Average Error: 4.7 → 2.6
Time: 4.4s
Precision: binary64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(z, t - x, x\right)\\ \mathbf{if}\;y \leq -7.414154157345965 \cdot 10^{+168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.0568740699100138 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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 (fma z (- t x) x)))
   (if (<= y -7.414154157345965e+168)
     t_1
     (if (<= y 1.0568740699100138e+135)
       (fma (* y z) (- (tanh (/ t y)) (tanh (/ x y))) x)
       t_1))))
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 = fma(z, (t - x), x);
	double tmp;
	if (y <= -7.414154157345965e+168) {
		tmp = t_1;
	} else if (y <= 1.0568740699100138e+135) {
		tmp = fma((y * z), (tanh((t / y)) - tanh((x / y))), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 = fma(z, Float64(t - x), x)
	tmp = 0.0
	if (y <= -7.414154157345965e+168)
		tmp = t_1;
	elseif (y <= 1.0568740699100138e+135)
		tmp = fma(Float64(y * z), Float64(tanh(Float64(t / y)) - tanh(Float64(x / y))), x);
	else
		tmp = t_1;
	end
	return tmp
end

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] + x), $MachinePrecision]}, If[LessEqual[y, -7.414154157345965e+168], t$95$1, If[LessEqual[y, 1.0568740699100138e+135], N[(N[(y * z), $MachinePrecision] * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

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 := \mathsf{fma}\left(z, t - x, x\right)\\
\mathbf{if}\;y \leq -7.414154157345965 \cdot 10^{+168}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;y \leq 1.0568740699100138 \cdot 10^{+135}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot z, \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), x\right)\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.7
Target2.1
Herbie2.6
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \]

Derivation

  1. Split input into 2 regimes

  2. if y < -7.41415415734596486e168 or 1.0568740699100138e135 < y

    1. Initial program 15.8

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

    2. Simplified15.8

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

    3. Taylor expanded in y around inf 6.5

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

    4. Simplified6.5

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

    5. Taylor expanded in z around 0 6.5

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

    6. Simplified6.5

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

    if -7.41415415734596486e168 < y < 1.0568740699100138e135

    1. Initial program 1.4

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

    2. Simplified1.4

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.414154157345965 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \mathbf{elif}\;y \leq 1.0568740699100138 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))))