SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.7% → 97.1%

Time: 13.8s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))

C

double code(double x, double y, double z, double t) {
	return x + ((y * z) * (tanh((t / y)) - tanh((x / y))));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y * z) * (tanh((t / y)) - tanh((x / y))))
end function

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))));
}

Python

def code(x, y, z, t):
	return x + ((y * z) * (math.tanh((t / y)) - math.tanh((x / y))))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y * z) * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y * z) * (tanh((t / y)) - tanh((x / y))));
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]

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))

C

double code(double x, double y, double z, double t) {
	return x + ((y * z) * (tanh((t / y)) - tanh((x / y))));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y * z) * (tanh((t / y)) - tanh((x / y))))
end function

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))));
}

Python

def code(x, y, z, t):
	return x + ((y * z) * (math.tanh((t / y)) - math.tanh((x / y))))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y * z) * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y * z) * (tanh((t / y)) - tanh((x / y))));
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]

Alternative 1: 97.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma y (* z (- (tanh (/ t y)) (tanh (/ x y)))) x))

C

double code(double x, double y, double z, double t) {
	return fma(y, (z * (tanh((t / y)) - tanh((x / y)))), x);
}

Julia

function code(x, y, z, t)
	return fma(y, Float64(z * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))), x)
end

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(z * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
2. Step-by-step derivation

   1. +-commutative 94.2%

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

   2. associate-*l* 97.9%

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

   3. fma-define 97.9%

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

3. Simplified 97.9%

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

Alternative 2: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+172}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.22e+172)
   (+ x (* (- (tanh (/ t y)) (tanh (/ x y))) (* y z)))
   (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.22e+172) {
		tmp = x + ((tanh((t / y)) - tanh((x / y))) * (y * z));
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.22e+172)
		tmp = Float64(x + Float64(Float64(tanh(Float64(t / y)) - tanh(Float64(x / y))) * Float64(y * z)));
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.22e+172], N[(x + N[(N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.21999999999999999e172

   1. Initial program 95.4%

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

   if 1.21999999999999999e172 < y

   1. Initial program 81.3%

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

   3. Taylor expanded in y around inf 91.1%

      \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 91.1%

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

      2. fma-define 91.2%

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

   5. Simplified 91.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+172}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5.2e+140) (fma y (* z (tanh (/ t y))) x) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.2e+140) {
		tmp = fma(y, (z * tanh((t / y))), x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.2e+140)
		tmp = fma(y, Float64(z * tanh(Float64(t / y))), x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5.2e+140], N[(y * N[(z * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.2000000000000002e140

   1. Initial program 95.7%

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
   2. Step-by-step derivation

      1. +-commutative 95.7%

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

      2. associate-*l* 98.9%

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

      3. fma-define 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around 0 22.2%

      \[\leadsto \mathsf{fma}\left(y, z \cdot \color{blue}{\left(\frac{e^{\frac{t}{y}}}{e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}} - \frac{1}{e^{\frac{t}{y}} \cdot \left(e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}\right)}\right)}, x\right) \]
   6. Step-by-step derivation

      1. associate-/r* 22.2%

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

      2. div-sub 22.2%

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

      3. rec-exp 22.2%

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

      4. rec-exp 22.2%

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

      5. tanh-def-a 84.5%

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

   7. Simplified 84.5%

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

   if 5.2000000000000002e140 < y

   1. Initial program 83.6%

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

   3. Taylor expanded in y around inf 88.0%

      \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 88.0%

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

      2. fma-define 88.0%

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

   5. Simplified 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t - x, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+141}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 7.2e+141) (+ x (* (tanh (/ t y)) (* y z))) (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.2e+141) {
		tmp = x + (tanh((t / y)) * (y * z));
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7.2e+141)
		tmp = Float64(x + Float64(tanh(Float64(t / y)) * Float64(y * z)));
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 7.2e+141], N[(x + N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.2000000000000003e141

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0 22.2%

      \[\leadsto x + \color{blue}{y \cdot \left(z \cdot \left(\frac{e^{\frac{t}{y}}}{e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}} - \frac{1}{e^{\frac{t}{y}} \cdot \left(e^{\frac{t}{y}} + \frac{1}{e^{\frac{t}{y}}}\right)}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 22.0%

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

      2. associate-/r* 22.0%

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

      3. div-sub 22.0%

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

      4. rec-exp 22.0%

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

      5. rec-exp 22.0%

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

      6. tanh-def-a 83.2%

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

   5. Simplified 83.2%

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

   if 7.2000000000000003e141 < y

   1. Initial program 83.6%

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

   3. Taylor expanded in y around inf 88.0%

      \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 88.0%

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

      2. fma-define 88.0%

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

   5. Simplified 88.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+141}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{+28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 3.4e+28) x (fma z (- t x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.4e+28) {
		tmp = x;
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.4e+28)
		tmp = x;
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.4e+28], x, N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4e28

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf 60.7%

      \[\leadsto \color{blue}{x} \]

   if 3.4e28 < y

   1. Initial program 87.7%

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

   3. Taylor expanded in y around inf 80.5%

      \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 80.5%

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

      2. fma-define 80.5%

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

   5. Simplified 80.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t - x, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 64.6% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.9e-20) x (if (<= y 3.8e+152) (+ x (* z t)) (* x (- 1.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.9e-20) {
		tmp = x;
	} else if (y <= 3.8e+152) {
		tmp = x + (z * t);
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.9d-20) then
        tmp = x
    else if (y <= 3.8d+152) then
        tmp = x + (z * t)
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.9e-20) {
		tmp = x;
	} else if (y <= 3.8e+152) {
		tmp = x + (z * t);
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.9e-20:
		tmp = x
	elif y <= 3.8e+152:
		tmp = x + (z * t)
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.9e-20)
		tmp = x;
	elseif (y <= 3.8e+152)
		tmp = Float64(x + Float64(z * t));
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.9e-20)
		tmp = x;
	elseif (y <= 3.8e+152)
		tmp = x + (z * t);
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.9e-20], x, If[LessEqual[y, 3.8e+152], N[(x + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.8999999999999999e-20

   1. Initial program 95.6%

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

   3. Taylor expanded in x around inf 60.8%

      \[\leadsto \color{blue}{x} \]

   if 1.8999999999999999e-20 < y < 3.8e152

   1. Initial program 92.5%

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

   3. Taylor expanded in y around inf 63.4%

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

   4. Taylor expanded in t around inf 63.0%

      \[\leadsto x + \color{blue}{t \cdot z} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

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

   6. Simplified 63.0%

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

   if 3.8e152 < y

   1. Initial program 86.4%

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

   3. Taylor expanded in y around inf 74.1%

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

   4. Taylor expanded in x around inf 74.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 74.0%

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

      2. unsub-neg 74.0%

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

   6. Simplified 74.0%

      \[\leadsto \color{blue}{x \cdot \left(1 - z\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 60.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{+178}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+254}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.2e+178) x (if (<= z 1.1e+254) (* z (- x)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.2e+178) {
		tmp = x;
	} else if (z <= 1.1e+254) {
		tmp = z * -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 2.2d+178) then
        tmp = x
    else if (z <= 1.1d+254) then
        tmp = z * -x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.2e+178) {
		tmp = x;
	} else if (z <= 1.1e+254) {
		tmp = z * -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 2.2e+178:
		tmp = x
	elif z <= 1.1e+254:
		tmp = z * -x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2.2e+178)
		tmp = x;
	elseif (z <= 1.1e+254)
		tmp = Float64(z * Float64(-x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2.2e+178)
		tmp = x;
	elseif (z <= 1.1e+254)
		tmp = z * -x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 2.2e+178], x, If[LessEqual[z, 1.1e+254], N[(z * (-x)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < 2.19999999999999997e178 or 1.1000000000000001e254 < z

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 62.8%

      \[\leadsto \color{blue}{x} \]

   if 2.19999999999999997e178 < z < 1.1000000000000001e254

   1. Initial program 75.9%

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

   3. Taylor expanded in y around inf 44.1%

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

   4. Taylor expanded in x around inf 42.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 42.9%

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

      2. unsub-neg 42.9%

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

   6. Simplified 42.9%

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

   7. Taylor expanded in z around inf 42.9%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot z\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 42.9%

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

      2. neg-mul-1 42.9%

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

   9. Simplified 42.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{+178}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+254}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.65e+27) x (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.65e+27) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.65d+27) then
        tmp = x
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.65e+27) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.65e+27:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.65e+27)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.65e+27)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.65e+27], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6499999999999999e27

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf 60.7%

      \[\leadsto \color{blue}{x} \]

   if 1.6499999999999999e27 < y

   1. Initial program 87.7%

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

   3. Taylor expanded in y around inf 80.5%

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

Alternative 9: 64.4% accurate, 21.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+43}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 1e+43) x (* x (- 1.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1e+43) {
		tmp = x;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1d+43) then
        tmp = x
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1e+43) {
		tmp = x;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1e+43:
		tmp = x
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1e+43)
		tmp = x;
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1e+43)
		tmp = x;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1e+43], x, N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000001e43

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf 60.4%

      \[\leadsto \color{blue}{x} \]

   if 1.00000000000000001e43 < y

   1. Initial program 87.4%

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

   3. Taylor expanded in y around inf 71.7%

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

   4. Taylor expanded in x around inf 69.3%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 69.3%

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

      2. unsub-neg 69.3%

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

   6. Simplified 69.3%

      \[\leadsto \color{blue}{x \cdot \left(1 - z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.9% accurate, 213.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 x)

C

double code(double x, double y, double z, double t) {
	return x;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

public static double code(double x, double y, double z, double t) {
	return x;
}

Python

def code(x, y, z, t):
	return x

Julia

function code(x, y, z, t)
	return x
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 94.2%

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

3. Taylor expanded in x around inf 57.8%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Developer Target 1: 97.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y)))))))

C

double code(double x, double y, double z, double t) {
	return x + (y * (z * (tanh((t / y)) - tanh((x / y)))));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + (y * (z * (tanh((t / y)) - tanh((x / y)))))
end function

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)))));
}

Python

def code(x, y, z, t):
	return x + (y * (z * (math.tanh((t / y)) - math.tanh((x / y)))))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(y * Float64(z * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y))))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (y * (z * (tanh((t / y)) - tanh((x / y)))));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(y * N[(z * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024116 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y)))))))

  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))