SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.4 → 0.8

Time: 49.8s

Precision: binary64

Cost: 27332

Math

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

↓

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

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 (- (tanh (/ t y)) (tanh (/ x y)))))
   (if (<= (+ x (* (* y z) t_1)) INFINITY)
     (+ x (* z (* y t_1)))
     (+ x (* 2.0 (/ z (/ 2.0 (- t 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 = tanh((t / y)) - tanh((x / y));
	double tmp;
	if ((x + ((y * z) * t_1)) <= ((double) INFINITY)) {
		tmp = x + (z * (y * t_1));
	} else {
		tmp = x + (2.0 * (z / (2.0 / (t - 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 = Math.tanh((t / y)) - Math.tanh((x / y));
	double tmp;
	if ((x + ((y * z) * t_1)) <= Double.POSITIVE_INFINITY) {
		tmp = x + (z * (y * t_1));
	} else {
		tmp = x + (2.0 * (z / (2.0 / (t - 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 = math.tanh((t / y)) - math.tanh((x / y))
	tmp = 0
	if (x + ((y * z) * t_1)) <= math.inf:
		tmp = x + (z * (y * t_1))
	else:
		tmp = x + (2.0 * (z / (2.0 / (t - 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(tanh(Float64(t / y)) - tanh(Float64(x / y)))
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * z) * t_1)) <= Inf)
		tmp = Float64(x + Float64(z * Float64(y * t_1)));
	else
		tmp = Float64(x + Float64(2.0 * Float64(z / Float64(2.0 / Float64(t - 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 = tanh((t / y)) - tanh((x / y));
	tmp = 0.0;
	if ((x + ((y * z) * t_1)) <= Inf)
		tmp = x + (z * (y * t_1));
	else
		tmp = x + (2.0 * (z / (2.0 / (t - 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[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + N[(N[(y * z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], Infinity], N[(x + N[(z * N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(2.0 * N[(z / N[(2.0 / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	4.4
Target	2.0
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 2 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 3.4

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

   2. Simplified 0.6

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

      [Start] 3.4	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
      rational.json-simplify-2 [=>] 3.4	\[ x + \color{blue}{\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)} \]
      rational.json-simplify-2 [=>] 3.4	\[ x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \color{blue}{\left(z \cdot y\right)} \]
      rational.json-simplify-43 [=>] 0.6	\[ x + \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} \]

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

   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 52.4

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

      [Start] 64.0	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ x + \color{blue}{\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)} \]
      rational.json-simplify-43 [=>] 52.4	\[ x + \color{blue}{y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} \]

   3. Taylor expanded in y around inf 17.1

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

   4. Simplified 17.3

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

      [Start] 17.1	\[ x + y \cdot \frac{\left(t - x\right) \cdot z}{y} \]
      rational.json-simplify-2 [<=] 17.1	\[ x + y \cdot \frac{\color{blue}{z \cdot \left(t - x\right)}}{y} \]
      rational.json-simplify-49 [=>] 17.3	\[ x + y \cdot \color{blue}{\left(\left(t - x\right) \cdot \frac{z}{y}\right)} \]

   5. Applied egg-rr 17.3

      \[\leadsto x + y \cdot \color{blue}{\frac{t - x}{\frac{y}{z}}} \]

   6. Applied egg-rr 18.3

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

   7. Applied egg-rr 10.2

      \[\leadsto x + \color{blue}{2 \cdot \frac{z}{\frac{2}{t - x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

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

Alternatives

Alternative 1
Error	1.7
Cost	13764

\[\begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+185}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right) \cdot z\\ \end{array} \]

Alternative 2
Error	12.4
Cost	7628

\[\begin{array}{l} t_1 := x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right)\right)\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right) \cdot z\\ \end{array} \]

Alternative 3
Error	21.7
Cost	848

\[\begin{array}{l} t_1 := \left(t - x\right) \cdot z\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-263}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-233}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	22.0
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-146}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-263}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-235}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	22.7
Cost	720

\[\begin{array}{l} t_1 := x \cdot \left(1 - z\right)\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{+202}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+198}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]

Alternative 6
Error	15.1
Cost	712

\[\begin{array}{l} t_1 := x + \left(t - x\right) \cdot z\\ \mathbf{if}\;y \leq -5 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	18.4
Cost	584

\[\begin{array}{l} t_1 := t \cdot z + x\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+180}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	22.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023064 
(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))))))