SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 7.24% → 2.16%

Time: 16.9s

Precision: binary64

Cost: 41033

Math

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

↓

\[\begin{array}{l} t_1 := x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+299}\right):\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 5e+299))) (* 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 <= 5e+299)) {
		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 <= 5e+299)) {
		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 <= 5e+299):
		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 <= 5e+299))
		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 <= 5e+299)))
		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, 5e+299]], $MachinePrecision]], N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision], t$95$1]]

Target

Original	7.24%
Target	3.08%
Herbie	2.16%

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

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 5.0000000000000003e299 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 85.95

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

   2. Simplified 20.02

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

      [Start] 85.95	\[ x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
      +-commutative [=>] 85.95	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x} \]
      *-commutative [=>] 85.95	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) + x \]
      associate-*l* [=>] 20.02	\[ \color{blue}{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} + x \]
      fma-def [=>] 20.02	\[ \color{blue}{\mathsf{fma}\left(z, y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \]

   3. Taylor expanded in z around inf 98.12

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

   4. Simplified 97.25

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

      [Start] 98.12

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

      *-commutative [=>] 98.12

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

   5. Taylor expanded in y around inf 17.9

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 5.0000000000000003e299

   1. Initial program 0.88

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

4. Final simplification 2.16

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq -\infty \lor \neg \left(x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right) \leq 5 \cdot 10^{+299}\right):\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.27%
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	14.08%
Cost	14032

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y, z \cdot \tanh \left(\frac{t}{y}\right), x \cdot \left(1 - z\right)\right)\\ t_2 := \tanh \left(\frac{x}{y}\right)\\ t_3 := x - z \cdot \left(y \cdot t_2\right)\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{-59}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(z, t_2 \cdot \left(-y\right), x\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	14.37%
Cost	13708

\[\begin{array}{l} t_1 := x + z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right)\right)\\ t_2 := \tanh \left(\frac{x}{y}\right)\\ t_3 := x - z \cdot \left(y \cdot t_2\right)\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-59}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 320000:\\ \;\;\;\;\mathsf{fma}\left(z, t_2 \cdot \left(-y\right), x\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	2.28%
Cost	13632

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

Alternative 5
Error	14.41%
Cost	7762

\[\begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-59} \lor \neg \left(x \leq 10^{-33}\right) \land \left(x \leq 70000000000 \lor \neg \left(x \leq 4.2 \cdot 10^{+84}\right)\right):\\ \;\;\;\;x - z \cdot \left(y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \frac{x}{y}\right)\right)\\ \end{array} \]

Alternative 6
Error	15.1%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y \leq -1.48 \cdot 10^{+131} \lor \neg \left(y \leq 3.8 \cdot 10^{+78}\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \end{array} \]

Alternative 7
Error	32.28%
Cost	1376

\[\begin{array}{l} t_1 := x + z \cdot t\\ t_2 := x - z \cdot x\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-136}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	23.11%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+61} \lor \neg \left(y \leq 8 \cdot 10^{+76}\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	34.21%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+194} \lor \neg \left(y \leq 4.5 \cdot 10^{+231}\right):\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	27.99%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-16} \lor \neg \left(y \leq 40000000000000\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	36.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+194}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+232}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]

Alternative 12
Error	34.96%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023089 
(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))))))