SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.3% → 98.4%

Time: 10.7s

Alternatives: 12

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: 93.3% 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: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 9.5e+191)
   (fma (* y (- (tanh (/ t y)) (tanh (/ x y)))) z x)
   (fma z (- t x) x)))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.5e+191) {
		tmp = fma((y * (tanh((t / y)) - tanh((x / y)))), z, x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9.5e+191)
		tmp = fma(Float64(y * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))), z, x);
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 9.5e+191], N[(N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.4999999999999998e191

   1. Initial program 92.3%

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

      1. +-commutative 92.3%

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

      2. *-commutative 92.3%

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

      3. associate-*r* 97.3%

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

      4. fma-def 97.3%

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

   3. Applied egg-rr 97.3%

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

   if 9.4999999999999998e191 < y

   1. Initial program 70.1%

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

   2. Taylor expanded in y around inf 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-def 93.0%

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

   4. Simplified 93.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Alternative 2: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+187}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.85e+187)
   (+ x (* (- (tanh (/ t y)) (tanh (/ x y))) (* y z)))
   (fma z (- t x) x)))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.85e+187) {
		tmp = x + ((tanh((t / y)) - tanh((x / y))) * (y * z));
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.85e+187)
		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

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 1.85e+187], 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 < 1.85000000000000009e187

   1. Initial program 92.3%

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

   if 1.85000000000000009e187 < y

   1. Initial program 70.1%

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

   2. Taylor expanded in y around inf 93.0%

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

      1. +-commutative 93.0%

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

      2. fma-def 93.0%

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

   4. Simplified 93.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+187}:\\ \;\;\;\;x + \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Alternative 3: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-96} \lor \neg \left(t \leq 3.8 \cdot 10^{+36}\right):\\ \;\;\;\;x + y \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4e-96) (not (<= t 3.8e+36)))
   (+ x (* y (* (tanh (/ t y)) z)))
   (+ x (* z (- t (* y (tanh (/ x y))))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4e-96) || !(t <= 3.8e+36)) {
		tmp = x + (y * (tanh((t / y)) * z));
	} else {
		tmp = x + (z * (t - (y * tanh((x / y)))));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 ((t <= (-4d-96)) .or. (.not. (t <= 3.8d+36))) then
        tmp = x + (y * (tanh((t / y)) * z))
    else
        tmp = x + (z * (t - (y * tanh((x / y)))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4e-96) || !(t <= 3.8e+36)) {
		tmp = x + (y * (Math.tanh((t / y)) * z));
	} else {
		tmp = x + (z * (t - (y * Math.tanh((x / y)))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if (t <= -4e-96) or not (t <= 3.8e+36):
		tmp = x + (y * (math.tanh((t / y)) * z))
	else:
		tmp = x + (z * (t - (y * math.tanh((x / y)))))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4e-96) || !(t <= 3.8e+36))
		tmp = Float64(x + Float64(y * Float64(tanh(Float64(t / y)) * z)));
	else
		tmp = Float64(x + Float64(z * Float64(t - Float64(y * tanh(Float64(x / y))))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4e-96) || ~((t <= 3.8e+36)))
		tmp = x + (y * (tanh((t / y)) * z));
	else
		tmp = x + (z * (t - (y * tanh((x / y)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4e-96], N[Not[LessEqual[t, 3.8e+36]], $MachinePrecision]], N[(x + N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - N[(y * N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.9999999999999996e-96 or 3.80000000000000025e36 < t

   1. Initial program 94.3%

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

   2. Taylor expanded in x around 0 13.1%

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

      1. associate-/r* 13.1%

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

      2. div-sub 13.1%

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

      3. rec-exp 13.2%

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

      4. rec-exp 13.2%

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

      5. tanh-def-a 85.3%

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

   4. Simplified 85.3%

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

   if -3.9999999999999996e-96 < t < 3.80000000000000025e36

   1. Initial program 85.1%

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

   2. Taylor expanded in t around 0 79.3%

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

   3. Taylor expanded in t around -inf 37.6%

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

      1. +-commutative 37.6%

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

      2. mul-1-neg 37.6%

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

      3. unsub-neg 37.6%

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

      4. associate-*r* 35.0%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in z around 0 37.6%

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

      1. cancel-sign-sub-inv 37.6%

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

      2. associate-/r* 37.6%

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

      3. div-sub 37.6%

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

   8. Simplified 89.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-96} \lor \neg \left(t \leq 3.8 \cdot 10^{+36}\right):\\ \;\;\;\;x + y \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - y \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \end{array} \]

Alternative 4: 85.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+136}:\\ \;\;\;\;x + y \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.5e+136) (+ x (* y (* (tanh (/ t y)) z))) (fma z (- t x) x)))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.5e+136) {
		tmp = x + (y * (tanh((t / y)) * z));
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.5e+136)
		tmp = Float64(x + Float64(y * Float64(tanh(Float64(t / y)) * z)));
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 3.5e+136], N[(x + N[(y * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.50000000000000001e136

   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. Taylor expanded in x around 0 25.5%

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

      1. associate-/r* 25.5%

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

      2. div-sub 25.5%

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

      3. rec-exp 25.5%

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

      4. rec-exp 25.5%

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

      5. tanh-def-a 78.9%

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

   4. Simplified 78.9%

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

   if 3.50000000000000001e136 < y

   1. Initial program 72.7%

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

   2. Taylor expanded in y around inf 90.3%

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

      1. +-commutative 90.3%

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

      2. fma-def 90.3%

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

   4. Simplified 90.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+136}:\\ \;\;\;\;x + y \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Alternative 5: 76.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t) :precision binary64 (if (<= y 2.1e-49) x (fma z (- t x) x)))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.1e-49) {
		tmp = x;
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.1e-49)
		tmp = x;
	else
		tmp = fma(z, Float64(t - x), x);
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 2.1e-49], x, N[(z * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0999999999999999e-49

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 62.6%

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

   if 2.0999999999999999e-49 < y

   1. Initial program 81.2%

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

   2. Taylor expanded in y around inf 76.4%

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

      1. +-commutative 76.4%

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

      2. fma-def 76.4%

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

   4. Simplified 76.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Alternative 6: 64.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+206} \lor \neg \left(y \leq 1.42 \cdot 10^{+251}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.2e+14)
   x
   (if (or (<= y 1.2e+206) (not (<= y 1.42e+251))) (* x (- 1.0 z)) (* t z))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e+14) {
		tmp = x;
	} else if ((y <= 1.2e+206) || !(y <= 1.42e+251)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = t * z;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 <= 2.2d+14) then
        tmp = x
    else if ((y <= 1.2d+206) .or. (.not. (y <= 1.42d+251))) then
        tmp = x * (1.0d0 - z)
    else
        tmp = t * z
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e+14) {
		tmp = x;
	} else if ((y <= 1.2e+206) || !(y <= 1.42e+251)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 2.2e+14:
		tmp = x
	elif (y <= 1.2e+206) or not (y <= 1.42e+251):
		tmp = x * (1.0 - z)
	else:
		tmp = t * z
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.2e+14)
		tmp = x;
	elseif ((y <= 1.2e+206) || !(y <= 1.42e+251))
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.2e+14)
		tmp = x;
	elseif ((y <= 1.2e+206) || ~((y <= 1.42e+251)))
		tmp = x * (1.0 - z);
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 2.2e+14], x, If[Or[LessEqual[y, 1.2e+206], N[Not[LessEqual[y, 1.42e+251]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(t * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.2e14

   1. Initial program 93.6%

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

   2. Taylor expanded in x around inf 62.7%

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

   if 2.2e14 < y < 1.2e206 or 1.41999999999999995e251 < y

   1. Initial program 76.7%

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

   2. Taylor expanded in y around inf 75.7%

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

   3. Taylor expanded in x around inf 53.1%

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

      1. mul-1-neg 53.1%

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

   5. Simplified 53.1%

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

   6. Taylor expanded in x around 0 53.1%

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

   if 1.2e206 < y < 1.41999999999999995e251

   1. Initial program 84.0%

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

   2. Taylor expanded in y around inf 84.9%

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

   3. Taylor expanded in z around inf 52.9%

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

   4. Taylor expanded in t around inf 52.8%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+206} \lor \neg \left(y \leq 1.42 \cdot 10^{+251}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]

Alternative 7: 76.3% accurate, 19.3× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x + t \cdot z\right) - x \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.5e-49) x (- (+ x (* t z)) (* x z))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.5e-49) {
		tmp = x;
	} else {
		tmp = (x + (t * z)) - (x * z);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 <= 2.5d-49) then
        tmp = x
    else
        tmp = (x + (t * z)) - (x * z)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.5e-49) {
		tmp = x;
	} else {
		tmp = (x + (t * z)) - (x * z);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 2.5e-49:
		tmp = x
	else:
		tmp = (x + (t * z)) - (x * z)
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.5e-49)
		tmp = x;
	else
		tmp = Float64(Float64(x + Float64(t * z)) - Float64(x * z));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.5e-49)
		tmp = x;
	else
		tmp = (x + (t * z)) - (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 2.5e-49], x, N[(N[(x + N[(t * z), $MachinePrecision]), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4999999999999999e-49

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 62.6%

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

   if 2.4999999999999999e-49 < y

   1. Initial program 81.2%

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

   2. Taylor expanded in t around 0 63.6%

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

   3. Taylor expanded in t around -inf 49.3%

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

      1. +-commutative 49.3%

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

      2. mul-1-neg 49.3%

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

      3. unsub-neg 49.3%

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

      4. associate-*r* 43.2%

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

   5. Simplified 67.4%

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

   6. Taylor expanded in y around inf 76.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x + t \cdot z\right) - x \cdot z\\ \end{array} \]

Alternative 8: 65.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+98}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+253}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.6e+98) x (if (<= y 1.6e+253) (* z (- t x)) (* x (- 1.0 z)))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+98) {
		tmp = x;
	} else if (y <= 1.6e+253) {
		tmp = z * (t - x);
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 <= 2.6d+98) then
        tmp = x
    else if (y <= 1.6d+253) then
        tmp = z * (t - x)
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+98) {
		tmp = x;
	} else if (y <= 1.6e+253) {
		tmp = z * (t - x);
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 2.6e+98:
		tmp = x
	elif y <= 1.6e+253:
		tmp = z * (t - x)
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.6e+98)
		tmp = x;
	elseif (y <= 1.6e+253)
		tmp = Float64(z * Float64(t - x));
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.6e+98)
		tmp = x;
	elseif (y <= 1.6e+253)
		tmp = z * (t - x);
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 2.6e+98], x, If[LessEqual[y, 1.6e+253], N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.6e98

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 62.6%

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

   if 2.6e98 < y < 1.6000000000000002e253

   1. Initial program 72.7%

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

   2. Taylor expanded in y around inf 81.0%

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

   3. Taylor expanded in z around inf 58.1%

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

   if 1.6000000000000002e253 < y

   1. Initial program 67.8%

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

   2. Taylor expanded in y around inf 93.9%

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

   3. Taylor expanded in x around inf 55.3%

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

      1. mul-1-neg 55.3%

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

   5. Simplified 55.3%

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

   6. Taylor expanded in x around 0 55.3%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+98}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+253}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 9: 76.8% accurate, 23.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.55e-50) x (+ x (* z (- t x)))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.55e-50) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 <= 1.55d-50) then
        tmp = x
    else
        tmp = x + (z * (t - x))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.55e-50) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 1.55e-50:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.55e-50)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * Float64(t - x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.55e-50)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 1.55e-50], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5500000000000001e-50

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 62.6%

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

   if 1.5500000000000001e-50 < y

   1. Initial program 81.2%

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

   2. Taylor expanded in y around inf 76.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 10: 60.9% accurate, 29.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-280}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-169}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.7e-280) x (if (<= x 3e-169) (* t z) x)))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.7e-280) {
		tmp = x;
	} else if (x <= 3e-169) {
		tmp = t * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 (x <= (-1.7d-280)) then
        tmp = x
    else if (x <= 3d-169) then
        tmp = t * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.7e-280) {
		tmp = x;
	} else if (x <= 3e-169) {
		tmp = t * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if x <= -1.7e-280:
		tmp = x
	elif x <= 3e-169:
		tmp = t * z
	else:
		tmp = x
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.7e-280)
		tmp = x;
	elseif (x <= 3e-169)
		tmp = Float64(t * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.7e-280)
		tmp = x;
	elseif (x <= 3e-169)
		tmp = t * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[x, -1.7e-280], x, If[LessEqual[x, 3e-169], N[(t * z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6999999999999999e-280 or 2.9999999999999999e-169 < x

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf 65.2%

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

   if -1.6999999999999999e-280 < x < 2.9999999999999999e-169

   1. Initial program 76.4%

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

   2. Taylor expanded in y around inf 70.8%

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

   3. Taylor expanded in z around inf 59.5%

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

   4. Taylor expanded in t around inf 55.0%

      \[\leadsto \color{blue}{t \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-280}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-169}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 70.0% accurate, 30.2× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t) :precision binary64 (if (<= y 2.8e-51) x (+ x (* t z))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.8e-51) {
		tmp = x;
	} else {
		tmp = x + (t * z);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 <= 2.8d-51) then
        tmp = x
    else
        tmp = x + (t * z)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.8e-51) {
		tmp = x;
	} else {
		tmp = x + (t * z);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 2.8e-51:
		tmp = x
	else:
		tmp = x + (t * z)
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.8e-51)
		tmp = x;
	else
		tmp = Float64(x + Float64(t * z));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.8e-51)
		tmp = x;
	else
		tmp = x + (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 2.8e-51], x, N[(x + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8e-51

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 62.6%

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

   if 2.8e-51 < y

   1. Initial program 81.2%

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

   2. Taylor expanded in y around inf 76.4%

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

   3. Taylor expanded in t around inf 67.8%

      \[\leadsto x + \color{blue}{t \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot z\\ \end{array} \]

Alternative 12: 59.6% accurate, 213.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ x \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t) :precision binary64 x)

C

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

Fortran

NOTE: y should be positive before calling this function
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

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	return x;
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := x

Derivation

1. Initial program 90.0%

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

2. Taylor expanded in x around inf 58.3%

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

3. Final simplification 58.3%

   \[\leadsto x \]

Developer target: 96.7% 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 2023271 
(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))))))