SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.3% → 97.8%

Time: 13.0s

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: 93.3% 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: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in y around 0 18.3%

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

   1. *-commutative 18.3%

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

   2. associate-*l* 18.3%

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

4. Simplified 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 85.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-77} \lor \neg \left(x \leq 2.8 \cdot 10^{+107}\right):\\ \;\;\;\;x - y \cdot \left(z \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.32e-77) (not (<= x 2.8e+107)))
   (- x (* y (* z (tanh (/ x y)))))
   (+ x (* z (* y (tanh (/ t y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.32e-77) || !(x <= 2.8e+107)) {
		tmp = x - (y * (z * tanh((x / y))));
	} else {
		tmp = x + (z * (y * tanh((t / y))));
	}
	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 ((x <= (-1.32d-77)) .or. (.not. (x <= 2.8d+107))) then
        tmp = x - (y * (z * tanh((x / y))))
    else
        tmp = x + (z * (y * tanh((t / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.32e-77) || !(x <= 2.8e+107)) {
		tmp = x - (y * (z * Math.tanh((x / y))));
	} else {
		tmp = x + (z * (y * Math.tanh((t / y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.32e-77) or not (x <= 2.8e+107):
		tmp = x - (y * (z * math.tanh((x / y))))
	else:
		tmp = x + (z * (y * math.tanh((t / y))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.32e-77) || !(x <= 2.8e+107))
		tmp = Float64(x - Float64(y * Float64(z * tanh(Float64(x / y)))));
	else
		tmp = Float64(x + Float64(z * Float64(y * tanh(Float64(t / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.32e-77) || ~((x <= 2.8e+107)))
		tmp = x - (y * (z * tanh((x / y))));
	else
		tmp = x + (z * (y * tanh((t / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.32e-77], N[Not[LessEqual[x, 2.8e+107]], $MachinePrecision]], N[(x - N[(y * N[(z * N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.31999999999999993e-77 or 2.79999999999999985e107 < x

   1. Initial program 96.5%

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

   2. Taylor expanded in t around 0 17.6%

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

      1. sub-neg 17.6%

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

      2. +-commutative 17.6%

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

      3. neg-sub0 17.6%

         \[\leadsto x + y \cdot \left(z \cdot \left(\color{blue}{\left(0 - \frac{e^{\frac{x}{y}}}{e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}}\right)} + \frac{1}{e^{\frac{x}{y}} \cdot \left(e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}\right)}\right)\right) \]

      4. associate--r- 17.6%

         \[\leadsto x + y \cdot \left(z \cdot \color{blue}{\left(0 - \left(\frac{e^{\frac{x}{y}}}{e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}} - \frac{1}{e^{\frac{x}{y}} \cdot \left(e^{\frac{x}{y}} + \frac{1}{e^{\frac{x}{y}}}\right)}\right)\right)}\right) \]

   4. Simplified 89.6%

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

   if -1.31999999999999993e-77 < x < 2.79999999999999985e107

   1. Initial program 96.6%

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

   2. Taylor expanded in y around 0 21.8%

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

      1. *-commutative 21.8%

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

      2. associate-*l* 21.8%

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

   4. Simplified 97.5%

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

   5. Taylor expanded in x around 0 22.9%

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

      1. associate-/r* 23.0%

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

      2. div-sub 22.9%

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

      3. rec-exp 22.9%

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

      4. rec-exp 22.9%

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

      5. tanh-def-a 88.3%

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

   7. Simplified 88.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-77} \lor \neg \left(x \leq 2.8 \cdot 10^{+107}\right):\\ \;\;\;\;x - y \cdot \left(z \cdot \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \end{array} \]

Alternative 3: 84.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq 1.9 \cdot 10^{+39}:\\ \;\;\;\;x + z \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t_1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (tanh (/ t y)))))
   (if (<= y 1.9e+39) (+ x (* z t_1)) (+ x (* z (- t_1 x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * tanh((t / y));
	double tmp;
	if (y <= 1.9e+39) {
		tmp = x + (z * t_1);
	} else {
		tmp = x + (z * (t_1 - 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) :: t_1
    real(8) :: tmp
    t_1 = y * tanh((t / y))
    if (y <= 1.9d+39) then
        tmp = x + (z * t_1)
    else
        tmp = x + (z * (t_1 - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * Math.tanh((t / y));
	double tmp;
	if (y <= 1.9e+39) {
		tmp = x + (z * t_1);
	} else {
		tmp = x + (z * (t_1 - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * math.tanh((t / y))
	tmp = 0
	if y <= 1.9e+39:
		tmp = x + (z * t_1)
	else:
		tmp = x + (z * (t_1 - x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * tanh(Float64(t / y)))
	tmp = 0.0
	if (y <= 1.9e+39)
		tmp = Float64(x + Float64(z * t_1));
	else
		tmp = Float64(x + Float64(z * Float64(t_1 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * tanh((t / y));
	tmp = 0.0;
	if (y <= 1.9e+39)
		tmp = x + (z * t_1);
	else
		tmp = x + (z * (t_1 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.9e+39], N[(x + N[(z * t$95$1), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t$95$1 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8999999999999999e39

   1. Initial program 97.5%

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

   2. Taylor expanded in y around 0 14.5%

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

      1. *-commutative 14.5%

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

      2. associate-*l* 14.5%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in x around 0 25.5%

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

      1. associate-/r* 25.5%

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

      2. div-sub 25.5%

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

      3. rec-exp 25.5%

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

      4. rec-exp 25.5%

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

      5. tanh-def-a 80.4%

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

   7. Simplified 80.4%

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

   if 1.8999999999999999e39 < y

   1. Initial program 92.7%

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

   2. Taylor expanded in x around 0 66.7%

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

      1. +-commutative 66.7%

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

   4. Simplified 98.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+39}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right) - x\right)\\ \end{array} \]

Alternative 4: 83.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{+136}:\\ \;\;\;\;x + y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.12e+136) (+ x (* y (* z (tanh (/ t y))))) (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.12e+136) {
		tmp = x + (y * (z * tanh((t / y))));
	} 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 <= 1.12d+136) then
        tmp = x + (y * (z * tanh((t / y))))
    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 <= 1.12e+136) {
		tmp = x + (y * (z * Math.tanh((t / y))));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.12e+136:
		tmp = x + (y * (z * math.tanh((t / y))))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.12e+136)
		tmp = Float64(x + Float64(y * Float64(z * tanh(Float64(t / y)))));
	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 <= 1.12e+136)
		tmp = x + (y * (z * tanh((t / y))));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.12e+136], N[(x + N[(y * N[(z * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.12000000000000001e136

   1. Initial program 97.7%

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

   2. Taylor expanded in x around 0 25.9%

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

      1. associate-/r* 25.9%

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

      2. div-sub 25.9%

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

      3. rec-exp 25.9%

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

      4. rec-exp 25.9%

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

      5. tanh-def-a 80.6%

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

   4. Simplified 80.6%

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

   if 1.12000000000000001e136 < y

   1. Initial program 89.3%

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

   2. Taylor expanded in y around inf 93.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{+136}:\\ \;\;\;\;x + y \cdot \left(z \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 5: 83.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+134}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 9e+134) (+ x (* z (* y (tanh (/ t y))))) (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9e+134) {
		tmp = x + (z * (y * tanh((t / y))));
	} 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 <= 9d+134) then
        tmp = x + (z * (y * tanh((t / y))))
    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 <= 9e+134) {
		tmp = x + (z * (y * Math.tanh((t / y))));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 9e+134:
		tmp = x + (z * (y * math.tanh((t / y))))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9e+134)
		tmp = Float64(x + Float64(z * Float64(y * tanh(Float64(t / y)))));
	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 <= 9e+134)
		tmp = x + (z * (y * tanh((t / y))));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9e+134], N[(x + N[(z * N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.9999999999999995e134

   1. Initial program 97.7%

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

   2. Taylor expanded in y around 0 14.3%

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

      1. *-commutative 14.3%

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

      2. associate-*l* 14.3%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around 0 25.9%

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

      1. associate-/r* 25.9%

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

      2. div-sub 25.9%

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

      3. rec-exp 25.9%

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

      4. rec-exp 25.9%

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

      5. tanh-def-a 81.0%

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

   7. Simplified 81.0%

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

   if 8.9999999999999995e134 < y

   1. Initial program 89.3%

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

   2. Taylor expanded in y around inf 93.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+134}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 6: 82.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+134}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 7.8e+134) (+ x (* (tanh (/ t y)) (* z y))) (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.8e+134) {
		tmp = x + (tanh((t / y)) * (z * y));
	} 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 <= 7.8d+134) then
        tmp = x + (tanh((t / y)) * (z * y))
    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 <= 7.8e+134) {
		tmp = x + (Math.tanh((t / y)) * (z * y));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 7.8e+134:
		tmp = x + (math.tanh((t / y)) * (z * y))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7.8e+134)
		tmp = Float64(x + Float64(tanh(Float64(t / y)) * Float64(z * y)));
	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 <= 7.8e+134)
		tmp = x + (tanh((t / y)) * (z * y));
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 7.8e+134], N[(x + N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.79999999999999967e134

   1. Initial program 97.7%

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

   2. Taylor expanded in x around 0 25.9%

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

      1. associate-*r* 25.7%

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

      2. associate-/r* 25.7%

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

      3. div-sub 25.7%

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

      4. rec-exp 25.7%

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

      5. rec-exp 25.7%

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

      6. tanh-def-a 82.2%

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

      7. *-commutative 82.2%

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

   4. Simplified 82.2%

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

   if 7.79999999999999967e134 < y

   1. Initial program 89.3%

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

   2. Taylor expanded in y around inf 93.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+134}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 7: 69.5% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 8e+38) x (+ x (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8e+38) {
		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 <= 8d+38) 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 <= 8e+38) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 8e+38:
		tmp = x
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 8e+38)
		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 <= 8e+38)
		tmp = x;
	else
		tmp = x + (z * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 8e+38], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999982e38

   1. Initial program 97.5%

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

   2. Taylor expanded in y around inf 51.9%

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

   3. Taylor expanded in z around 0 62.5%

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

   if 7.99999999999999982e38 < y

   1. Initial program 92.7%

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

   2. Taylor expanded in y around inf 87.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 8: 65.5% accurate, 30.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 8.5e+95) x (+ x (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8.5e+95) {
		tmp = x;
	} else {
		tmp = x + (z * t);
	}
	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 <= 8.5d+95) then
        tmp = x
    else
        tmp = x + (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8.5e+95) {
		tmp = x;
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 8.5e+95:
		tmp = x
	else:
		tmp = x + (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 8.5e+95)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 8.5e+95)
		tmp = x;
	else
		tmp = x + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 8.5e+95], x, N[(x + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000002e95

   1. Initial program 97.7%

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

   2. Taylor expanded in y around inf 52.2%

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

   3. Taylor expanded in z around 0 61.3%

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

   if 8.5000000000000002e95 < y

   1. Initial program 90.8%

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

   2. Taylor expanded in y around inf 94.6%

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

   3. Taylor expanded in t around inf 76.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \]

Alternative 9: 64.2% accurate, 30.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 9.5e+38) x (- x (* x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.5e+38) {
		tmp = x;
	} else {
		tmp = x - (x * 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 <= 9.5d+38) then
        tmp = x
    else
        tmp = x - (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.5e+38) {
		tmp = x;
	} else {
		tmp = x - (x * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 9.5e+38:
		tmp = x
	else:
		tmp = x - (x * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9.5e+38)
		tmp = x;
	else
		tmp = Float64(x - Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 9.5e+38)
		tmp = x;
	else
		tmp = x - (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9.5e+38], x, N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.4999999999999995e38

   1. Initial program 97.5%

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

   2. Taylor expanded in y around inf 51.9%

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

   3. Taylor expanded in z around 0 62.5%

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

   if 9.4999999999999995e38 < y

   1. Initial program 92.7%

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

   2. Taylor expanded in y around inf 87.0%

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

   3. Taylor expanded in x around inf 72.8%

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

      1. distribute-lft-in 72.8%

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

      2. *-rgt-identity 72.8%

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

      3. mul-1-neg 72.8%

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

      4. distribute-rgt-neg-in 72.8%

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

      5. unsub-neg 72.8%

         \[\leadsto \color{blue}{x - x \cdot z} \]

   5. Simplified 72.8%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \]

Alternative 10: 60.7% 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 96.5%

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

2. Taylor expanded in y around inf 59.2%

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

3. Taylor expanded in z around 0 58.9%

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

4. Final simplification 58.9%

   \[\leadsto x \]

Developer target: 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 2023275 
(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))))))