SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 7.11% → 1.59%

Time: 15.0s

Precision: binary64

Cost: 41032

Math

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

↓

\[\begin{array}{l} t_1 := x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot t - z \cdot x\\ \end{array} \]

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+304) 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+304) {
		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+304) {
		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+304:
		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+304)
		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+304)
		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+304], t$95$1, N[(N[(z * t), $MachinePrecision] - N[(z * x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.11%
Target	3.05%
Herbie	1.59%

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

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 100

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

   2. Simplified 1.74

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

      [Start] 100	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
      +-commutative [=>] 100	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
      *-commutative [=>] 100	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
      associate-*l* [=>] 1.74	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
      fma-def [=>] 1.74	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

   3. Taylor expanded in z around inf 97.13

      \[\leadsto \color{blue}{y \cdot \left(\left(\left(\frac{1}{e^{\frac{x}{y}} \cdot \left(e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}\right)} + \frac{e^{\frac{t}{y}}}{e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}}\right) - \left(\frac{e^{\frac{x}{y}}}{e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}} + \frac{1}{e^{\frac{t}{y}} \cdot \left(e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}\right)}\right)\right) \cdot z\right)} \]

   4. Simplified 100

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

      [Start] 97.13

      \[ y \cdot \left(\left(\left(\frac{1}{e^{\frac{x}{y}} \cdot \left(e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}\right)} + \frac{e^{\frac{t}{y}}}{e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}}\right) - \left(\frac{e^{\frac{x}{y}}}{e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}} + \frac{1}{e^{\frac{t}{y}} \cdot \left(e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}\right)}\right)\right) \cdot z\right) \]

      *-commutative [=>] 97.13

      \[ y \cdot \color{blue}{\left(z \cdot \left(\left(\frac{1}{e^{\frac{x}{y}} \cdot \left(e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}\right)} + \frac{e^{\frac{t}{y}}}{e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}}\right) - \left(\frac{e^{\frac{x}{y}}}{e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}} + \frac{1}{e^{\frac{t}{y}} \cdot \left(e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}\right)}\right)\right)\right)} \]

   5. Taylor expanded in y around inf 0.04

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 4.9999999999999997e304

   1. Initial program 1.1

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

   if 4.9999999999999997e304 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 92.72

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

   2. Simplified 31.56

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

      [Start] 92.72	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
      +-commutative [=>] 92.72	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
      *-commutative [=>] 92.72	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
      associate-*l* [=>] 31.56	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
      fma-def [=>] 31.56	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

   3. Taylor expanded in z around inf 97.96

      \[\leadsto \color{blue}{y \cdot \left(\left(\left(\frac{1}{e^{\frac{x}{y}} \cdot \left(e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}\right)} + \frac{e^{\frac{t}{y}}}{e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}}\right) - \left(\frac{e^{\frac{x}{y}}}{e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}} + \frac{1}{e^{\frac{t}{y}} \cdot \left(e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}\right)}\right)\right) \cdot z\right)} \]

   4. Simplified 97.98

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

      [Start] 97.96

      \[ y \cdot \left(\left(\left(\frac{1}{e^{\frac{x}{y}} \cdot \left(e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}\right)} + \frac{e^{\frac{t}{y}}}{e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}}\right) - \left(\frac{e^{\frac{x}{y}}}{e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}} + \frac{1}{e^{\frac{t}{y}} \cdot \left(e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}\right)}\right)\right) \cdot z\right) \]

      *-commutative [=>] 97.96

      \[ y \cdot \color{blue}{\left(z \cdot \left(\left(\frac{1}{e^{\frac{x}{y}} \cdot \left(e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}\right)} + \frac{e^{\frac{t}{y}}}{e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}}\right) - \left(\frac{e^{\frac{x}{y}}}{e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}} + \frac{1}{e^{\frac{t}{y}} \cdot \left(e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}\right)}\right)\right)\right)} \]

   5. Taylor expanded in y around inf 99.91

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

   6. Taylor expanded in t around 0 14.16

      \[\leadsto \color{blue}{t \cdot z + -1 \cdot \left(z \cdot x\right)} \]

   7. Applied egg-rr 14.16

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

4. Final simplification 1.59

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -\infty:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{elif}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t - z \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	2.28%
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	16.41%
Cost	13645

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+139}:\\ \;\;\;\;x + \left(z \cdot t - z \cdot x\right)\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+15} \lor \neg \left(y \leq -3.4 \cdot 10^{-62}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tanh \left(\frac{x}{y}\right) \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 3
Error	30.98%
Cost	7508

\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-204}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-304}:\\ \;\;\;\;x + \left(z \cdot t - z \cdot x\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	15.87%
Cost	7504

\[\begin{array}{l} t_1 := x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;x + \left(z \cdot t - z \cdot x\right)\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-62}:\\ \;\;\;\;x - \tanh \left(\frac{x}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 5
Error	14.55%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+139}:\\ \;\;\;\;x + \left(z \cdot t - z \cdot x\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+178}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 6
Error	29.34%
Cost	849

\[\begin{array}{l} t_1 := x + z \cdot t\\ \mathbf{if}\;y \leq -9 \cdot 10^{+208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+119}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{+94} \lor \neg \left(y \leq 580000\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	24.14%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+92} \lor \neg \left(y \leq 2050\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	24.15%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+92}:\\ \;\;\;\;x + \left(z \cdot t - z \cdot x\right)\\ \mathbf{elif}\;y \leq 1850:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 9
Error	36.41%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+157} \lor \neg \left(y \leq 9 \cdot 10^{+122}\right):\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	28.32%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+92} \lor \neg \left(y \leq 122000\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	35.05%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-227}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-241}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	35.99%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023102 
(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))))))