SynthBasics:moogVCF from YampaSynth-0.2

Percentage Accurate: 93.0% → 97.4%

Time: 12.9s

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: 93.0% 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: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq 1.2 \cdot 10^{+125}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(t_1 - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot t_1 - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y))))
   (if (<= y 1.2e+125)
     (+ x (* y (* z (- t_1 (tanh (/ x y))))))
     (+ x (* z (- (* y t_1) x))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y));
	double tmp;
	if (y <= 1.2e+125) {
		tmp = x + (y * (z * (t_1 - tanh((x / y)))));
	} else {
		tmp = x + (z * ((y * t_1) - 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) :: t_1
    real(8) :: tmp
    t_1 = tanh((t / y))
    if (y <= 1.2d+125) then
        tmp = x + (y * (z * (t_1 - tanh((x / y)))))
    else
        tmp = x + (z * ((y * t_1) - 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 t_1 = Math.tanh((t / y));
	double tmp;
	if (y <= 1.2e+125) {
		tmp = x + (y * (z * (t_1 - Math.tanh((x / y)))));
	} else {
		tmp = x + (z * ((y * t_1) - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	t_1 = math.tanh((t / y))
	tmp = 0
	if y <= 1.2e+125:
		tmp = x + (y * (z * (t_1 - math.tanh((x / y)))))
	else:
		tmp = x + (z * ((y * t_1) - x))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	t_1 = tanh(Float64(t / y))
	tmp = 0.0
	if (y <= 1.2e+125)
		tmp = Float64(x + Float64(y * Float64(z * Float64(t_1 - tanh(Float64(x / y))))));
	else
		tmp = Float64(x + Float64(z * Float64(Float64(y * t_1) - x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	t_1 = tanh((t / y));
	tmp = 0.0;
	if (y <= 1.2e+125)
		tmp = x + (y * (z * (t_1 - tanh((x / y)))));
	else
		tmp = x + (z * ((y * t_1) - x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.2e+125], N[(x + N[(y * N[(z * N[(t$95$1 - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(N[(y * t$95$1), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e125

   1. Initial program 95.9%

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

      1. associate-*l* 98.1%

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

   3. Simplified 98.1%

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

   if 1.2e125 < y

   1. Initial program 90.1%

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

      1. associate-*l* 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in x around 0 66.9%

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

      1. mul-1-neg 66.9%

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

      2. unsub-neg 66.9%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in z around 0 66.9%

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

      1. *-commutative 66.9%

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

   9. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+125}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right) - x\right)\\ \end{array} \]

Alternative 2: 97.7% accurate, 0.7× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (fma z (* y (- (tanh (/ t y)) (tanh (/ x y)))) x))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. +-commutative 95.0%

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

   2. *-commutative 95.0%

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

   3. associate-*l* 98.1%

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

   4. fma-def 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 3: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-37} \lor \neg \left(t \leq 5.5 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - y \cdot \tanh \left(\frac{x}{y}\right), x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.7e-37) (not (<= t 5.5e-42)))
   (fma z (* y (tanh (/ t y))) x)
   (fma z (- t (* y (tanh (/ x y)))) x)))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.7e-37) || !(t <= 5.5e-42)) {
		tmp = fma(z, (y * tanh((t / y))), x);
	} else {
		tmp = fma(z, (t - (y * tanh((x / y)))), x);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.7e-37) || !(t <= 5.5e-42))
		tmp = fma(z, Float64(y * tanh(Float64(t / y))), x);
	else
		tmp = fma(z, Float64(t - Float64(y * tanh(Float64(x / y)))), x);
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.7e-37], N[Not[LessEqual[t, 5.5e-42]], $MachinePrecision]], N[(z * N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(t - N[(y * N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.7e-37 or 5.5e-42 < t

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-*l* 99.4%

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

      4. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 8.6%

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

      1. associate-/r* 8.6%

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

      2. div-sub 8.6%

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

      3. rec-exp 8.6%

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

      4. rec-exp 8.6%

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

      5. tanh-def-a 86.2%

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

   6. Simplified 86.2%

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

   if -3.7e-37 < t < 5.5e-42

   1. Initial program 93.7%

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

      1. +-commutative 93.7%

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

      2. *-commutative 93.7%

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

      3. associate-*l* 96.4%

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

      4. fma-def 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in t around 0 90.4%

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

   5. Taylor expanded in t around -inf 36.7%

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

      1. +-commutative 36.7%

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

      2. mul-1-neg 36.7%

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

      3. unsub-neg 36.7%

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

      4. associate-/r* 36.7%

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

   7. Simplified 91.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-37} \lor \neg \left(t \leq 5.5 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - y \cdot \tanh \left(\frac{x}{y}\right), x\right)\\ \end{array} \]

Alternative 4: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-52} \lor \neg \left(t \leq 1.15 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.6e-52) (not (<= t 1.15e-42)))
   (fma z (* y (tanh (/ t y))) x)
   (+ x (* z (* y (- (/ t y) (tanh (/ x y))))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.6e-52) || !(t <= 1.15e-42)) {
		tmp = fma(z, (y * tanh((t / y))), x);
	} else {
		tmp = x + (z * (y * ((t / y) - tanh((x / y)))));
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.6e-52) || !(t <= 1.15e-42))
		tmp = fma(z, Float64(y * tanh(Float64(t / y))), x);
	else
		tmp = Float64(x + Float64(z * Float64(y * Float64(Float64(t / y) - tanh(Float64(x / y))))));
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.6e-52], N[Not[LessEqual[t, 1.15e-42]], $MachinePrecision]], N[(z * N[(y * N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(z * N[(y * N[(N[(t / y), $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.5999999999999999e-52 or 1.15000000000000002e-42 < t

   1. Initial program 96.2%

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

      1. +-commutative 96.2%

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

      2. *-commutative 96.2%

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

      3. associate-*l* 99.4%

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

      4. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 9.8%

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

      1. associate-/r* 9.8%

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

      2. div-sub 9.8%

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

      3. rec-exp 9.8%

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

      4. rec-exp 9.8%

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

      5. tanh-def-a 86.6%

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

   6. Simplified 86.6%

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

   if -2.5999999999999999e-52 < t < 1.15000000000000002e-42

   1. Initial program 93.4%

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

      1. +-commutative 93.4%

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

      2. *-commutative 93.4%

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

      3. associate-*l* 96.2%

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

      4. fma-def 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in t around 0 90.1%

      \[\leadsto \mathsf{fma}\left(z, y \cdot \left(\color{blue}{\frac{t}{y}} - \tanh \left(\frac{x}{y}\right)\right), x\right) \]
   5. Step-by-step derivation

      1. fma-udef 90.1%

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

   6. Applied egg-rr 90.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-52} \lor \neg \left(t \leq 1.15 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \tanh \left(\frac{t}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \left(\frac{t}{y} - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array} \]

Alternative 5: 87.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right)\\ \mathbf{if}\;y \leq 1.15 \cdot 10^{+78}:\\ \;\;\;\;x + t_1 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot t_1 - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (tanh (/ t y))))
   (if (<= y 1.15e+78) (+ x (* t_1 (* z y))) (+ x (* z (- (* y t_1) x))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y));
	double tmp;
	if (y <= 1.15e+78) {
		tmp = x + (t_1 * (z * y));
	} else {
		tmp = x + (z * ((y * t_1) - 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) :: t_1
    real(8) :: tmp
    t_1 = tanh((t / y))
    if (y <= 1.15d+78) then
        tmp = x + (t_1 * (z * y))
    else
        tmp = x + (z * ((y * t_1) - 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 t_1 = Math.tanh((t / y));
	double tmp;
	if (y <= 1.15e+78) {
		tmp = x + (t_1 * (z * y));
	} else {
		tmp = x + (z * ((y * t_1) - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	t_1 = math.tanh((t / y))
	tmp = 0
	if y <= 1.15e+78:
		tmp = x + (t_1 * (z * y))
	else:
		tmp = x + (z * ((y * t_1) - x))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	t_1 = tanh(Float64(t / y))
	tmp = 0.0
	if (y <= 1.15e+78)
		tmp = Float64(x + Float64(t_1 * Float64(z * y)));
	else
		tmp = Float64(x + Float64(z * Float64(Float64(y * t_1) - x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	t_1 = tanh((t / y));
	tmp = 0.0;
	if (y <= 1.15e+78)
		tmp = x + (t_1 * (z * y));
	else
		tmp = x + (z * ((y * t_1) - x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.15e+78], N[(x + N[(t$95$1 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(N[(y * t$95$1), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1500000000000001e78

   1. Initial program 95.6%

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

      1. associate-*l* 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in x around 0 20.8%

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

      1. *-commutative 20.8%

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

      2. associate-/r* 20.8%

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

      3. div-sub 20.8%

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

      4. rec-exp 20.8%

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

      5. rec-exp 20.8%

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

   6. Simplified 80.4%

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

   if 1.1500000000000001e78 < 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. Step-by-step derivation

      1. associate-*l* 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around 0 60.9%

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

      1. mul-1-neg 60.9%

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

      2. unsub-neg 60.9%

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

   6. Simplified 78.2%

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

   7. Taylor expanded in z around 0 60.9%

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

      1. *-commutative 60.9%

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

   9. Simplified 93.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+78}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot \tanh \left(\frac{t}{y}\right) - x\right)\\ \end{array} \]

Alternative 6: 85.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+166}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.8e+166) (+ x (* (tanh (/ t y)) (* z y))) (+ x (* z (- t x)))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.8e+166) {
		tmp = x + (tanh((t / y)) * (z * y));
	} 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 <= 3.8d+166) then
        tmp = x + (tanh((t / y)) * (z * y))
    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 <= 3.8e+166) {
		tmp = x + (Math.tanh((t / y)) * (z * y));
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 3.8e+166:
		tmp = x + (math.tanh((t / y)) * (z * y))
	else:
		tmp = x + (z * (t - x))
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.8e+166)
		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

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.8e+166)
		tmp = x + (tanh((t / y)) * (z * y));
	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, 3.8e+166], 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 < 3.80000000000000007e166

   1. Initial program 96.1%

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

      1. associate-*l* 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in x around 0 22.8%

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

      1. *-commutative 22.8%

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

      2. associate-/r* 22.8%

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

      3. div-sub 22.8%

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

      4. rec-exp 22.8%

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

      5. rec-exp 22.8%

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

   6. Simplified 79.7%

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

   if 3.80000000000000007e166 < y

   1. Initial program 87.2%

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

      1. associate-*l* 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around inf 95.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+166}:\\ \;\;\;\;x + \tanh \left(\frac{t}{y}\right) \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 7: 68.9% accurate, 13.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_1 := x - z \cdot x\\ \mathbf{if}\;y \leq 6 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.3 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+121} \lor \neg \left(y \leq 8 \cdot 10^{+236}\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (* z x))))
   (if (<= y 6e+38)
     x
     (if (<= y 1.55e+87)
       t_1
       (if (<= y 8.3e+96)
         x
         (if (or (<= y 1.35e+121) (not (<= y 8e+236))) (+ x (* z t)) t_1))))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double t_1 = x - (z * x);
	double tmp;
	if (y <= 6e+38) {
		tmp = x;
	} else if (y <= 1.55e+87) {
		tmp = t_1;
	} else if (y <= 8.3e+96) {
		tmp = x;
	} else if ((y <= 1.35e+121) || !(y <= 8e+236)) {
		tmp = x + (z * t);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x - (z * x)
    if (y <= 6d+38) then
        tmp = x
    else if (y <= 1.55d+87) then
        tmp = t_1
    else if (y <= 8.3d+96) then
        tmp = x
    else if ((y <= 1.35d+121) .or. (.not. (y <= 8d+236))) then
        tmp = x + (z * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z, double t) {
	double t_1 = x - (z * x);
	double tmp;
	if (y <= 6e+38) {
		tmp = x;
	} else if (y <= 1.55e+87) {
		tmp = t_1;
	} else if (y <= 8.3e+96) {
		tmp = x;
	} else if ((y <= 1.35e+121) || !(y <= 8e+236)) {
		tmp = x + (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	t_1 = x - (z * x)
	tmp = 0
	if y <= 6e+38:
		tmp = x
	elif y <= 1.55e+87:
		tmp = t_1
	elif y <= 8.3e+96:
		tmp = x
	elif (y <= 1.35e+121) or not (y <= 8e+236):
		tmp = x + (z * t)
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	t_1 = Float64(x - Float64(z * x))
	tmp = 0.0
	if (y <= 6e+38)
		tmp = x;
	elseif (y <= 1.55e+87)
		tmp = t_1;
	elseif (y <= 8.3e+96)
		tmp = x;
	elseif ((y <= 1.35e+121) || !(y <= 8e+236))
		tmp = Float64(x + Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	t_1 = x - (z * x);
	tmp = 0.0;
	if (y <= 6e+38)
		tmp = x;
	elseif (y <= 1.55e+87)
		tmp = t_1;
	elseif (y <= 8.3e+96)
		tmp = x;
	elseif ((y <= 1.35e+121) || ~((y <= 8e+236)))
		tmp = x + (z * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(z * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6e+38], x, If[LessEqual[y, 1.55e+87], t$95$1, If[LessEqual[y, 8.3e+96], x, If[Or[LessEqual[y, 1.35e+121], N[Not[LessEqual[y, 8e+236]], $MachinePrecision]], N[(x + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.0000000000000002e38 or 1.55e87 < y < 8.2999999999999997e96

   1. Initial program 96.0%

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

      1. associate-*l* 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in y around inf 47.8%

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

   5. Taylor expanded in z around 0 61.9%

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

   if 6.0000000000000002e38 < y < 1.55e87 or 1.3500000000000001e121 < y < 8.00000000000000043e236

   1. Initial program 94.8%

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

      1. associate-*l* 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x around 0 51.6%

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

      1. mul-1-neg 51.6%

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

      2. unsub-neg 51.6%

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

   6. Simplified 83.3%

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

   7. Taylor expanded in y around 0 60.3%

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

   if 8.2999999999999997e96 < y < 1.3500000000000001e121 or 8.00000000000000043e236 < y

   1. Initial program 87.4%

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

      1. associate-*l* 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in x around 0 40.3%

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

      1. *-commutative 40.3%

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

      2. associate-/r* 40.3%

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

      3. div-sub 40.3%

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

      4. rec-exp 40.3%

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

      5. rec-exp 40.3%

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

   6. Simplified 62.4%

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

   7. Taylor expanded in y around inf 65.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+87}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{elif}\;y \leq 8.3 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+121} \lor \neg \left(y \leq 8 \cdot 10^{+236}\right):\\ \;\;\;\;x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot x\\ \end{array} \]

Alternative 8: 78.2% accurate, 23.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{+33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.4e+33) x (+ x (* z (- t x)))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.4e+33) {
		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 <= 3.4d+33) 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 <= 3.4e+33) {
		tmp = x;
	} else {
		tmp = x + (z * (t - x));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 3.4e+33:
		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 <= 3.4e+33)
		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 <= 3.4e+33)
		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, 3.4e+33], x, N[(x + N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3999999999999999e33

   1. Initial program 95.9%

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

      1. associate-*l* 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in y around inf 47.4%

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

   5. Taylor expanded in z around 0 61.4%

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

   if 3.3999999999999999e33 < y

   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. Step-by-step derivation

      1. associate-*l* 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in y around inf 82.2%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{+33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 9: 71.0% accurate, 30.2× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z t) :precision binary64 (if (<= y 4.5e+33) x (+ x (* z t))))

C

y = abs(y);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.5e+33) {
		tmp = x;
	} else {
		tmp = x + (z * t);
	}
	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 <= 4.5d+33) then
        tmp = x
    else
        tmp = x + (z * t)
    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 <= 4.5e+33) {
		tmp = x;
	} else {
		tmp = x + (z * t);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z, t):
	tmp = 0
	if y <= 4.5e+33:
		tmp = x
	else:
		tmp = x + (z * t)
	return tmp

Julia

y = abs(y)
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.5e+33)
		tmp = x;
	else
		tmp = Float64(x + Float64(z * t));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.5e+33)
		tmp = x;
	else
		tmp = x + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[y, 4.5e+33], x, N[(x + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5e33

   1. Initial program 95.9%

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

      1. associate-*l* 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in y around inf 47.4%

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

   5. Taylor expanded in z around 0 61.4%

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

   if 4.5e33 < y

   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. Step-by-step derivation

      1. associate-*l* 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in x around 0 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-/r* 33.6%

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

      3. div-sub 33.6%

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

      4. rec-exp 33.6%

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

      5. rec-exp 33.6%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in y around inf 59.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot t\\ \end{array} \]

Alternative 10: 60.0% 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 95.0%

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

   1. associate-*l* 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in y around inf 56.3%

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

5. Taylor expanded in z around 0 56.8%

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

6. Final simplification 56.8%

   \[\leadsto x \]

Developer target: 97.0% 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 2023230 
(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))))))