		tmp = z * (t - x)
	elif t_1 <= 5e+306:
		tmp = t_1
	else:
		tmp = (z * t) - (z * 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)
	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))
		tmp = Float64(z * Float64(t - x));
	elseif (t_1 <= 5e+306)
		tmp = t_1;
	else
		tmp = Float64(Float64(z * t) - Float64(z * x));
	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)
		tmp = z * (t - x);
	elseif (t_1 <= 5e+306)
		tmp = t_1;
	else
		tmp = (z * t) - (z * x);
	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[LessEqual[t$95$1, (-Infinity)], N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], t$95$1, N[(N[(z * t), $MachinePrecision] - N[(z * x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	4.5
Target	1.9
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 1.3

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

      [Start] 64.0	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
      +-commutative [=>] 64.0	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
      *-commutative [=>] 64.0	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
      associate-*l* [=>] 1.3	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
      fma-def [=>] 1.3	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

   3. Taylor expanded in y around inf 0.0

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

   4. Taylor expanded in z around inf 0.2

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 4.99999999999999993e306

   1. Initial program 0.5

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

   if 4.99999999999999993e306 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 61.9

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

   2. Simplified 20.7

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

      [Start] 61.9	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
      +-commutative [=>] 61.9	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
      *-commutative [=>] 61.9	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
      associate-*l* [=>] 20.7	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
      fma-def [=>] 20.7	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

   3. Taylor expanded in y around inf 6.2

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

   4. Taylor expanded in z around inf 7.2

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

   5. Applied egg-rr 7.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -\infty:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{elif}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t - z \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
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	7.5
Cost	13769

\[\begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-69} \lor \neg \left(t \leq 1.4 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	8.3
Cost	13769

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-76} \lor \neg \left(x \leq 4.3 \cdot 10^{-52}\right):\\ \;\;\;\;x - z \cdot \left(y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, x, y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	7.5
Cost	13513

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-69} \lor \neg \left(t \leq 1.4 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	16.3
Cost	7768

\[\begin{array}{l} t_1 := x + z \cdot \left(t - x\right)\\ t_2 := x - z \cdot \left(y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-304}:\\ \;\;\;\;y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	9.6
Cost	7497

\[\begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-82} \lor \neg \left(x \leq 1.28 \cdot 10^{-51}\right):\\ \;\;\;\;x - z \cdot \left(y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right)\\ \end{array} \]

Alternative 7
Error	16.4
Cost	7244

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+66}:\\ \;\;\;\;z \cdot t + x \cdot \left(1 - z\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 8
Error	15.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+67} \lor \neg \left(y \leq 1.62 \cdot 10^{+71}\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	15.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+66}:\\ \;\;\;\;z \cdot t + x \cdot \left(1 - z\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 10
Error	18.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+66} \lor \neg \left(y \leq 3.6 \cdot 10^{-10}\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	23.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-91}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	24.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-165}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-90}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	22.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022356 
(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))))))