SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 5.0 → 1.0

Time: 20.6s

Precision: binary64

Cost: 41288

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot \left(t - x\right)\\ t_2 := \tanh \left(\frac{t}{y}\right)\\ t_3 := \tanh \left(\frac{x}{y}\right)\\ t_4 := x + \left(t_2 - t_3\right) \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;x + \left(t_2 \cdot \left(y \cdot z\right) - t_3 \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* z (- t x)))
        (t_2 (tanh (/ t y)))
        (t_3 (tanh (/ x y)))
        (t_4 (+ x (* (- t_2 t_3) (* y z)))))
   (if (<= t_4 (- INFINITY))
     t_1
     (if (<= t_4 2e+304) (+ x (- (* t_2 (* y z)) (* t_3 (* y z)))) 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 = z * (t - x);
	double t_2 = tanh((t / y));
	double t_3 = tanh((x / y));
	double t_4 = x + ((t_2 - t_3) * (y * z));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_4 <= 2e+304) {
		tmp = x + ((t_2 * (y * z)) - (t_3 * (y * z)));
	} 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 = z * (t - x);
	double t_2 = Math.tanh((t / y));
	double t_3 = Math.tanh((x / y));
	double t_4 = x + ((t_2 - t_3) * (y * z));
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_4 <= 2e+304) {
		tmp = x + ((t_2 * (y * z)) - (t_3 * (y * z)));
	} 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 = z * (t - x)
	t_2 = math.tanh((t / y))
	t_3 = math.tanh((x / y))
	t_4 = x + ((t_2 - t_3) * (y * z))
	tmp = 0
	if t_4 <= -math.inf:
		tmp = t_1
	elif t_4 <= 2e+304:
		tmp = x + ((t_2 * (y * z)) - (t_3 * (y * z)))
	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(z * Float64(t - x))
	t_2 = tanh(Float64(t / y))
	t_3 = tanh(Float64(x / y))
	t_4 = Float64(x + Float64(Float64(t_2 - t_3) * Float64(y * z)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_4 <= 2e+304)
		tmp = Float64(x + Float64(Float64(t_2 * Float64(y * z)) - Float64(t_3 * Float64(y * z))));
	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 = z * (t - x);
	t_2 = tanh((t / y));
	t_3 = tanh((x / y));
	t_4 = x + ((t_2 - t_3) * (y * z));
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = t_1;
	elseif (t_4 <= 2e+304)
		tmp = x + ((t_2 * (y * z)) - (t_3 * (y * z)));
	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[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(x + N[(N[(t$95$2 - t$95$3), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$1, If[LessEqual[t$95$4, 2e+304], N[(x + N[(N[(t$95$2 * N[(y * z), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Target

Original	5.0
Target	2.2
Herbie	1.0

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

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 1.9999999999999999e304 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 61.1

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

   2. Taylor expanded in y around inf 4.1

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

   3. Simplified 4.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t - x, x\right)} \]
      Proof
      (fma.f64 z (-.f64 t x) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (-.f64 t x)) x)): 3 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 z (-.f64 t x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in z around inf 5.6

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

   if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.9999999999999999e304

   1. Initial program 0.6

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

   2. Applied egg-rr 0.6

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right) \leq -\infty:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{elif}\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right) \leq 2 \cdot 10^{+304}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right) - \tanh \left(\frac{x}{y}\right) \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	41032

\[\begin{array}{l} t_1 := z \cdot \left(t - x\right)\\ t_2 := x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	1.7
Cost	19904

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

Alternative 3
Error	10.8
Cost	13896

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+98}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;y \leq 3.6712928990852255 \cdot 10^{+25}:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Alternative 4
Error	12.8
Cost	7760

\[\begin{array}{l} t_1 := x + \left(\tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right) - x \cdot z\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+91}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;y \leq -1.984073577184792 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.6633035098895167 \cdot 10^{-149}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Alternative 5
Error	15.1
Cost	7628

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+91}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;y \leq 4.83856897880989 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.892076011332116 \cdot 10^{+23}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Alternative 6
Error	15.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+91}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;y \leq 0.49900884014679986:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Alternative 7
Error	18.8
Cost	848

\[\begin{array}{l} t_1 := x - x \cdot z\\ t_2 := x + t \cdot z\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.9107770814651894 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	19.4
Cost	848

\[\begin{array}{l} t_1 := x - x \cdot z\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+251}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.9107770814651894 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot z\\ \end{array} \]

Alternative 9
Error	15.1
Cost	712

\[\begin{array}{l} t_1 := x + z \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.49900884014679986:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	23.1
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+251}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+262}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+294}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	18.4
Cost	584

\[\begin{array}{l} t_1 := x + t \cdot z\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+141}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	22.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022297 
(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))))))